Cut and conjugate points of the exponential map, with applications
UUNIVERSIDAD AUTONOMA
Facultad de CienciasDepartamento de Matemáticas
Cut and conjugate points of the exponential map, with applications
A study on the singularities of the exponential maps ofRiemann and Finsler manifolds, with applications toHamilton-Jacobi equations, and the Ambrose conjecture
Pablo Angulo Ardoy
Tesis doctoral dirigida porLuis Guijarro Santamaría a r X i v : . [ m a t h . A P ] N ov ontents Introducción iiiAcknowledgements vIntroduction viiI. Relation between Hamilton-Jacobi equations and Finsler Geometry viiII. Singularities of the exponential map viiiIII. Structure of the cut locus ixIV. Characterization of the cut locus and the balanced split loci xiiV. The Ambrose conjecture xiiiVI. Summary of results xvChapter 1. Preliminaries 11.1. A little background 11.2. Exponential maps of Finsler Manifolds 31.3. The Cut Locus 51.4. Hamilton-Jacobi equations and Finsler geometry 7Chapter 2. A new way to look at Cut and Singular Loci 92.1. The relation between Finsler geometry and Hamilton-Jacobi BVPs 92.2. Split locus and balanced split locus 122.3. Balanced property of the Finsler cut locus 14Chapter 3. Local structure of cut and singular loci up to codimension 3 173.1. Statements of results 173.2. Conjugate points in a balanced split locus 193.3. Structure up to codimension 3 23Chapter 4. Balanced split sets and Hamilton-Jacobi equations 274.1. Introduction 274.2. Statement of results. 274.3. Preliminaries 304.4. ρ S is Lipschitz 334.5. Proof of the main theorems. 384.6. Proof that ∂T = 0 -manifolds 535.1. Introduction 535.2. Notation and preliminaries 555.3. Generic exponential maps 575.4. Proof of the conjecture for generic -manifold 62 ii CHAPTER 0. CONTENTS Chapter 6. Further questions 936.1. Order k conjugate cut points 936.2. HJBVP and balanced split loci 946.3. Poincaré conjecture 946.4. Magician’s hat 966.5. Proof for a 3-manifold with an arbitrary metric 96Chapter 7. Conclusiones 101Bibliography 103 ntroducción En esta tesis estudiamos las singularidades de la aplicación exponencial envariedades Riemannianas y Finslerianas, y el objeto conocido en inglés como cut locus , ridge , medial axis o skeleton , de los cuales sólo el último término sueletraducirse al castellano. En primer lugar mejoramos los resultados existentessobre las singularidades de la aplicación exponencial y la estructura del cutlocus, y después aplicamos estos resultados a los problemas de frontera paraecuaciones de Hamilton-Jacobi y a la conjectura de Ambrose.El cut locus es un objeto de interés para muchas disciplinas: geometría difer-encial, teoría de control óptimo, teoría de transporte óptimo, procesamiento deimágenes, estadística y una herramienta útil en algunas demostraciones de re-sultados en otras disciplinas en las que el cut locus en sí no es un objeto deinterés directo.Durante la primera fase recogimos resultados sobre la estructura del cutlocus provenientes de muchas de estas disciplinas, encontrando resultados du-plicados, y mucho desconocimiento en cada área del trabajo que sobre esteobjeto se hacía desde las otras disciplinas. Cuando aportamos nuestros pro-pios resultados, tuvimos que elegir una notación que no podía ser compatiblecon toda la literatura existente.Nuestros resultados sobre estructura en los capítulos 3 y 4 generalizan resul-tados bien conocidos y demostrados muchas veces de forma independiente, quedescriben la estructura del cut locus excepto por un conjunto de codimensión , lo que es útil para muchas aplicaciones, pero no para todas, aumentando elconocimiento del cut locus hasta codimensión .Estos resultados de estructura son esenciales para nuestras aportaciones ala teoría de Problemas de Frontera para Ecuaciones de Hamilton-Jacobi en elcapítulo 4, donde conectamos la noción de solución de viscosidad con la soluciónclásica por características caracterizando el lugar singular de la primera comoun cut locus, o como un balanced split locus , noción que identificamos en estetrabajo aunque estaba implícito en trabajos previos.Creemos que los resultados sobre las singularidades de la aplicación expo-nencial del capítulo 3 podrían ser útiles para extender la demostración de laconjetura de Ambrose que aportamos a todas las métricas riemannianas. En elcapítulo 5, damos una demostración nueva de la conjetura de Ambrose quecubre un conjunto genérico de variedades riemannianas, pero en el capítulo 6,pergeñamos una estrategia que podría servir para dar una demostración másgeneral que usa de forma esencial los resultados de estructura mencionados.El resto de esta disertación doctoral será en inglés para ser útil a un públicomás amplio, esperamos que este hecho no suponga un impedimento al lectorinteresado. iii cknowledgements Yanyan Li introduced me to the Ambrose conjecture and sparked my in-terest in the conjecture during my stay at Rutgers University. He is a greatteacher, mathematician, and human being.During the preparation of the papers [
AG1 ] and [
AG2 ], I had many helpfulconversations with young and senior mathematicians. The list includes BiaoYin, Luc Nguyen, Juan Carlos Álvarez Paiva, Ireneo Peral, Yanyan Li andMarco Fontelos. Finally, the referee of [
AG2 ] was very helpful. Whoever thatwas, please receive my warmest regards.I have also talked with many people about the Ambrose conjecture. Thelist includes Juan Carlos Álvarez Paiva, Paolo Piccione, Herman Gluck.Other resources were more impersonal but equally useful. Neil Stricklandand Ben Wieland answered a question I posted in the algebraic topology listALGTOP-L. The Wikipedia helped save a lot of time by providing quick an-swers to many simple questions. The site mathoverflow already containedanswers to a few questions before we could even word them correctly. Someanonymous mathematicians scanned, uploaded and shared a big mathematicallibrary. They helped a lot, and they have my respect and my full support.I hold even more respect for all the mathematicians that released their workdirectly to the public. The wonderful book of Allen Hatcher was particularlyhelpful. You even can find one picture from his book in this thesis (with per-mission of the author). A special place goes to the Sage community, an opensource mathematical software that I’ve used mainly for teaching, but also todo computations and explore some hypothesis related to this thesis. I’m proudto have been part of that community for several years.Luis Guijarro was always helpful and respectful as a thesis advisor. Ourfriendship has grown during these years and will survive this work. His patienceknows no bounds.During most of the years I worked on this thesis, I was lucky to work onthe Mathematics Department of the Universidad Autónoma de Madrid. Thedifferent universities and research centers at Madrid make it a great place tostay tuned with the latests advances in mathematics, but it is the warmth andfellowship of the people in the department that made those years so pleasant.In particular, I want to recall my office mates Pedro Caro and Carlos Vinuesa,with whom I had many laughs and interesting mathematical conversations, andDaniel Ortega, who taught me by example that is more satisfying to be usefulto the department than to improve the cv. Daniel is also the latex guru of thedepartment, and is responsible for fixing all the badboxes and other stylisticerrors in this thesis.Besides my positions in the department, I was partially supported duringthe preparation of this work by grants MTM2007-61982 and MTM2008-02686 vi ACKNOWLEDGEMENTS of the MEC and the MCINN respectively. At the end of this work, I wassupported by the Instituto Nacional de Empleo.Agradecimientos. Let me switch now to spanish: the rest is more personal.La casualidad quiso que mi amigo Daniel estuviera en la UAM preparandosu tesis cuando entré como profesor ayudante. Dani me abrió muchas puertas,y durante estos años, al igual que siempre, fue una persona dispuesta a escucharcualquier problema en cualquier área y a proponer soluciones creativas.María y Clara han sido mis compañeras durante toda mi vida matemática.Con ellas he compartido matemáticas y muchas otras cosas. Creo que me hanhecho crecer como persona de un modo tan profundo que listar sus contribu-ciones a esta tesis en particular sería frívolo. De no haberlas conocido, otrapersona habría escrito este trabajo.Clara y mis dos hijos César y Héctor son mi pasión y fuerza vital. Subido ahombros de estos gigantes, afronto el futuro con alegría, incluso con optimismo,aún intuyendo algunos de los momentos difíciles que nos esperan, confiado deque podrán superar todas las dificultades.Pero quiero dedicar esta tesis a mis padres. Desde niño me he sentidosiempre orgulloso de mi madre. Incluso ahora, siento que tengo mucho queaprender de ella para poder ser el padre que mis hijos merecen. Mi padre esahora el abuelo de mis hijos, y es un orgullo haber contribuido a elevarle a estestatus. En un hombre con muchas virtudes, su estilo y saber hacer como abuelose elevan sobre todas las demás. De todo el legado de mis padres, sin dudasu impronta sobre sus hijos y sus nietos será la más importante y duradera.Su gran humanidad me acompaña y me da fuerza en los momentos fáciles ydifíciles. ntroduction
The goal of this thesis is to study the singularities of the exponentialmap of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes ), and the object known as the cut locus (aka ridge , medial axis or skeleton , with applications to differential geometry, control theory, statistics,image processing...), to improve existing results about its structure, to look atit in new ways, and to derive applications to the Ambrose conjecture andthe
Hamilton-Jacobi equations . I. Relation between Hamilton-Jacobi equations and FinslerGeometry
Boundary Value Problems of Hamilton-Jacobi (HJBVP) are intimately re-lationed to Finsler Geometry. In such problems, we look for an unknownfunction u : M → R satisfying the following equations: H ( p, du ( p )) = 1 p ∈ Mu ( p ) = g ( p ) p ∈ ∂M where the first equation is a non-linear first order partial differential equation and the second equation prescribes the boundary values for u .We ask for the following conditions: • M is a smooth compact manifold of dimension n with boundary • H : T ∗ M → RR is a smooth function defined on the cotangent space to M , H − (1) ∩ T ∗ p M strictly convex for every p • g : ∂M → R smoothFurthermore, the boundary data g and the equation coefficients H mustsatisfy a compatibility condition : | g ( y ) − g ( z ) | < d ( y, z ) ∀ y, z ∈ ∂M where d is the distance on M induced by the following Finsler metric : ϕ p ( v ) = sup (cid:110) (cid:104) v, α (cid:105) p : α ∈ T ∗ p M, H ( p, α ) = 1 (cid:111) The above definition gives a norm in every tangent space T p M . Indeed, H can be redefined so that H is positively homogeneous of order : H ( p, λα ) = λH ( p, α ) for λ > , and the HJBVP is the same. Then, H is a norm at everycotangent space and ϕ is the dual norm in the tangent space.A classical solution to these equations has been known for a long time,and it admits a geometrical interpretation in terms of Finsler geometry. viiiii INTRODUCTION
First, using the definition of dual form in Finsler geometry (see 1.1.6), wedefine the characteristic vector field at points p ∈ ∂M : ϕ p ( X p ) = 1 (cid:99) X p | T ( ∂M ) = dgX p points inwardsThe (projected) characteristic curves are the geodesics M with initial point z ∈ ∂M and initial speed given by the characteristic vector field.A local smooth solution u to the HJBVP can be computed near ∂M fol-lowing characteristic curves: Definition.
Let U be a neighborhood of ∂M such that every point q ∈ U belongs to a unique (projected) characteristic contained in U and starting at apoint p ∈ ∂M (the point p is often called the footpoint of q ).The solution by characteristics u : U → R is defined as follows: if γ : [0 , t ] → M is the unique (projected) characteristic from a point p ∈ ∂M to q = γ ( t ) that does not intersect Sing , then u ( q ) = g ( p ) + t In this way, the classical solution can be defined in a neighborhood of ∂M ,but not in all of M .A different notion of solution appeared later (see [ L ]). The solution (in the viscosity sense) to the above HJBVP is given by the Lax-Oleinik formula: u ( p ) = inf q ∈ ∂M { d ( p, q ) + g ( q ) } where d is again the distance function induced from the Finsler metric. Wedefer the definition to 1.4.2, because the actual definition of the viscosity so-lution plays no role in this thesis. All we need to know is that the viscositysolution is given by the above formula.Thus, when g = 0 , the solution to the equations is the distance to theboundary.In theorem 2.1.6, we prove that when g (cid:54) = 0 , the viscosity solution is also adistance function, but to the boundary of a larger manifold ˜ M ⊃ M . II. Singularities of the exponential map
The exponential map from a point or submanifold in a Finsler manifoldis defined in the same way as that of Riemannian manifolds and has similarproperties.Let M be a smooth Finsler manifold, p a point in M and v ∈ T p M a tangentvector to M at p . Then the exponential of v is the point exp p ( v ) = γ (1) ∈ M ,for the unique geodesic γ that starts at p and has initial speed vector v . Theexponential map from p is a diffeomorphism from a small ball near the origin,but it can develop singularities as we move far away from the origin.The exponential map from a submanifold L ⊂ M is defined for (some)vectors of the normal bundle of L in M : let p be a point in L and v ∈ T p M a vector orthogonal to the subspace T p L , the exponential of v is the point exp L ( v ) = γ (1) ∈ M , for the unique geodesic γ that starts at p and has initialspeed vector v . If the submanifold is the boundary ∂M of a closed manifold II. STRUCTURE OF THE CUT LOCUS ix M , the definition is the same, but the exponential is only defined in the innernormal bundle .This time, the exponential map is a diffeomorphism from a tubular neigh-borhood of the zero section of the normal bundle of L into a tubular neighbor-hood of L in M and again, it can develop singularities and self-intersections ifwe consider larger vectors.The singularities, however, are not those of an arbitrary smooth map be-tween n -dimensional manifolds. Let us restrict for a moment to the exponentialmap from a point p in a manifold M without boundary. For a point x ∈ T p M where exp p is singular, the order of conjugacy of x is the corank of the linearmap d x exp p . Along a radial line in T p M , the singularities cannot cluster: ifwe add the orders of all the singularities along a radial line, in a small neigh-borhood of a point x ∈ T p M of order k , the number is always k .It makes sense, thus, to talk about the k -th conjugate point in the direction x , for a point x in the unit ball in T p M . This is the point t · x , for t > ,such that t · x is a conjugate point of order j and such that the sum of theorders of all the singularities along the radial segment t → t · x , for t < t is aninteger between k − j and k − .This allows us to define the function λ k that maps x ∈ B ( T p M ) to theparameter t = λ k ( x ) such that t · x is the the k -th conjugate point in thedirecton x . It follows from the above that λ k is continuous.In [ IT98 ], J. I. Itoh and M. Tanaka proved that for Riemannian manifolds,all the functions λ k are locally Lipschitz continuous. In theorem 4.2.5, we provethat all λ k are locally Lipschitz continuous in Finsler manifolds. M. Castelpi-etra and L. Rifford were working on this result simultaneously, and shortlyafter our proof appeared, they gave a proof that λ is locally semiconcave, astronger property that Lipschitz. This property does not hold for λ k , for k > .The three proofs are different. III. Structure of the cut locus
Let M be a Riemannian or Finsler manifold, and let S be a smooth subman-ifold of any dimension. S can also be a point, and we also consider S = ∂M .It is easy to see that this latter case contains the others, substracting a tubularneighborhood of the submanifold S . The cut locus Cut S of S in M can bedefined in several equivalent ways: • Every unit speed geodesic γ with initial point in S and initial speed or-thogonal to S will minimize the distance between γ (0) and γ ( t ) , for small t . Define t cut = sup { t : d ( γ (0) , γ ( t )) = t } . The cut locus is the set ofall points γ ( t cut ( γ )) for all geodesics γ starting at S with initial speedorthogonal to S . • For a point p ∈ M , let Q p be the set of points q ∈ S such that d ( p, q ) = d ( p, S ) . Then Cut S is the closure of the set of points such that Q p hasmore than one element. • The distance function to S is singular exactly when Q p has more than oneelement, so we can also define Cut S as the closure of the set where thedistance function to S is singular. INTRODUCTION
Figure III.1.
Two open sets in R with their respective cutloci (with respect to the boundary). The second set has analyticboundary, and its cut locus is a subanalytic set. • Cut S is also the set of points such that either Q p has more than oneelement, or Q p = { q } , and q is conjugate to p along one geodesic thatminimizes the distance between them.These sets have been studied by mathematicians from many different fields: • Cut ∂M is a deformation retract of M (and Cut S is a deformation retractof M \ S ) (obvious). • Cut S is the union of a ( n − )-dimensional smooth manifold consisting ofpoints with two minimizing non-conjugate geodesics and a set of Haus-dorff dimension at most n − (Hebda83, Itoh-Tanaka98, Barden-Le97,Mantegazza-Menucci03 for the riemannian case). • Cut S is stratified by the dimension of the subdifferential of the function distance to S : ∂d S ( Alberti-Ambrosio-Cannarsa-Etcetera92-94). • It has finite Hausdorff measure H n − (Itoh-Tanaka00 for the riemanniancase, Li-Nirenberg05 and Castelpietra-Rifford10 for Finsler manifolds). • If all the data is analytic,
Cut S is a stratified subanalytic set ( Buchner77). • If we add a generic perturbation to H or M , Cut S becomes a stratifiedsmooth manifold (Buchner78).Despite these facts, cut loci can be non-triangulable, even for surfaces ofrevolution (see [ GS ]). Also, their combinatorial topology can be complicated:even though Cut S is homotopic to M \ S , there are metrics in the -dimensionalsphere whose cut locus is a simplicial complex equivalent to the house with tworooms (see figure III.2). Figure III.2.
The house with two roomsWe have improved the previous knowledge about cut loci and the singularset of solutions to static Hamilton-Jacobi equations in the following ways:
II. STRUCTURE OF THE CUT LOCUS xi
Figure III.3.
Different types of points in the cut locus, exceptfor a set of small Haussdorff dimension • In theorem 2.1.8, we show that the singular set of a solution to a Hamilton-Jacobi BVP is the cut locus of a Finsler manifold. This was only knownwhen the boundary data was identically zero. • In the same chapter, in subsection 2.3, we also show that cut loci are whatwe call balanced split loci . In general, there are many balanced split locibesides the cut locus, but there is only one such set on a simply connectedmanifold with connected boundary. • In theorem 3.1.2, we prove a local structure theorem for balanced splitloci. The theorem states that the cut locus consists only of cleave points , edge points , degenerate cleave points , crossing points , and a remainder, ofHausdorff codimension at least 3 (see figure III.3). ii INTRODUCTION • In chapter 4, we provide more detailed descriptions of a balanced splitloci near the different types of points listed above (see 4.6.2, 4.6.3, 4.6.5,4.6.6 and 4.6.7).We believe that our description of the cut locus can be useful in othercontexts. For instance, brownian motion on manifolds is often studied on thecomplement of the cut locus from a point, and then the results have to beadapted to take care of the situation when the brownian motion hits the cutlocus. As brownian motion almost never hits a set with null H n − measure(but will almost surely hit any set with positive H n − measure), we think ourresult can be useful in that field. IV. Characterization of the cut locus and the balanced split loci
As we mentioned in the previous section all cut loci, and thus the singularset of HJBVP, are balanced split loci . In chapter 4, we study and classify allpossible balanced split loci. The following list is a summary of theorems 4.2.1,4.2.2 and 4.2.4: M is simply connected → The singular set is the unique and ∂M connected balanced split locus M is simply connected, → We can add a different constant ∂M is not connected to g at each component of ∂M and getdifferent balanced split loci (see fig IV.4)General case → Balanced split loci are parametrized by aneighborhood of in H n − ( M, R ) (see fig IV.5) Figure IV.4.
A balanced split locus when ∂M consists of twoconcentric spheres . THE AMBROSE CONJECTURE xiii Figure IV.5.
A balanced split locus of a non-simply-connectedmanifold (a torus with a disc removed)
V. The Ambrose conjecture
Let ( M , g ) and ( M , g ) be two Riemannian manifolds of the same dimen-sion , with selected points p ∈ M and p ∈ M . We’ll speak about the pointedmanifolds ( M , p ) and ( M , p ). Any linear map L : T p M → T p M inducesthe map ϕ = exp ◦ L ◦ (exp | O ) − , defined in any domain O ⊂ T p M suchthat e | O is injective (for example, if O is a normal neighborhood of p ). M p V g M p L(V) g L(V) V φ Figure V.6.
The map ϕ induced by the linear map L This idea was introduced by E. Cartan [ C ], who proved that under some(strong) hypothesis on the curvature of M and M , this map is a local or evena global isometry. We prefer to rephrase it in the following terms: Definition.
Let ( M , p ) and ( M , p ) be complete Riemannian manifoldsof the same dimension with base points, and L : T p M → T p M a linearisometry.Let γ and γ be the geodesics defined in the interval [0 , , with γ startingat p with initial speed vector x ∈ T p M and γ starting at p with initial speed L ( x ) .For any three vectors v , v , v in T p M , define: • R ( x, v , v , v ) is the vector of T p M obtained by performing paralleltransport of v , v , v along γ , computing the Riemann curvature ten-sor at the point γ (1) ∈ M acting on those vectors, and then performingparallel transport backwards up to the point p . iv INTRODUCTION • R ( x, v , v , v ) is the vector of T p M obtained by performing paralleltransport of L ( v ) , L ( v ) , L ( v ) along γ , computing the Riemann curva-ture tensor at the point γ (1) ∈ M acting on those vectors, then perform-ing parallel transport backwards up to the point p , and finally applying L − to get a vector in T p M .If R ( x, v , v , v ) = R ( x, v , v , v ) ∀ v , v , v ∈ T p M for any two geodesics γ and γ as above, we say that the curvature tensors of ( M , p ) and ( M , p ) are L -related . The usual way to express that the curvature tensors of ( M , p ) and ( M , p ) are L -related is to say that the parallel traslation of curvature along radialgeodesics of ( M , p ) and ( M , p ) coincides.We say ( M , p ) and ( M , p ) are L -related iff they have the same dimen-sion and, whenever exp | O is injective for some domain O ⊂ T p M , then themap ϕ = exp ◦ L ◦ (exp | O ) − is an isometric inmersion.Cartan’s theorem states that if the curvature tensors of ( M , p ) and ( M , p ) are L -related, then ( M , p ) and ( M , p ) are L -related. (see lemma 1.35 of [ CE ]).In 1956 (see [ A ]), W. Ambrose proved a global version of the above theorem,but with stronger hypothesis. A broken geodesic is the concatenation of a finiteamount of geodesic segments.The theorem of Ambrose states that if the parallel traslation of curvaturealong broken geodesics on M and M coincide, then there is a global isometry ϕ : M → M whose differential at p is L . It is simple to prove that ϕ can beconstructed as above. It is enough if the hypothesis holds for broken geodesicswith only one “ elbow ” (the reader can find more details in [ CE ]).However, he conjectured that the same hypothesis of the Cartan’s lemmashould suffice, except for the obvious counterexamples of covering spaces:The Ambrose conjecture states that if the curvature tensors of ( M , p ) and ( M , p ) are L -related, and M and M are simply connected , there is a global isometry ψ : M → M such that ψ ◦ exp = exp ◦ L .Ambrose himself was able to prove the conjecture if all the data is analytic.In [ Hi ], in 1959, the conjecture was generalized to parallel transport for affineconnections, and in [ BH ], in 1987, to Cartan connections. Also in 1987, in thepaper [ H87 ], James Hebda proved that the conjecture was true for surfacesthat satisfy a certain regularity hypothesis, that he was able to prove true in1994 in [
H94 ]. J.I. Itoh also proved the regularity hypothesis independently in[ I ]. The latest advance came in 2010, after we had started our research on theAmbrose conjecture, when James Hebda proved in [ H10 ] that the conjectureholds if M is a heterogeneous manifold . Such manifolds are generic.In 5.1.6 we provide a new proof that works for surfaces and for a genericclass of manifolds in dimension . James Hebda’s proof in [ H10 ] is shorterthan ours and works for any dimension. However, his proof does not extendto arbitrary metrics and we think that our proof might, even though we havebeen unable to complete all the details to this day. Indeed, the proof presentedhere extends to some manifolds that are not covered by the result of J. Hebda,as this is truly a different approach. In chapter 6 we also provide some hintson how our proof might extend to a -dimensional manifold with an arbitrarymetric. I. SUMMARY OF RESULTS xv
We remark that in both our proof and James Hebda’s, it is only used that ( M , p ) and ( M , p ) are L -related, and there is no need to use the originalhypothesis that the curvarture tensors are L -related. VI. Summary of results • Reduction of a HJBVP with g (cid:54) = 0 to a HJBVP with g = 0 : theorem 2.1.6,published in [ AG1 ]. • Proof that all λ k are locally Lipschitz continuous in Finsler manifolds: the-orem 4.2.5, published in [ AG1 ]. • Proof that the singular set of a solution to a Hamilton-Jacobi BVP is thecut locus of a different manifold: theorem 2.1.8, published in [
AG1 ]. • Proof that cut loci in Finsler manifolds are balanced split loci : section 2.3,published in [
AG1 ]. • Local structure theorem for balanced split loci: theorem 3.1.2, published in[
AG1 ]. • Detailed descriptions of a balanced split loci near the different types ofpoints, except for a set of Hausdorff codimension 3: 4.6.2, 4.6.3, 4.6.5, 4.6.6and 4.6.7, published in [
AG2 ]. • Characterization of the cut locus and the balanced split loci: theorems 4.2.1,4.2.2 and 4.2.4, published in [
AG2 ]. • Proof of the Ambrose conjecture for a metric in G M : theorem 5.1.6, publi-cation pending. • A whole selection of fresh conjectures for the new generations in chapter 6...HAPTER 1
Preliminaries
Notation.
We fix the following notation for the rest of the thesis: • A C ∞ manifold M with boundary ∂M . • A Finsler metric ϕ on M . • The distance d induced on M by ϕ , and the “distance to the boundary” d ∂M ( p ) = inf q ∈ ∂M d ( q, p ) . • The geodesic vector field ρ in T M . • The time- t flow of ρ : Φ t : T M → T M . For a non-complete manifold, suchas a manifold with boundary, these maps are not defined in all of
T M . • A smooth map
Γ : ∂M → T M that is a section of the projection map π : T M → M of the tangent to M , and such that Γ( p ) points to the insideof M for every p ∈ ∂M . Definition . For a pair of points p, q ∈ M such that q belongs to aconvex neighborhood of p , we define, following [ IT00 ] , (1.1.1) v p ( q ) = ˙ γ (0) as the speed at of the unique unit speed minimizing geodesic γ from p to q . Definition . The approximate tangent cone to a subset E ⊂ M at p is: T ( E, p ) = (cid:26) rv : v = lim v p ( p n ) | v p ( p n ) | , ∃{ p n } ⊂ E, p n → p, r > (cid:27) and the approximate tangent space T an ( E, p ) to E at p is the vector spacegenerated by T ( E, p ) . We remark that the definition is independent of the Finsler metric, despiteits apparent dependence on the vectors v p ( p n ) . Concave (or convex)functions u may not be differentiable, but they are differentiable almost every-where. This allows for a simple definition of the subdifferential of a convex orsemiconvex function (see [ CS ] for different definitions): Definition . The subdifferential ∂u ( p ) of a concave function u at p can be defined as the convex hull of all the one forms that are limits ofdifferentials of u at points p n where u is differentiable. Definition . A function u : S → R is semiconcave if there ex-ists a nondecreasing upper semicontinuous function ω : R + → R + such that lim ρ → ω ( ρ ) = 0 and, for any x, y ∈ R and λ ∈ [0 , : λu ( x ) + (1 − λ ) u ( y ) − u ( λu + (1 − λ ) y ) ≤ λ (1 − λ ) | x − y | ω ( | x − y | ) The function ω is called the modulus of semiconcavity . The concave functions are those for which ω is zero. The functions with a linear modulus of semiconcavity are exactly those that can be written as thesum of a concave and a smooth function.It turns out that viscosity solutions to HJBVP (to be defined later), anddistance functions to the boundary in Finsler geometry, are semiconcave func-tions. Those functions share many of the regularity results of concave functions.For example, they are differentiable almost everywhere. Indeed, this statementcan be refined: let u : S → R be a semiconcave function and define the sets Σ k = { x ∈ S : dim( ∂u ( p )) ≥ k } Then Σ k is countably H n − k rectifiable: it is contained in the union of countablymany C hypersurfaces of dimension n − k , plus a H n − k -negligible set.Check [ CS ] and [ AAC ] for more details.
Definition . The orthogonal hyperplane to a vector v ∈ T p M isthe hyperplane tangent at v to the level set { v (cid:48) ∈ T p M : ϕ ( v (cid:48) ) = ϕ ( v ) } The orthogonal distribution to a vector field is defined pointwise.
Remark.
There are two unit vectors with a given hyperplane as orthogonalhyperplane. The first need not to be the opposite of the second unless H issymmetric ( H ( − v ) = H ( v ) ). We thus define two unit normal vectors to ahypersurface (the inner normal and outer normal). Definition . The dual one form to a vector v ∈ T p M with respectto a Finsler metric ϕ is the unique one form ω ∈ T ∗ p M such that ω ( v ) = ϕ ( v ) and ω | H = 0 , where H is the orthogonal hyperplane to v .For a vector field X , the dual differential one-form is obtained by applyingthe above construction at every point. We will often use the notation (cid:98) X for thedual one-form to the vector field X . Remark.
In Riemannian geometry, a different scaling is often used: ω ( v ) = 1 instead of ω ( v ) = ϕ ( v ) . We have chosen this definition because it makes theduality map v → ω continuous. Remark.
In coordinates, the dual one form w to the vector v is given by: w j = ϕ ( v ) ∂ϕ∂v j ( p, v ) Actually ϕ is 1-homogeneous, so Euler’s identity yields: w j v j = ϕ ( v ) ∂ϕ∂v j ( p, v ) v j = ϕ ( v ) .2. EXPONENTIAL MAPS OF FINSLER MANIFOLDS 3 and, for a curve γ ( − ε, ε ) → T p M such that γ (0) = v , ϕ ( γ ( t )) = ϕ ( v ) and γ (cid:48) (0) = z , ϕ ( v ) w j z j = ϕ ( v ) ∂∂t | t =0 ϕ ( γ ( t )) = 0 Remark.
The hypothesis on ϕ imply that the orthogonal form to a vector isunique, and the correspondence between a vector and its dual one form is oneto one, but it is only linear for riemannian metrics. Definition . Let D (Φ t ) be the domain of the time- t flow of the geo-desic vector field in T M . We introduce the sets V and W : (1.2.1) V = { ( t, p ) : t ≥ , p ∈ ∂M, Γ( p ) ∈ D (Φ t ) } ⊂ R × ∂M (1.2.2) W = { Φ t (Γ( p )) : t ≥ , p ∈ ∂M, Γ( p ) ∈ D (Φ t ) } ⊂ T MV and W are diffeomorphic through the map G ( t, p ) = Φ t (Γ( p )) . We definethe exponential map F associated to ( M, Γ) as F = π ◦ G : V → M .The interior of W is locally invariant under Φ t . This is equivalent to sayingthat ρ is tangent to W . The radial vector r = ∂∂t is mapped to ρ by G ∗ . Remark . The map G is injective in its domain; its inverse can becomputed walking a geodesic backwards until we hit the boundary for the firsttime. In other words, W is a smooth manifold and G is a diffeomorphism from V ⊂ R × ∂M . Remark . The map G can also be written G ( t, p ) = ( F ( p ) , dF p ( r )) ,as follows from the geodesic equations. In the particular case when Γ( p ) is the inner unit normal vector to ∂M , thisis the standard “exponential map from ∂M ”. This, in turn, includes the expo-nential map from a point p in a manifold without boundary, in the followingway:Let p be a point in a manifold without boundary, and remove a ball B ofsmall radius around p . The result is a manifold with boundary ∂M = ∂B , andthe exponential map from ∂B ⊂ M \ B coincides with the exponential mapfrom p ∈ M .The same trick works for the exponential from a submanifold of any codi-mension inside a manifold without boundary, removing a tubular neighborhoodaround the submanifold.The motivation for working in the above setting is that it also works for theHamilton-Jacobi BVP with non-trivial boundary data, and it does not makethe proofs more complicated. Definition . Let x = ( t, z ) ∈ V .We say x is conjugate iff F is not a local diffeomorphism at x . The order of conjugacy is the dimension of the kernel of dF .We say x is a first conjugate vector iff no point ( s, z ) , for s < t , is conjugate.We also call the image of the radial vector d x F ( r x ) a conjugate vector (for F ) whenever x is conjugate. CHAPTER 1. PRELIMINARIES
In differential geometry, it is more usual to use the term focal instead of conjugate , when studying the distance function from a hypersurface, but otherauthors do otherwise (see for instance [ LN ]). We have decided to use this termbecause our results about the Ambrose conjecture, that will appeal the mostto differential geometers, deal only with the exponential map from a singlepoint in a Riemannian manifold, while the other results appeal more to peopleworking in PDEs, which prefer the term conjugate . The following proposition states someproperties of a Finsler exponential map that correspond approximately to thedefinition of regular exponential map introduced in [ Wa ]. The second propertyis the only one that is not standard, but we need it to prove the existence ofthe special coordinates (and only for that). Proposition . The exponential map F has the following properties: • dF x ( r x ) is a non zero vector in T F ( x ) M . • at every point x ∈ V there is a basis B = { v , .., v n } of T x V where v = r and v n − k +1 , .., v n span ker dF x , and such that: B (cid:48) = (cid:110) dF ( v ) , . . . , dF ( v n − k ) , (cid:94) d F ( r(cid:93)v n − k +1 ) , . . . (cid:94) d F ( r(cid:93)v n ) (cid:111) is a basis of T F ( x ) M , where (cid:94) d F ( r(cid:93)v j ) is a representative of d F ( r(cid:93)v j ) ∈ T F ( x ) M/dF ( T V x ) , for n − k + 1 ≤ j ≤ n . • Any point x ∈ V has a neighborhood U such that for any ray γ (an integralcurve of r ), the sum of the dimensions of the kernels of dF at points in γ ∩ U is constant. • For any two points x (cid:54) = x in V with F ( x ) = F ( x ) , dF x ( r x ) (cid:54) = dF x ( r x ) Proof. The first three properties follow from the work of Warner [ Wa , Theorem4.5] for a Finsler exponential map. We emphasize that they are local properties.The last one follows from the uniqueness property for second order ODEs. Weremark that the second property implies the last one locally.Indeed, properties 1 and 3 are found in standard textbooks such as [ M ].Let us recall some of the notation in [ Wa ] and [ APS ] and show the equivalenceof the second property with his condition (R2) on page 577: • Second order tangent vectors at a point in an n -dimensional manifold arewritten in coordinates in the following way ( a ij is symmetric): σ ( f ) = (cid:88) i,j a ij ∂ f∂x i ∂x j + (cid:88) i b i ∂f∂x i • T x V is the set of second order tangent vectors at the point x in themanifold V . • The second order differential of F : V → M at x is the map d x F : T x V → T x M defined by: d x F ( σ ) f = σ ( f ◦ F ) .3. THE CUT LOCUS 5 • The symmetric product v(cid:93)w of v ∈ T x V and w ∈ T x V is a well definedelement of T x V /T x V with a representative given by the formula: ( v(cid:93)w ) f = 12 ( v ( w ( f )) + w ( v ( f ))) for arbitrary extensions of v to w to vector fields near x . • The map d x F induces the map d F : T V x /T V x → T F ( x ) M/dF ( T V x ) bythe standard procedure in linear algebra. • For x ∈ V , v ∈ T x V and w ∈ ker dF x , d F ( v(cid:93)w ) makes sense as a vectorin the space T F ( x ) M/dF ( T V x ) . For any extension of v and w , the vector d F ( v(cid:93)w ) is a first order vector.Thus, our condition is equivalent to property (R2) of Warner:At any point x where ker dF x (cid:54) = 0 , the map d F : T V x /T V x → T M F ( x ) /dF ( T V x ) sends (cid:104) r x (cid:105) (cid:93) ker dF x isomorphically onto T F ( x ) M/dF ( T V x ) .We recall that d x F ( v ) can be computed as a Jacobi field when we consideronly the exponential map from a point, but this interpretation is somewhatdiluted when we work with the exponential map from the boundary. Remark.
Warner defines a regular exponential map as any map from T p M into M thast satisfies the properties of proposition 1.2.5. We do not need towork in that generality, as it does not include any new application. In order to study the map F more comfortably,we define the special coordinates , a pair of coordinates near a conjugate point z of order k and its image F ( z ) that make F specially simple. They can bedefined for any regular exponential map .Let B = { v , . . . , v n } be the basis of T z V indicated in the second part ofProposition 1.2.5, and B (cid:48) F ( z ) the corresponding basis at F ( z ) ∈ M formed byvectors d z F ( v i ) , i ≤ n − k , and (cid:94) d z F ( v (cid:93)v n − k +1 ) , . . . , (cid:94) d z F ( v (cid:93)v n ) .Make a linear change of coordinates in a neighborhood of F ( z ) taking B (cid:48) F ( z ) to the canonical basis. The coordinate functions F i ( x ) − F i ( z ) of F for i ≤ n − k can be extended to a coordinate system near z with the help of k functionshaving v n − k +1 , . . . , v n as their respective gradients at z . In these coordinates F looks:(1.2.3) F ( x , . . . , x n ) = ( x , . . . , x n − k , F n − k +1 z ( x ) , . . . , F nz ( x )) and • ∂∂x i F j ( x ) is for any i and j ≥ n − k + 1 , • ∂∂x i ∂∂x F j ( x ) is δ ij , for i, j ≥ n − k + 1 . • ∂∂x ( x ) = r x Let M be a Finsler manifold with boundary ∂M . We mentioned earlierthat the study of the exponential map from a point or submanifold can bereduced to a exponential map of a manifold with boundary ∂M . The same CHAPTER 1. PRELIMINARIES principle applies to the cut locus, so we will consider only the cut locus fromthe boundary
Cut = Cut ∂M . It can be defined in several equivalent ways: Definition . Let M be a Finsler manifold with boundary ∂M : • For any p ∈ ∂M , let γ p be the unit speed geodesic γ with initial point in ∂M and initial speed orthogonal to ∂M (and inner-pointing). γ p minimizes thedistance between γ (0) and γ ( t ) , for small t . Define t cut ( p ) = sup { t : d ( γ p (0) , γ p ( t )) = t } Then
Cut = { γ p ( t cut ( p )) : p ∈ ∂M }• For a point p ∈ M , let Q p be the set of points q ∈ ∂M such that d ( q, p ) = d ∂M ( p ) . Then Cut is the closure of the set of points such that Q p hasmore than one element. • The function d ∂M is singular exactly when Q p has more than one element,so we can also define Cut as the closure of the set where the distancefunction to ∂M is singular. • Cut is also the set of points such that either Q p has more than one element,or Q p = { q } , and ( d ( q, p ) , q ) ∈ V is conjugate. The reader can find the proof of those facts for Riemann manifolds instandard textbooks in Riemannian geometry (see for example chapter 13 in[ dC ]). The proof for Finsler manifolds can be found in [ LN ], for example. Forbasic information about the distance function, such as its differentiabilty, thereader can use [ CS ].Much is known about the set Cut : • Cut is a deformation retract of M (obvious). • It is the union of a ( n − )-dimensional smooth manifold consisting ofpoints with two minimizing geodesics and a set of Hausdorff dimension atmost n − . This easy but important lemma appears to have been provenat least in [ H83 ], [
IT98 ], [ BL ] and [ MM ], always for the Riemanniancase. We give a proof of this result in lemma 3.2.3 that is also true forFinsler manifolds. • It is stratified by the dimension of the subdifferential of the distance to theboundary ∂d ∂M . This follows from the properties of semiconcave functionsmentioned in 1.1.2, as d ∂M is semiconcave. This result can be found in[ AAC ], and their proof works verbatim for balanced split locus in our3.3.3. • The local homology of the cut locus of p ∈ M at a point q is related tothe set of minimizing geodesics from p to q (see [ Oz ] and [ H83 ]). • The cut locus has finite Hausdorff measure H n − . This result can be foundin [ IT00 ] for Riemannian manifolds, and in [ LN ] for Finsler manifolds.We provide a new proof of this result for Finsler manifolds in 4.4. M.Castelpietra and L. Rifford also gave a proof of this result that appearedshortly after the one we present here. • If all the data is analytic,
Cut is a stratified analytic manifold (see [
B77II ]).We will not use this result. • If we add a generic perturbation to H or M , Sing becomes a stratifiedsmooth manifold. Furthermore, for dimension up to , the cut locus is .4. HAMILTON-JACOBI EQUATIONS AND FINSLER GEOMETRY 7 generically stable, in the sense that adding a small perturbation to themetric, the new cut locus would still be diffeomorphic to the original one.This is a very deep result from M. A. Buchner, a student of J. Matter,and it is beyond the scope of this work to include a proof of his results(see [ B ] and [ B77 ]), but we will make use of them in chapter 5.On the other hand, H. Gluck and D. Singer proved that there are non-triangulable cut loci in [ GS ] and, in [ GSII ], that there are surfaces of revolutionsuch that the cut locus from any point is non-triangulable. Another difficultyis mentioned by J. Hebda in [
H87 ]: even though the homotopy of
Cut isknown, and even if the cut locus is a simplicial complex, that simplicial complexmay not descend simplicially to one point, and this was a major obstacle inextending his proof of the Ambrose conjecture for surfaces to manifolds ofhigher dimension.
Here we review the relationship between Hamilton-Jacobi equations andFinsler geometry. The reader can find more details in [ LN ], [ L ] and [ CS ].Here M is a manifold with possibly non-compact boundary. We are inter-ested on solutions to the system (which we will refer to as a Hamilton-JacobiBoundary Value Problem or HJBVP for short): H ( p, du ( p )) = 1 p ∈ M (1.4.1) u ( p ) = g ( p ) p ∈ ∂M (1.4.2)where H : T ∗ M → R is a smooth function that is -homogeneous and sub-additive for linear combinations of covectors lying over the same point p , and g : ∂M → R is a smooth function that satisfies the compatibility condition:(1.4.3) | g ( p ) − g ( q ) | < kd ( p, q ) ∀ p, q ∈ ∂M for some k < . Here d is the distance induced by the Finsler metric ϕ that isthe pointwise dual of the metric in T ∗ M given by H :(1.4.4) ϕ p ( v ) = sup (cid:110) (cid:104) v, α (cid:105) p : α ∈ T ∗ p M, H ( p, α ) = 1 (cid:111) Remark.
As mentioned in the introduction, we can ask that H − (1) ∩ T ∗ p M is strictly convex for every p instead of asking that H is -homogeneous andsubadditive for linear combinations of covectors lying over the same point p .The properties are not equivalent for a function H , but the equations that weconsider are the same, because the only thing we use about H is the -level set.If the sets H − (1) ∩ T ∗ p M are convex for every p , we can replace H with a newone that is -homogeneous, subadditive for linear combinations of covectorslying over the same point p , and has the same -level set. Using the definition 1.1.6 of dualform in Finsler geometry, we can restate the usual equations for the character-istic vector field at points p ∈ ∂M : ϕ p ( X p ) = 1 (cid:99) X p | T ( ∂M ) = dgX p points inwards(1.4.5) CHAPTER 1. PRELIMINARIES
We define the characteristic vector field as a map
Γ : ∂M → T M , by theformula Γ( p ) = X p . The characteristic curves are the integral curves of thegeodesic vector field in T M with initial point Γ( z ) for z ∈ ∂M . The projectedcharacteristics are the projection to M of the characteristics.A local classical solution u to the HJBVP can be computed near ∂M fol-lowing characteristic curves: Definition . Let U be a neighborhood of ∂M such that every point q ∈ U belongs to a unique (projected) characteristic contained in U and startingat a point p ∈ ∂M (the point p is often called the footpoint of q ).The solution by characteristics u : U → R is defined as follows: if γ : [0 , t ] → M is the unique (projected) characteristic from a point p ∈ ∂M to q = γ ( t ) that does not intersect Sing , then u ( q ) = g ( p ) + t The solutionfound above using characteristic curves is only defined in a neighborhood of ∂M .There is a different notion of solution to these equations. The inspirationcame from the following observation: if we add a small viscosity term like − ε ∆ to (1.4.1), that equation becomes semilinear elliptic, and admits a globalsolution. So the idea appeared to add that viscosity term, and later let ε converge to . Even though this was the inspiration, it later became clear thatit was better to use a different definition, using comparison functions. Definition . A function u : M → R is a viscosity subsolution (resp. supersolution) to the HJBVP given by (1.4.1) and (1.4.2) iff for any φ ∈ C ( M ) such that u − φ has a local maximum at p (resp., a local minimum),we have: (1.4.6) H ( p, Dφ ( p )) ≤ (resp. H ( p, Dφ ( p )) ≥ )It is a viscosity solution to the HJBVP iff it is both a subsolution and asupersolution. The reader can find more details in [ L ] or [ CS ].However, in this thesis we will not be concerned neither with the inspirationthat gave them the name “viscosity solutions”, nor with the actual definition.We only need to know that the unique viscosity solution is given by theLax-Oleinik formula (see theorem 5.2 in [ L ]):(1.4.7) u ( p ) = inf q ∈ ∂M { d ( p, q ) + g ( q ) } The viscosity solution can be thought of as a way to extend the classicalsolution by characteristics to the whole M . When g = 0 , the solution (1.4.7)is the distance to the boundary .As we mentioned earlier, the viscosity solution to a HJBVP is a semiconcavefunction . It is interesting to remark that a semiconcave function that satifiesthe equation 1.4.1 at the points at which it is differentiable is the viscositysolution to the HJBVP.HAPTER 2 A new way to look at Cut and Singular Loci
Let us consider the HJBVP given by (1.4.1) and (1.4.2) when g = 0 . Onthe one hand, Γ( p ) is the inner pointing unit normal to ∂M at p . On the otherhand, the viscosity solution u given by (1.4.7) is the distance to the boundary.This has nice consequences: for example, the singular set of u is a cut locus,and we can apply the various structure results about the cut locus mentionedin section 1.3.Our intention in this section is to adapt this result to the case g > . If ∂M is compact, a global constant can be added to an arbitrary g so that this issatisfied and S is unchanged. We still require that g satisfies the compatibilitycondition 1.4.3.Subject to these conditions, our goal is to show that the Finsler manifold ( M, ϕ ) can be embedded in a new manifold with boundary ( N, ˜ ϕ ) such that u is the restriction of the unique solution ˜ u to the problem ˜ H ( p, d ˜ u ( p )) = 1 p ∈ N ˜ u ( p ) = 0 p ∈ ∂N thus reducing to the original problem ( ˜ H and ˜ ϕ are dual to one another as in1.4.4). This allows us to characterize the singular set of (1.4.7) as a cut locus,which automatically implies that all the structure results about the cut locusin section 1.3 apply to the more general case. Definition . The indicatrix of a Finsler metric ϕ at the point p isthe set I p = { v ∈ T p M : ϕ ( p, v ) = 1 } Lemma . Let ϕ and ϕ be two Finsler metrics in an open set U , andlet X be a vector field in U such that: • The integral curves of X are geodesics for ϕ . • ϕ ( p, X p ) = ϕ ( p, X p ) = 1 • At every p ∈ U , the tangent hyperplanes to the indicatrices of ϕ and ϕ in T p U coincide.Then the integral curves of X are also geodesics for ϕ Proof.
Let p be a point in U . Take bundle coordinates of T p U around p suchthat X is one of the vertical coordinate vectors. An integral curve α of X satisfies: ( ϕ ) p ( α ( t ) , α (cid:48) ( t )) = ( ϕ ) p ( α ( t ) , α (cid:48) ( t )) = 1
90 CHAPTER 2. A NEW WAY TO LOOK AT CUT AND SINGULAR LOCI because of the second hypothesis. The third hypothesis imply: ( ϕ ) v ( α ( t ) , α (cid:48) ( t )) = ( ϕ ) v ( α ( t ) , α (cid:48) ( t )) So inspection of the geodesic equation:(2.1.1) ϕ p ( α ( t ) , α (cid:48) ( t )) = ddt ( ϕ v ( α ( t ) , α (cid:48) ( t ))) shows that α is a geodesic for ϕ . (cid:3) Corollary . Let ϕ be a Finsler metric and X a vector field whoseintegral curves are geodesics. Then there is a Riemannian metric for whichthose curves are also geodesics. Proof.
The Riemannian metric g ij ( p ) = ∂∂v i v j ϕ ( p, X ) is related to ϕ as in thepreceeding lemma. (cid:3) Lemma . Let X be a non-zero norm- geodesic vector field in a Finslermanifold and ω its dual differential one-form. Then the integral curves of X aregeodesics if and only if the Lie derivative of ω in the direction of X vanishes. Proof.
The integral curves of X are geodesics for ϕ iff they are geodesics forthe Riemannian metric g ij ( p ) = ∂∂v i v j ϕ ( p, X ) , but the dual one-form to X withrespect to both metrics is the same one-form ω , and the vanishing of L X ω hasnothing to do with the metric.We have thus reduced the problem to a Riemannian metric, when this resultis standard: L X ( Y ) = X ( w ( Y )) − ω ([ X, Y ])= X ( (cid:104) X, Y (cid:105) ) − (cid:104) X, D X Y − D Y X (cid:105) = (cid:104) D X X, Y (cid:105) + (cid:104) X, D X Y (cid:105) − (cid:104) X, D X Y (cid:105) + (cid:104) X, D Y X (cid:105) = (cid:104) D X X, Y (cid:105) + Y (cid:104) X, X (cid:105) (cid:3)
Proposition . Let M be an open manifold with smooth boundary anda Finsler metric ϕ . Let X be a smooth transversal vector field in ∂M pointinginwards (resp. outwards). Then M is contained in a larger open manifoldadmitting a smooth extension ˜ ϕ of ϕ to this open set such that the geodesicsstarting at points p ∈ ∂M with initial vectors X p can be continued indefinitelybackward (resp. forward) without intersecting each other. Proof.
We will only complete the proof for a compact manifold with bound-ary M and inward pointing vector X , as the other cases require only minormodifications.We start with an arbitrary smooth extension ϕ (cid:48) of ϕ to a larger open set M ⊃ M . The geodesics with initial speed X can be continued backwardsto M , and there is a small ε for which they do not intersect each other fornegative values of time before the parameter reaches − ε .Define P : ∂M × ( − ε, → M , P ( q, t ) := α q ( t ) where α q : ( − ε, → M is the geodesic of ϕ (cid:48) starting at the point q ∈ ∂M with initial vector X q . When p ∈ U ε := Image( P ) there is a unique value of t .1. THE RELATION BETWEEN FINSLER GEOMETRY AND HAMILTON-JACOBI BVPS 11 such that p = P ( q, t ) for some q ∈ ∂M . We will denote such t by d ( p ) . Extendalso the vector X to U ε as X p = ˙ α q ( t ) where p = P ( q, t ) .Let c : ( − ε, → [0 , be a smooth function such that • c is non-decreasing • c ( t ) = 1 for − ε/ ≤ t • c ( t ) = 0 for t ≤ − ε/ and finally define ˜ X p = c ( d ( p )) X p in the set U ε .Let ω be the dual one form of ˜ X with respect to ϕ for points in ∂M , andlet ω be the one form in U ε whose Lie derivative in the direction ˜ X is zeroand which coincides with ω in ∂M . Then we take any metric ϕ (cid:48)(cid:48) in U ε (whichcan be chosen Riemannian) such that ˜ X has unit norm and the kernel of ω istangent to the indicatrix at ˜ X .By lemma 2.1.4, the integral curves of ˜ X are geodesics for ϕ (cid:48)(cid:48) . Now let ρ bea smooth function in U ε ∪ M such that ρ | M = 1 , ρ | U ε \ U ε/ = 0 and ≤ ρ ≤ ,and define the metric: ˜ ϕ = ρ ( p ) ϕ ( p, v ) + (1 − ρ ( p )) ϕ (cid:48)(cid:48) ( p, v ) This metric extends ϕ to the open set U ε and makes the integral curvesof ˜ X geodesics. As the integral curves of X do not intersect for small t , theintegral curves of ˜ X reach infinite length before they approach ∂U ε and thelast part of the statement follows. (cid:3) Application of this proposition to M and the characteristic, inwards-pointingvector field v yields a new manifold N containing M , and a metric for N thatextends ϕ (so we keep the same letter) such that the geodesics departing from ∂M which correspond to the characteristic curves continue indefinitely back-wards without intersecting.This allows the definition, for small δ of ˜ P : ∂M × ( −∞ , δ ] → N, P ( q, t ) := ˜ α q ( t ) where ˜ α are the geodesics with initial condition X , continued backwards if t isnegative. Finally, define ˜ u : U → R by:(2.1.2) ˜ u ( p ) = (cid:40) g ( q ) + t p = ˜ P ( q, t ) , p ∈ N \ Mu ( p ) p ∈ M We notice that both definitions agree in an inner neighborhood of ∂M , so thefunction ˜ u is a smooth extension of u to N . Theorem . Let
Λ = ˜ u − (0) . Then the following identity holds in { ˜ u ≥ } : (2.1.3) ˜ u ( p ) = d (Λ , p ) Proof. Λ is smooth because it is contained in N \ M , where ˜ u is smooth andhas non-vanishing gradient.In order to show that ˜ u and d Λ agree in U , we use the uniqueness propertiesof viscosity solutions. Let N be the open set where ˜ u > . The distancefunction to Λ is characterized as the unique viscosity solution to: • ˜ u = 0 in Λ • H ( p, d ˜ u ( p )) = 1 in N , in the viscosity senseClearly ˜ u satisfies the first condition. It also satisfies the second for pointsin the set M because it coincides with u , and for points in N \ M because ˜ u issmooth and H ( p, d ˜ u ( p )) = 1 holds in the classical sense there. (cid:3) The following fact is well known but we provide a geometric proof.
Corollary . The differential du of the solution by characteristics isFinsler dual to the tangent to the (projected) characteristics. Proof.
By the above, we can assume that g = 0 . Let U be a neighborhoodof ∂M where a solution by characteristics u is defined, as in 1.4.1. Let q ∈ U , X be the tangent to the characteristic that goes through q , with footpoint p .The claim can be checked easily if q ∈ ∂M , because both ˜ X and du arelinear forms and they agree on the hyperplane T q ∂M and Γ( q ) , by the definition1.4.5 of the characteristic vector field.For the rest of points in U , we notice that the level curves of u are Lieparallel with respect to X , and so L X ( du ) = 0 . But 2.1.4 says that L X ( ˜ X ) = 0 ,and thus we have two -forms that agree on ∂M and are parallel with respectto X , so they agree everywhere. (cid:3) The following theorem is an extension of Theorem 1.1 of [ LN ]. In thisresult ∂M may not be compact. Theorem . Let S be the closure of the singular set of the viscositysolution to the following HJBVP: H ( p, du ( p )) = 1 p ∈ Mu ( p ) = g ( p ) p ∈ ∂M where g : ∂M → R satisfies the usual compatibility condition 1.4.3.If µ is the function whose value at p ∈ ∂M is the distance to S along theunique characteristic departing from q , then (1) µ is Lipschitz. (2) If in addition ∂M is compact, then the ( n − -dimensional Hausdorffmeasure of S ∩ K is finite for any compact K . (3) In general, S is a Finsler cut locus from the boundary of some Finslermanifold, so all the regularity results for cut loci apply to S (see section1.3). Proof.
The first part follows immediately from Theorem 2.1.6 and Theorem1.1 in [ LN ]. The second is an easy consequence of the first, while the last iscontained in the results of this section. (cid:3) Remark.
The regularity hypothesis on M can be softened. In order to applythe results in [ LN ], it is enough that M is C , , which implies that Λ is C , . We study a Hamilton-Jacobi equation given by (1.4.1) and (1.4.2) in a C ∞ compact manifold with boundary M , with the hypothesis stated there. .2. SPLIT LOCUS AND BALANCED SPLIT LOCUS 13 Let
Sing be the closure of the singular set of the viscosity solution u toaHamilton-Jacobi BVP. Sing has a key property: any point in M \ Sing canbe joined to ∂M by a unique characteristic curve that does not intersect Sing .A set with this property is said to split M along characteristics of the HJBVP or simply to split M for short. Once characteristic curves are known, if wereplace Sing by any set S that splits M , we can use the value of u that thecharacteristics carry along with them, to obtain a function, defined in M \ S with some resemblance to the viscosity solution (see definition 2.2.5).Looking at the cut locus from this new perspective, we wonder what dis-tinguishes the cut locus from all the other sets that split M . Definition . For a set S ⊂ M , let A ( S ) ⊂ V be the set of all x =( t, z ) ∈ V such that F ( s, z ) / ∈ S, ∀ ≤ s < t . We say that a set S ⊂ M splits M iff F restricts to a bijection between A ( S ) and M \ S . Whenever S splits M , we can define a vector field R p in M \ S to be dF x ( r x ) for the unique x in A ( S ) such that F ( x ) = p . Definition . For a point a ∈ S , we define the limit set R a as the setof vectors in T a M that are limits of sequences of the vectors R p defined aboveat points p ∈ M \ S . Remark . If S is a cut locus, the set R p is the set of all vectors tangentto the minimizing geodesics from p to ∂M . Definition . If S splits M , we also define a set Q p ⊂ V for p ∈ M by Q p = (cid:16) F | A ( S ) (cid:17) − ( p ) The following relation holds between the sets R p and Q p : R p = { dF x ( r x ) : x ∈ Q p } Definition . If S splits M , we can define a real-valued function h in M \ S by setting: h ( p ) = g ( z ) + t where ( t, z ) is the unique point in A ( S ) with F ( t, z ) = p . If we start with the viscosity solution u to the Hamilton-Jacobi equations,and let S = Sing be the closure of the set where u is not C , then S splits M .If we follow the above definition involving A ( S ) to get a new function h , thenwe find h = u . Definition . A set S that splits M is a split locus iff S = { p ∈ S : (cid:93)R p ≥ } The role of this condition is to restrict S to its essential part. A set thatmerely splits M could be too big: actually M itself splits M . The followinglemma may clarify this condition. Lemma . A set S that splits M is a split locus if and only if S is closedand it has no proper closed subsets that split M . Proof.
The “if” part is trivial, so we will only prove the other implication.Assume S is a split locus and let S (cid:48) ⊂ S be a closed set splitting M . Let q ∈ S \ S (cid:48) be a point with (cid:93)R q ≥ . Since S (cid:48) is closed, there is a neighborhoodof q away from S (cid:48) ; so, if γ is a segment of a geodesic in M \ S (cid:48) joining ∂M with q , there is a point q in γ lying beyond q . Furthermore, we can choose the point q not lying in S , so there is a second geodesic γ contained in M \ S ⊂ M \ S (cid:48) from ∂M to q . As q ∈ S , we see γ is necessarily different from γ , which is acontradiction if S (cid:48) splits M . Therefore we learn S (cid:48) ⊃ { p ∈ S : (cid:93)R p ≥ } , so S = { p ∈ S : (cid:93)R p ≥ } ⊂ S (cid:48) . (cid:3) Finally, we introduce the following more restrictive condition (see 1.1.1 forthe definition of v p ( q ) , the vector from p to q , and 1.1.6 for the Finsler dual ofa vector). Definition . We say a split locus S ⊂ M is balanced for given M , H and g (or simply that it is balanced if there is no risk of confusion) iff forall p ∈ S , all sequences p i → p with v p i ( p ) → v ∈ T p M , and any sequence ofvectors X i ∈ R p i → X ∞ ∈ R p , then w ∞ ( v ) = max { w ( v ) , w is dual to some R ∈ R p } where w ∞ is the dual of X ∞ . Figure 2.2.1.
An arbitrary split locus and a balanced split locus
In this section we show that the cut locus of a Finsler exponential map is abalanced set. We provide two proofs, none of which is original. Although thehypothesis look different, our result 2.1.6 show that they are equivalent.The first proof is the same as in lemma 2.1 in [
IT00 ], but we adapt it sothat it also works for Finsler manifolds, where angles are not defined. .3. BALANCED PROPERTY OF THE FINSLER CUT LOCUS 15
Proposition . The cut locus of a Finsler manifold M with boundaryis a balanced split locus. Moreover, for p , p n , v and X ∞ as in the definition ofa balanced split locus, we have lim n →∞ d ( ∂M, p ) − d ( ∂M, p n ) d ( p, p n ) = w ∞ ( v ) Proof.
The cut locus S splits M , as follows from the first definition of cutlocus in 1.3.1.It is also a split locus, as follows from the second definition of cut locus.Next we show that S is balanced. Take any Y ∈ R p , and let γ be theminimizing geodesic segment joining ∂M to p with speed Y at p . Take anypoint q ∈ γ that lies in a convex neighborhood of p and use the triangleinequality to get: d ( ∂M, p ) − d ( ∂M, p n ) ≥ d ( q, p ) − d ( q, p n ) The first variation formula yields, for a constant C : d ( q, p ) − d ( q, p n ) ≥ w ( v p n ( p )) d ( p n , p ) − Cd ( p, p n ) and we get: lim inf n →∞ d ( ∂M, p ) − d ( ∂M, p n ) d ( p, p n ) ≥ w ( X ) for any w that is dual to a vector in R p .Then consider X ∞ , let γ be the minimizing geodesic segment joining ∂M to p with speed X ∞ at p , and let γ n be the minimizing geodesic segment joining ∂M to p n with speed X n at p n . Take points q n in γ n that lie in a fix convexneighborhood of p . Again: d ( ∂M, p ) − d ( ∂M, p n ) ≤ d ( q n , p ) − d ( q n , p n ) while the first variation formula yields, for a constant C : d ( q n , p ) − d ( q n , p n ) ≤ w ( v p n ( p )) d ( p n , p ) + Cd ( p, p n ) and thus: lim sup n →∞ d ( ∂M, p ) − d ( ∂M, p n ) d ( p, p n ) ≤ w ∞ ( X ) This proves the claim that S is balanced. (cid:3) We give now another proof that relates the balanced condition to the notionof semiconcave functions, which is now common in the study of Hamilton-Jacobi equations. More precisely, we simply translate theorem 3.3.15 in thebook [ CS ] to our language to get the following lemma: Lemma . The closure of the singular set of the viscosity solution to (1.4.1) and (1.4.2) is a balanced split locus.
Proof.
Let u be the viscosity solution to (1.4.1) and (1.4.2), and let Sing bethe closure of its singular set. We leave to the reader the proof that
Sing is asplit locus (otherwise, recall it is a cut locus).It is well known that u is semiconcave (see for example [ CS , 5.3.7]). Thesuperdifferential D + u ( p ) of u at p is the convex hull of the set of limits ofdifferentials of u at points where u is C (see [ CS , 3.3.6]). At a point where u is C , the dual of the speed vector of a characteristic is the differential of u . Thus, the superdifferential at p is the convex hull of the duals to the vectorsin R p . We deduce: max { w ( v ) , w is dual to some R ∈ R p } = max (cid:8) w ( v ) , w ∈ D + u ( p ) (cid:9) Given p ∈ M , and v ∈ T p M , the exposed face of D + u ( p ) in the direction v isgiven by: D + ( p, v ) = { ˜ w ∈ D + u ( p ) : ˜ w ( v ) ≤ w ( v ) ∀ w ∈ D + u ( p ) } The balanced condition can be rephrased in these terms as:Let p i → p ∈ S be a sequence with v p i ( p ) → v ∈ T p M , and let w i ∈ D + u ( p i ) be a sequence converging to w ∈ D + u ( p ) .Then w ∈ D + u ( p, − v ) which is exactly the statement of theorem [ CS , 3.3.15], with two minor remarks:(1) The condition is restricted to points p ∈ S . At points in M \ S , thebalanced condition is trivial.(2) In the balanced condition, we use the vectors v p i ( p ) from p i to p , contraryto the reference [ CS ]. Thus the minus sign in the statement. (cid:3) In the light of this new proof, we can regard the balanced condition asa differential version of the semiconcavity condition. A semiconcave functionthat is a solution to (1.4.1) is also a viscosity solution (see [ CS , 5.3.1]). We willlater study if the solution of (1.4.1) built by characteristics using a balancedsplit loci is also a viscosity solution.HAPTER 3 Local structure of cut and singular loci up tocodimension 3
Our main result in this chapter is a local description of the cut locus aroundany point of the cut locus except for a set of Hausdorff dimension n − (seeTheorem 3.1.2).Actually, our structure results hold for the more general balanced split loci (recall that in 2.3.1 and 2.1.8 we showed that cut loci, and singular sets ofsolutions to HJ equations, are balanced split loci). Working in this generalitycomplicates some proofs and, in particular, we have to prove some results for balanced split loci that are long known to be true for cut loci. However, we needto actually prove the structure results for balanced split loci for the applicationsto chapter 4, and all the new proofs of old facts are either short or interestingfor their own sake. For the results of this chapter, M is a C ∞ Finsler manifold with compact boundary ∂M , but M itself need not be compact. S ⊂ M is a balanced splitlocus. Recall 2.2.2 for the definition of R p . Our main result asserts that we can avoid conjugate pointsof order and above if we neglect a set of Hausdorff dimension n − : Theorem ) . There is a set N ⊂ S ofHausdorff dimension at most n − such that for any p ∈ S \ N and x ∈ V such that F ( x ) = p and d x F ( r x ) ∈ R p : dim(ker d x F ) ≤ Combining this new result with previous ones in the literature, we are ableto provide the following description of a cut locus. All the extra results requiredfor the proof of this result will be proved in this chapter.
Theorem H -codimension 3) . Let S be eitherthe cut locus of a point or submanifold in a Finsler manifold or the closureof the singular locus of a solution of 1.4.1 and 1.4.2. Then S consists of thefollowing types of points : • Cleave points:
Points at which R p consists of two non-conjugate vectors.The set of cleave points is a smooth hypersurface; • Edge points:
Points at which R p consists of exactly one conjugate vectorof order 1. This is a set of Hausdorff dimension at most n − ;
178 CHAPTER 3. LOCAL STRUCTURE OF CUT AND SINGULAR LOCI UP TOCODIMENSION 3 • Degenerate cleave points:
Points at which R p consists of two vectors,such that one of them is conjugate of order 1, and the other may be non-conjugate or conjugate of order 1. This is a set of Hausdorff dimensionat most n − ; • Crossing points:
Points at which R p consists of non-conjugate and con-jugate vectors of order 1, and R ∗ p spans an affine subspace of dimension . This is a rectifiable set of dimension at most n − ; • Remainder:
A set of Hausdorff dimension at most n − ; In the next chapter, we will provide more detailed descriptions of how doesa balanced split loci looks near each of these different points (see 4.6.2, 4.6.3,4.6.5, 4.6.6 and 4.6.7).In the next chapter and also in section 6.5 we show applications of thisresult, but we believe it can also be useful in other contexts. For instance,stochastic processes on manifolds is often studied on the complement of thecut locus from a point, and then the results have to be adapted to take care ofthe situation when the process hits the cut locus (see [ BL ]). Brownian motion,for example, almost never hits a set with null H n − measure, but will almostsurely hit any set with positive H n − measure, so we think our result can beuseful in that field. We provide examples of Riemannian manifolds and expo-nential maps which illustrate our results.First, consider a solid ellipsoid with two equal semiaxis and a third largerone. This is a 3D manifold with boundary, and the geodesics starting at thetwo points that lie further away from the center have a first conjugate vectorof order while remaining minimizing up to that point. This example showsthat our bound on the Hausdorff dimension of the points in the cut locus witha minimizing geodesic of order cannot be improved.Second, consider the surface of an ellipsoid with three different semiaxis(or any generic surface as in [ B ], with metric close to the standard sphere)and an arbitrary point on it. It is known that in the tangent space the set offirst conjugate points is a closed curve C bounding the origin, and at most ofthese points the kernel of the exponential map is transversal to the curve C .More explicitely, the set C ∗ of points of C where it is not transversal is finite.Consider then the product M of two such ellipsoids. The exponential maponto M has a conjugate point of order at any point in ( C \ C ∗ ) × ( C \ C ∗ ) ,and the kernel of the exponential map is transversal to the tangent to C × C .Thus the image of the set of conjugate points of order is a smooth manifoldof codimension .This example shows that theorem 3.1.1 does not hold for the image of allthe conjugate points of order , and only holds for the minimizing conjugatepoints.Finally, recall the construction in [ GS ], where the authors build a riemann-ian surface whose cut locus is not triangulable. Their example shows that theset of points with a conjugate minimizing geodesic can have infinite H n − mea-sure. A similar construction replacing the circle in their construction with a3d ball shows that the set of points with a minimizing geodesic conjugate oforder can have infinite H n − measure. .2. CONJUGATE POINTS IN A BALANCED SPLIT LOCUS 19 Our structure the-orem generalizes a standard result that has been proven several times by math-ematicians from different fields (see for example [ BL ] , [ H87 ], [ MM ] and[ IT98 ]):A cut locus in a Riemannian manifold is the union of a smooth ( n − -dimensional manifold C and a set of zero ( n − -dimensionalHausdorff measure (actually, a set of Hausdorff dimension at most n − ). The set C consists of cleave points, which are joined to theorigin or initial submanifold by exactly two minimizing geodesics,both of which are non-conjugate.This result follows from 3.1.2, since the union of edge, degenerate cleave,and crossing points is a set of Hausdorff dimension at most n − .The statement quoted above follows from lemmas 3.2.3, 3.3.2 and 3.3.3only. Theorem 3.1.1 is not necessary if a description is needed only up tocodimension . The proof of the three lemmas is simple and has many featuresin common with earlier results on the cut locus.In a previous paper, A. C. Mennucci studied the singular set of solutions tothe HJ equations with only C k regularity. Under this hypothesis, the set S \ C may have Hausdorff dimension strictly between n − and n − (see [ Me ]).We work only in a C ∞ setting, and under this stronger condition, the set S \ C has always Haussdorf dimension at most n − .Our result 3.1.2 uses the theory of singularities of semi-concave functionsthat can be found for example in [ AAC ]. Though their result can be appliedto a Finsler manifold, we had to give a new proof that applies to balanced splitloci instead of just the cut locus.
In this section we prove Theorem 3.1.1. Throughout this section, M , r , V and F are as in section 1.2 and S is a balanced split locus as defined in 2.2.8. Definition . A singular point x ∈ V of the map F is an A2 pointif ker ( dF x ) has dimension and is transversal to the tangent to the set ofconjugate vectors. Remark.
Warner shows in [ Wa ] that the set of conjugate points of order is a smooth (open) hypersurface inside V , and that for adequate coordinatefunctions in V and M , the exponential has the following normal form aroundany A2 point, ( x , x , . . . , x m ) −→ ( x , x , . . . , x m ) (3.2.1) Proposition . For any p ∈ M and X ∈ R p , the vector X is not ofthe form dF x ( r ) for any A2 point x . Proof.
The proof is by contradiction. Let p ∈ S be such that R p contains anA2 vector Z = dF c ( r c ) , for c ∈ Q p . By the normal form (3.2.1), we see there isa neighborhood U of c such that no other point in U maps to p . Furthermore, Although there is a mistake in their proof in a neighborhood B of p the image of the conjugate vectors is a hypersurface H such that all points at one side (call it B ) have two preimages of F | U , allpoints at the other side B of H have no preimages, and points at H haveone preimage, whose corresponding vector is A2-conjugate. It follows that Z is isolated in R p .We notice there is a sequence of points p n → p in B with vectors Y n ∈ R p n such that Y n → Y (cid:54) = X . Thus R a does not reduce to Z .The vector Z is tangent to H , so we can find a sequence of points p n ∈ B approaching p such that lim n →∞ v p n ( p ) = Z We can find a subsequence p n k of the p n and vectors X k ∈ R p nk such that X k converges to some X ∞ ∈ R p . By the above, X ∞ is different from Z , but ˆ Z ( X ) < Z ( Z ) (where ˆ Z is the dual form to Z ), so the balanced propertyis violated. (cid:3) The following is the analogous to Theorem 3.1.1 for conjugate points oforder . Proposition ) . There is a set N ⊂ S of Hausdorff dimension n − such that for all p ∈ S \ N , R p does not containconjugate vectors. Proof.
The proof is identical to the proof of lemma 2 in [
IT98 ] for a cut locus,but we include it here for completeness. First of all, at the set of conjugatevectors of order k ≥ we can apply directly the Morse-Sard-Federer theorem(see [ F ]) to show that the image of the set of conjugate cut vectors of order k ≥ has Hausdorff dimension at most n − .Let Q be the set of conjugate vectors of order (recall it is a smoothhypersurface in V ). Let G be the set of conjugate vectors such that the kernelof dF is tangent to the conjugate locus. Apply the Morse-Sard-Federer theoremagain to the map F | Q to show that the image of G has Hausdorff dimension atmost n − . Finally, the previous result takes cares of the A points. (cid:3) We now turn to the main result of this paper: we state and prove Theorem3.2.4 which has 3.1.1 as a direct consequence.
Theorem . Let M , V , F and r be as in section 1.2, and let S bea balanced split locus (2.2.8) . The set of conjugate points of order in V decomposes as the union of two subsets Q and Q such that: • No point in Q maps under dF to a vector in any of the R a (in setnotation: dF ( Q ) ∩ ( ∪ R p ) = ∅ ) • The image under F of Q has Hausdorff dimension at most n − . Proof.
Let z be a conjugate point of order and take special coordinates at U z near z . In the special coordinates near z (see 1.2.2), F is written:(3.2.2) F ( x , . . . , x n ) = ( x , . . . , x n − , F n − z ( x ) , F nz ( x )) for some functions F n − z and F nz , and x = ( x , . . . , x n ) in a neighborhood U z of z with F (0 , . . . ,
0) = (0 , . . . , . .2. CONJUGATE POINTS IN A BALANCED SPLIT LOCUS 21 The Jacobian of F is: J F = . . . ∗ ∗ ... . . . ... ... ... . . . ∗ ∗ . . . ∂F n − z ∂x n − ∂F nz ∂x n − . . . ∂F n − z ∂x n ∂F nz ∂x n A point x is of order if and only if the × submatrix for the x n − and x n coordinates (and the corresponding coordinates in F ( U z ) : y n − and y n ) vanish:(3.2.3) (cid:34) ∂F n − z ∂x n − ∂F nz ∂x n − ∂F n − z ∂x n ∂F nz ∂x n (cid:35) = 0 Near a point of order , we write: F n − z ( x ) = x x n − + q ( x n − , x n ) + T n − ( x ) F nz ( x ) = x x n + r ( x n − , x n ) + T n ( x ) where q ( x n − , x n ) and r ( x n − , x n ) are the quadratic terms in x n − and x n in aTaylor expansion, and T consists of terms of order ≥ in x n − and x n , andterms of order ≥ with at least one x i , i ≤ n − .The nature of the polynomials q and r in the special coordinates at z willdetermine whether z is in Q or in Q . We have the following possibilities:(1) either q or r is a sum of squares of homogeneous linear functions in x n − and x n (possibly with a global minus sign).(2) both q and r are products of distinct linear functionals (equivalently, theyare difference of squares). Later on, we will split this class further intothree types: 2a, 2b and 2c.(3) one of q and r is zero, the other is not.(4) both q and r are zero.We set Q to be the points of type 1 and 2c, and Q to be the points oftype 2a, 3 and 4. Points of type 2b do not appear under the hypothesis of thistheorem.Type 1. The proof is similar to Proposition 3.2.2. Assume z = (0 , . . . , is oftype 1. If, say, q is a sum of squares, then in the set { x = a, x = · · · = x n − =0 } ∩ F ( U z ) , x n − will reach a minimum value that will be greater than − Ca for some C > . We learn there is a sequence p k = ( t k , , . . . , x n − , − ( C +1)( t k ) , , for t k (cid:37) , approaching (0 , . . . , with incoming speed (1 , , . . . , and staying in the interior of the complement of F ( U z ) for k large enough. Pickup any vectors V k ∈ R p k converging to some V (passing to a subsequence ifnecessary). Then V is different from (1 , , . . . , ∈ R , and (cid:98) V ((1 , . . . , < (cid:92) (1 , . . . ,
0) ((1 , . . . , violating the balanced condition.
Type 2 and 3. If a point is of type 2 or 3, we can assume q (cid:54) = 0 . Beforewe proceed, we change coordinates to simplify the expression of F further.Consider a linear change of coordinates near x that mixes only the x n − and x n coordinates. (cid:18) x (cid:48) n − x (cid:48) n (cid:19) = A · (cid:18) x n − x n (cid:19) followed by the linear change of coordinates near p that mixes only the y n − and y n coordinates with the inverse of the matrix above: (cid:18) y (cid:48) n − y (cid:48) n (cid:19) = A − · (cid:18) y n − y n (cid:19) Straightforward but tedious calculations show that there is a matrix A suchthat the map F has the following expression in the coordinates above: F ( x , . . . , x n ) = ( x , . . . , x n − , x x n − + ( x n − − x n ) , x x n + r ( x n − , x n )) + T In other words, we can assume q ( x n − , x n ) = ( x n − − x n ) .Fix small values for all x i for ≤ i ≤ n − . At the origin, J F is a diagonalmatrix with zeros in the positions ( n − , n − and ( n, n ) . We recall that z isconjugate of order iff the submatrix (3.2.3) vanishes. This submatrix is thesum of (cid:20) x + 2 x n − r x n − − x n x + r x n (cid:21) (3.2.4)and some terms that either have as a factor one of the x i for ≤ i ≤ n − , orare quadratic in x n − and x n .We want to show that, near points of type 3 and some points of type 2, allconjugate points of order are contained in a submanifold of codimension .The claim will follow if we show that the gradients of the four entries span a -dimensional space at points in U . For convenience, write r ( x n − , x n ) = αx n − + βx n − x n + γx n . It is sufficient that the matrix with the partial derivatives withrespect to x i for i ∈ { , n − , n } of the four entries have rank : A = −
20 2 α β β γ The claim holds if all x i are small, for i (cid:54)∈ { , n − , n } , unless α = 0 and β = 2 .This covers points of type . We say a point of type 2 has type 2a if the rankof the above matrix is . Otherwise, the polynomial r looks: r ( x n − , x n ) = 2 x n − x n + γx n = 2 x n ( x n − + γ x n ) We say a point of type 2 has type 2b if r has the above form and − < γ < .We will show that there are integral curves of r arbitrarily close to the onethrough z without conjugate points near z , which contradicts property 3 ofexponential maps in Proposition 1.2.5.Take a ray t → ζ x n ( t ) passing through a point (0 , . . . , , x n ) . The determi-nant of 3.2.3 along the ray is: .3. STRUCTURE UP TO CODIMENSION 3 23 d ( t ) = ∂F n − z ∂x n − ( ζ ( t )) ∂F nz ∂x n ( ζ ( t )) − ∂F nz ∂x n − ( ζ ( t )) ∂F n − z ∂x n ( ζ ( t ))= t + t (4 x n − + 2 γx n ) + (4 x n − + 4 γx n − x n + 4 x n ) + R ( x n , t )= ( t + 2 x n − + γx n ) + (4 − γ ) x n + R ( x n , t ) ≥ c ( t + x n ) + R ( x n , t ) for a remainder R of order . Thus there is a δ > such that for any x n (cid:54) = 0 and | t | < δ , | x n | < δ , ζ x n ( t ) is not a conjugate point.We have already dealt with points of type 3, 2a and 2b. Now we turn tothe rest of points of type 2 (type 2c). We have either γ ≥ or γ ≤ − . Wenotice that x n − − x n ≤ iff | x n − | ≤ | x n | , but whenever | x n − | ≤ | x n | , thesign of r ( x n − , x n ) is the sign of γ . Thus the second order part of F maps U into the complement of points with negative second coordinate and whosethird coordinate has the opposite sign of γ .A similar argument as the one for type 1 points yields a contradiction withthe balanced condition. If, for example, γ ≥ , none of the following points x k = ( t k , − ( C + 1)( t k ) , − ( C + 1)( t k ) , .. , ) is in F ( U ) , for t k → . But then we can carry a vector other than (1 , , . . . , as we approach F ( x ) .Type 4. Let z be a conjugate point of order . We show that the image of thepoints of type 4 inside U z has Hausdorff dimension at most n − . U z is anopen set around an arbitrary point z of order , and thus the result follows.First, we find that for any point x of type 4, we have d x F ( v(cid:93)w ) = 0 for all v, w ∈ ker d x F , making the computation in the special coordinates at x ∈ U z (see section 1.2.1 for the definition of d F ).Then we switch to the special coordinates around z . In these coordinates,the kernel of dF at x is generated by ∂∂x n − and ∂∂x n . Thus ∂ F n − z ∂x i x j = 0 for i, j ≥ n − at any point x ∈ U z of type 4.The set of conjugate points of order is contained in the set H = { ∂F n − z ∂x n − ( x ) =0 } . This set is a smooth hypersurface: the second property in 1.2.5 impliesthat ∂ F n − z ∂x x n − (cid:54) = 0 at points of H . At every conjugate point of type 4, the kernelof dF is contained in the tangent to H . Thus conjugate points of type 4 areconjugate points of the restriction of F to H . The Morse-Sard-Federer theoremapplies, and the image of the set of points of type 4 has Hausdorff dimension n − . (cid:3) Proof of Theorem 3.1.1.
Follows immediately from the above, setting N = F ( Q ) . (cid:3) This section contains the proof of 3.1.2, splitted into several lemmas. All ofthem are known for cut loci in riemannian manifolds, but we repeat the proofso that it applies to balanced split loci in Finsler manifolds.
Definition . We say p ∈ S is a cleave point iff R p has two elements X and X , with ( p, X ) = ( F ( y ) , dF y ( r y )) and ( p, X ) = ( F ( y ) , dF y ( r y )) ,and both dF y and dF y are non-singular. In other words, p ∈ S is a cleave point iff R p consists of two non-conjugatevectors. Proposition . C is a ( n − -dimensional manifold. Proof.
Let p = F ( y ) = F ( y ) be a cleave point, with R p = { dF y ( r ) , dF y ( r ) } .We can find a small neighborhood U of p so that the following conditions aresatisfied:(1) U is the diffeomorphic image of neighborhoods U and U of the points y and y . Thus, the two smooth vector fields X q = dF | U ( r ) and X q = dF | U ( r ) are defined in points q ∈ U .(2) At all points q ∈ U , R q ⊂ { X q , X q } . Other vectors must be images of thevector r at points not in U or U , and if they accumulate near p we couldfind a subsequence converging to a vector that is neither X nor X . Wereduce U if necessary to achieve the property.(3) Let H be an hypersurface in U passing through y and transversal to X ,and let ˜ H = F ( H ) . We define local coordinates p = ( z, t ) in U , where z ∈ ˜ H and t ∈ R are the unique values for which p is obtained by followingthe integral curve of X that starts at x for time t . U is a cube in thesecoordinates.We will show that S is a graph in the coordinates ( z, t ) . Let A i be the set ofpoints q for which R q contains X iq , for i = 1 , . By the hypothesis, S = A ∩ A .Every tangent vector v to S at q ∈ S (in the sense of 1.1.2), satisfies thefollowing property (where ˆ X is the dual covector to a vector X ∈ T M .): ˆ X i ( v ) = max Y ∈ R p ˆ Y ( v ) which in this case amounts to ˆ X ( v ) = ˆ X ( v ) , or v ∈ ker( ˆ X − ˆ X ) We can define in U the smooth distribution D = ker( ˆ X − ˆ X ) . S is aclosed set whose approximate tangent space is contained in D .We first claim that for all z , there is at most one time t such that ( z, t ) is in S . If ( z, t ) is in A , R ( z,t ) contains X and, unless ( z, s ) is contained in A for s in an interval ( t − ε, t ) , we can find a sequence ( z n , t n ) converging to ( z, t ) with t n (cid:37) t and carrying vectors X . The incoming vector is X , but ˜ X ( X ) < ˜ X ( X ) = 1 which contradicts the balanced property. Analogously, if R ( z,t ) contains X there is an interval ( t, t + ε ) such that ( z, s ) is contained in A for all s in theinterval. Otherwise there is a sequence ( z n , t n ) converging to ( z, t ) with t n (cid:38) t and carrying vectors X . The incoming vector is − X , but − X ( − X ) < ˜ X ( − X ) which is again a contradiction. The claim follows easily.We show next that the set of p for which there is a t with ( z, t ) ∈ S isopen and closed in Γ , and thus S is the graph of a function h over Γ . Take ( z, t ) ∈ U ∩ S and choose a cone D ε around D p . We can assume the coneintersects ∂U only in the z boundary. There must be a point in S of the .3. STRUCTURE UP TO CODIMENSION 3 25 form ( z (cid:48) , t (cid:48) ) inside the cone for all z (cid:48) sufficiently close to z : otherwise thereis either a sequence ( z n , t n ) approaching ( z, t ) with t n > h + ( z ) ( h being theupper graph of the cone D ε ) and carrying vectors X or a similar sequencewith t n < h − ( z ) and carrying vectors X . Both options violate the balancedcondition. Closedness follows trivially from the definition of S .Define t = h ( z ) whenever ( z, t ) ∈ S . The tangent to the graph of h isgiven by D at every point, thus S is smooth and indeed an integral maximalsubmanifold of D . (cid:3) Remark.
It follows from the proof above that there cannot be any balancedsplit locus unless D is integrable. This is not strange, as the sister notion ofcut locus does not make sense if D is not integrable.We recall that the orthogonal distribution to a geodesic vector field is par-allel for that vector field, so the distribution is integrable at one point of thegeodesic if and only if it is integrable at any other point. In particular, ifthe vector field leaves a hypersurface orthogonally (which is the case for a cutlocus) the distribution D (which is the difference of the orthogonal distribu-tions to two geodesic vector fields) is integrable. It also follows from 2.1.8 thatthe characteristic vector field in a Hamilton-Jacobi problem has an integrableorthogonal distribution. Remark.
In the next chapter we study whether a balanced split locus isactually a cut locus. The proof of the above lemma showed there is a uniquesheet of cleave points near a given point in a balanced split loci.
Proposition . The set of points p ∈ S where co ( R ∗ p ) has dimension k is ( n − k ) -rectifiable. Proof.
Throughout the proof, let ˆ X be the dual covector to the vector X ∈ T M .Let p n be a sequence of points such that co ( R ∗ p n ) contains a k -dimensionalball of radius greater than δ . Suppose they converge to a point p and v p n ( p ) converges to a vector η .We take a neighborhood U of p and fix product coordinates in π − ( U ) ofthe form U × R n . Then, we extract a subsequence of p n and vectors X n ∈ R p n such that X n converge to a vector X in R p . Outside a ball of radius cδ at ˆ X n , where c is a fixed constant and n >> , there must be vectors in R p n , andwe can extract a subsequence of p n and vectors X n converging to a vector X such that ˆ X is at a distance at least cδ of ˆ X . Iteration of this process yieldsa converging sequence p n and k vectors X n , .., X kn ∈ R p n converging to vectors X , .., X k ∈ R p such that the distance between ˆ X k and the linear span of ˆ X , .. ˆ X k − is at least cδ , so that coV ∗ p contains a k -dimensional ball of radius at least c (cid:48) δ .The balanced property implies that the ˆ X j evaluate to the same value at η , which is also the maximum value of the ˆ Z ( η ) for a vector Z in R p . In other words, the convex hull of the ˆ X j belong to the face of R ∗ p that is exposed by η . If co R ∗ p is k -dimensional, η belongs to (cid:0) co R ∗ p (cid:1) ⊥ = (cid:8) v ∈ T p M : (cid:104) w, v (cid:105) is constant for w ∈ co R ∗ p (cid:9) = (cid:110) v ∈ T p M : (cid:104) ˆ X, v (cid:105) is constant for X ∈ R p (cid:111) which is a n − k dimensional subspace.Let Σ kδ be the set of points p ∈ S for which co R ∗ p is k -dimensional andcontains a k -dimensional ball of radius greater than or equal to δ . We haveshown that all tangent directions to Σ kδ at a point p are contained in a n − k dimensional subspace. We can apply theorem 3.1 in [ AAC ] to deduce Σ kδ is n − k rectifiable, so their union for all δ > is rectifiable too. (cid:3) HAPTER 4
Balanced split sets and Hamilton-Jacobi equations
In this chapter we consider the Hamilton-Jacobi boundary value problem(1.4.1) and (1.4.2) in a compact set M .A local classical solution can be computed near ∂M following characteristic curves as in section 1.4.1.A unique viscosity solution is given by the Lax-Oleinik formula (1.4.7).The viscosity solution can be thought of as a way to extend the classicalsolution to the whole M .Recall from section 2.3 that the singular set Sing is a balanced split locus .This notion was inspired originally by the paper [
IT00 ], but is also relatedto the notion of semiconcave functions that is now common in the study ofHamilton-Jacobi equations (see section 2.3). Our goal in this chapter is todetermine whether there is a unique balanced split locus. In the cases whenthis is not true, we also give an interpretation of the multiple balanced splitloci.
In section 4.2 we state our results, give examples, and com-ment on possible extensions. Section 4.3 gathers some of the results from theliterature we will need, and includes a few new lemmas that we use later.Section 4.4 contains our proof that the distance to a balanced split locus and distance to the k -th conjugate point are Lipschitz. Section 4.5 contains theproof of the main theorems, modulo a result that is proved in section 4.6. Thislast section also features detailed descriptions of a balanced split set at each ofthe points in the classification in theorem 3.1.2. For fixed M , H and g satisfying the conditions stated earlier,there is always at least one balanced split locus, namely the singular set of thesolution of (1.4.1) and (1.4.2). In general, there might be more than onebalanced split loci, depending on the topology of M .Our first theorem covers a situation where there is uniqueness. Theorem . Assume M is simply connected and ∂M is connected.Then there is a unique balanced split locus, which is the singular locus ofthe solution of (1.4.1) and (1.4.2) . The next theorem removes the assumption that ∂M is connected, anduniqueness goes away: Theorem . Assume M is simply connected and ∂M has several con-nected components. Let S ⊂ M be a balanced split locus.
278 CHAPTER 4. BALANCED SPLIT SETS AND HAMILTON-JACOBI EQUATIONS
Then S is the singular locus of the solution of (1.4.1) and (1.4.2) withboundary data g + a where the function a is constant at each connected com-ponent of ∂M . The above theorem describes precisely all the balanced split loci in a situa-tion where there is non-uniqueness. If M is not simply connected, the balancedsplit loci are more complicated to describe. We provide a somewhat involvedprocedure using the universal cover of the manifold. However, the final answeris very natural in the light of the examples. Theorem . There exists a bijection between balanced split loci forgiven M , H and g and an open subset of the homology space H ( M, ∂M ) containing zero. In fact, this theorem follows immediately from the next, where we constructsuch bijection:
Theorem . Let (cid:102) M be the universal cover of M , and lift both H and g to (cid:102) M .Let a : [ ∂ (cid:102) M ] → R be an assignment of a constant to each connected compo-nent of ∂ (cid:102) M that is equivariant for the action of the automorphism group of thecovering and such that (cid:101) g ( z ) + a ( z ) satisfies the compatibility condition (1.4.3) in (cid:102) M . Then the singular locus (cid:101) S of the solution (cid:101) u to: (cid:101) H ( x, d (cid:101) u ( x )) = 1 x ∈ (cid:102) M (cid:101) u ( x ) = (cid:101) g ( x ) + a ( z ) x ∈ ∂ (cid:102) M is invariant by the automorphism group of the covering, and its quotient is aset S that is a balanced split locus for M , H and g . Furthermore: (1) The procedure above yields a bijection between balanced split loci for given M , H and g and equivariant compatible functions a : [ ∂ (cid:102) M ] → R . (2) Among the set of equivariant functions a : [ ∂ (cid:102) M ] → R (that can be identi-fied naturally with H ( M, ∂M ) ), those compatible correspond to an opensubset of H ( M, ∂M ) that contains . Remark. The space H ( M, ∂M ) is dual to H n − ( M ) by Lefschetz theorem.The proof of the above theorems rely on the construction from S of a ( n − -dimensional current T S that is shown to be closed and thus represents acohomology class in H n − ( M ) . The proof of the above theorem also showsthat the map sending S to the homology class of T S is a bijection from the setof balanced split loci onto a subset of H n − ( M ) .In order to prove these theorems we will make heavy use of some structureresults for balanced split loci. To begin with, we start with the results fromthe previous chapter, specifically theorem 3.1.2. In the last section, we prove new structure results in order to improve the description of balanced split locinear each of these types of points (see 4.6.2, 4.6.3, 4.6.5, 4.6.6 and 4.6.7).We also study some very important functions for the study of the cut locus.Recall the global coordinates in V given by z ∈ ∂M and t ∈ R . Let λ j ( z ) bethe value of t at which the geodesic s → Φ( s, z ) has its j -th conjugate point(counting multiplicities), or ∞ if there is no such point. Let ρ S : ∂M → R bethe minimum t such that F ( t, z ) ∈ S . .2. STATEMENT OF RESULTS. 29 Lemma . All functions λ j : ∂M → R are Lipschitz continuous. Lemma . The function ρ S : ∂M → R is Lipschitz continuous if S isbalanced. Both results were proven in [
IT00 ] for Riemannian manifolds, and thesecond one was given in [ LN ]. Thus, our results are not new for a cut locus,but the proof is different from the previous ones and may be of interest. Wehave recently known of another proof that ρ and λ are Lipschitz ([ CR ]). Take as M any ring in a euclidean n -space bounded by twoconcentric spheres. Solve the Hamilton-Jacobi equations with H ( x, p ) = | p | and g = 0 . The solution is the distance to the spheres, and the cut locusis the sphere concentric to the other two and equidistant from each of them.However, any sphere concentric to the other two and lying between them is abalanced split set, so there is a one parameter family of split balanced sets.When n > , this situation is a typical application of 4.2.2. In the n = 2 case,there is also only one free parameter, which is in accord with 4.2.4, as the rankof the H homology space of the ring is one.For a more interesting example, we study balanced split sets with respect toa point in a euclidean torus. We take as a model the unit square in the euclideanplane, centered at the origin, with its borders identified. It is equivalent tostudy the distance with respect to a point in this euclidean torus, or the solutionto Hamilton-Jacobi equations with respect to a small disc centered at the originwith the Hamiltonian H ( p ) = | p | and g = 0 . Figure 4.2.1.
Balanced split set in a torusA branch of cleave points (see 3.1.2) must keep constant the difference of thedistances from either sides (recall the proof of prop 3.3.2, or read the beginningof section 4.5). Moving to the covering plane of the torus, we see they mustbe segments of hyperbolas. A balanced split locus is the union of the cleavesegments and a few triple or quadruple points. The set of all balanced splitloci is a -parameter family, as predicted by our theorem 4.2.4. T ∗ M . Definition . The canonical symplectic form in T ∗ M is given incanonical coordinates by (cid:88) i dp i ∧ dq i A submanifold L ⊂ T ∗ M is Lagrangian iff the restriction of the canonicalsymplectic form to L vanishes. Let D be the duality homeomorphism between T M and T ∗ M induced bythe Finsler metric as in definition 1.1.6 ( D is actually a C ∞ diffeomorphismaway from the zero section). We define a map:(4.3.1) ∆( t, z ) = D (Φ t (Γ( z ))) and a subset of T ∗ M :(4.3.2) Θ = ∆( V ) = D ( W ) where Φ t is the geodesic flow in T M . This is a smooth n -submanifold of T ∗ M with boundary.It is a standard fact that, for a smooth function u : M → R , the graphof its differential du is a Lagrangian submanifold of T ∗ M , for the canonicalsymplectic structure in T ∗ M . The subset of Θ corresponding to small t is thegraph of the differential of the solution by characteristics u to the HJ equations.Indeed, all of Θ is a lagrangian submanifold of T ∗ M when Γ comes from anexponential map. As we have seen, this covers HJBVPs as well.We can also carry over the geodesic vector field from T M into T ∗ M (outsidethe zero sections). This vector field in T ∗ M is tangent to Θ . Then, as we followan integral curve γ ( t ) within Θ , the tangent space to Θ describes a curve λ ( t ) in the bundle G of lagrangian subspaces of T ∗ M . It is a standard fact thatthe vector subspace λ ( t ) ⊂ T ∗ γ ( t ) M intersects the vertical subspace of T ∗ γ ( t ) M ina non-trivial subspace for a discrete set of times. We will review this fact, inelementary terms, and prove a lemma that will be important for the proof oflemma 4.2.6.Let η ( t ) be an integral curve of r with x = η (0) a conjugate point of order k . In special coordinates near x , for t close to , the differential of F along η has the form: dF ( η ( t )) = (cid:18) I n − k ∗ ∗ (cid:19) = (cid:18) I n − k
00 0 (cid:19) + t (cid:18) I k (cid:19) + (cid:18) ∗ E ( t ) (cid:19) where | E | < ε , with E = 0 if γ (0) = x .Let w ∈ ker dF ( η ( t )) and v ∈ ker dF ( η ( t )) be unit vectors in the kernelof dF for t < t close to . It follows that both v and w are spanned by thelast k coordinates. We then find: w · dF ( η ( t )) · v − v · dF ( η ( t )) · w = ( t − t ) w · v + w ( E ( t ) − E ( t )) v and it follows (for some t < t ∗ < t ): ( t − t ) w · v < ε | w || v | ( t − t ) .3. PRELIMINARIES 31 and so:(4.3.3) w · v < ε This also shows that the set of t ’s such that dF ( η ( t )) is singular is discrete.Say the point x = ( z , t ) is the j -th conjugate point along the integralcurve of r through x from z , and recall that it is of order k as conjugate point.As z moves towards z , all functions λ j ( z ) , . . . , λ j + k ( z ) converge to t . Let z i be a sequence of points converging to z such that the integral curve through z i meets its k conjugate points near z at M linear subspaces (e.g. λ j ( z i ) = · · · = λ j + k ( z i ) ; λ j + k +1 ( z i ) = · · · = λ j + k ( z i ) ; ...; λ j + k M − +1 ( z i ) = · · · = λ j + k M ( z i ) ).we get the following result (see also lemma 1.1 in [ IT00 ]):
Lemma . The subspaces ker d ( λ j + kl ( z i ) ,z i ) F for l = 1 , . . . , M converge toorthogonal subspaces of ker d ( λ j ( z ) ,z ) F , for the standard inner product in thespecial coordinates at the point ( λ j ( z ) , z ) . Lemma . Let U be an open set in R n , A ⊂ U a proper open set, C + ⊂ R n an open cone, V ⊂ U an arbitrary open set and ε > such that atany point q ∈ ∂A ∩ V , we have ( q + C + ) ∩ ( q + B ε ) ⊂ A .Then ∂A ∩ V is a Lipschitz hypersurface. Moreover, for any vector X ∈ C + ,take coordinates so that X = ∂∂x . Then ∂A ∩ V is a graph S = { ( h ( x , .., x n ) , x ,.., x n ) } for a Lipschitz function h . Proof.
Choose the vector X ∈ C + and coordinate system in the statement.Assume X has norm , so that q + tX ∈ q + B t for small positive t . Take anypoint p ∈ ∂A ∩ V . We claim that all points p + t ∂∂x for < t < ε belong to A ,and all points p + t ∂∂x for − ε < t < belongs to U \ A . Indeed, there cannotbe a point p + t ∂∂x ∈ A for − ε < t < because the set ( p + t ∂∂x ) + ( C + ∩ B ε ) would contain an open neighborhood of p , which contains points not in A . Inparticular, there is at most one point of ∂A ∩ V in each line with directionvector ∂∂x .Take two points q , q ∈ R n − sufficiently close and consider the lines L = { ( t, q ) , t ∈ R } and L = { ( t, q ) , t ∈ R } . Assume there is a t such that ( t , q ) belongs to ∂A . If there is no point of ∂A in L then either all pointsof L belong to A or they belong to U \ A . Both of these options lead toa contradiction if (( t , q ) + C + ) ∩ (( t , q ) + B ε ) ∩ L (cid:54) = ∅ (this condition isequivalent to K | q − q | < ε for a constant K that depends on C + and thechoice of X ∈ C + and the coordinate system).Thus there is a point ( t , q ) ∈ ∂A . For the constant K above and t ≥ t + K | q − q | , the point ( t, q ) lies in the set ( t , q ) + C + , so we have t < t + K | q − q | The points q and q are arbitrary, and the lemma follows. (cid:3) Remark.
We are working in a paper about some limitations of the techniqueof 3d printing known as fused desposition modeling. The above lemma is usedto prove that all current 3d printers using this technique will print pieces thatare specially fragile in some directions.
Lemma . For fixed M and H , two functions g, g (cid:48) : ∂M → R have thesame characteristic vector field in ∂M iff g (cid:48) can be obtained from g by additionof a constant at each connected component of ∂M . Proof.
It follows from (1.4.5) that g and g (cid:48) have the same characteristic vectorfield at all points if and only dg = dg (cid:48) at all points. (cid:3) For our next definition, observe that given M , H and g , we can define amap ˜ u : V → R by ˜ u ( t, z ) = t + g ( z ) . Definition . We say that a function u : M → R is made fromcharacteristics iff u | ∂M = g and u can be written as u ( p ) = ˜ u ◦ s for a (notnecessarily continuous) section s of F : V → M . Remark. In the paper [ Me ], the same idea is expressed in different terms:all characteristics are used to build a multi-valued solution, and then somecriterion is used to select a one-valued solution. The criterion used there is toselect the characteristic with the minimum value of ˜ u . Lemma . The viscosity solution to (1.4.1) and (1.4.2) is the uniquecontinuous function that is made from characteristics.
Proof.
Let h be a function made from characteristics, and u be the viscositysolution given by formula (1.4.7). Let Sing be the closure of the singular setof u .Take a point z ∈ ∂M . Define: t ∗ z = sup { t ≥ h ( F ( τ, z )) = u ( F ( τ, z )) ∀ ≤ τ < t } Let p = F ( t ∗ z , z ) . Assume for simplicity that h ( p ) = u ( p ) . Claim : t ∗ z < ρ Sing ( z ) implies h is discontinuous at F ( t ∗ z , z ) . Proof of the claim : Assume that t ∗ z < ρ Sing ( z ) and h is continuous at F ( t ∗ z , z ) for some z ∈ ∂M .As t ∗ z < ρ Sing ( z ) ≤ λ ( z ) , there is an open neighborhood O of ( t ∗ z , z ) suchthat F | O is a diffeomorphism onto a neighborhood of p = F ( t ∗ z , z ) .By hypothesis, there is a sequence t n → t ∗ z and p n = F ( t n , z ) such that h ( p n ) (cid:54) = u ( p n ) . As h is built from characteristics using a section s , we have h ( p n ) = ˜ u ( s ( p n )) = ˜ u (( s n , y n )) = s n + g ( y n ) , for ( s n , y n ) (cid:54) = ( t n , z ) .For n big enough, the point ( s n , y n ) does not belong to O , as ( t n , z ) isthe only preimage of p n in O . As h ( p n ) → h ( p ) , and ∂M is compact, wededuce the s n are bounded. We can take a subsequence of ( s n , y n ) convergingto ( s ∞ , y ∞ ) (cid:54)∈ O . So we have p = F ( t ∗ z , z ) = F ( s ∞ , y ∞ ) . If p (cid:54)∈ Sing , we deducethat lim n →∞ h ( p n ) = ˜ u ( s ∞ , y ∞ ) > h ( p ) = u ( p ) = ˜ u ( t ∗ z , z ) , so h is discontinuousat p .Using the claim, we conclude the proof: if h is continuous, then ρ Sing ( z ) ≤ t ∗ z for all z ∈ ∂M , and u = h , as any point in M can be expressed as F ( t, z ) forsome z , and some t ≤ ρ Sing ( z ) . (cid:3) .4. ρ S IS LIPSCHITZ 33
We will need later the following version of the same principle:
Lemma . Let S be a split locus, and h be the function associated to S as in definition 2.2.5. If ρ S is continuous, and h can be extended to M so thatit is continuous except for a set of null H n − measure, then S = Sing . Proof.
Define Y = { z ∈ ∂M : h ( F ( t, z )) (cid:54) = u ( F ( t, z )) for some t ∈ [0 , ρ Sing ( z )) } By the claim in the previous lemma, Y is contained in: Y = { z ∈ ∂M : h discontinuous at F ( t, z ) for some t ∈ [0 , ρ Sing ( z )) } Let A = A ( Sing ) be the set in definition 2.2.1. The map F restricts to adiffeomorphism from A onto M \ Sing . The set Y can be expressed as: Y = π ◦ ( F | A ) − ( { p ∈ M \ Sing : h discontinuous at p } )) and thus by the hypothesis has null H n − measure. Therefore, ∂M \ Y is densein ∂M .We claim now that S ⊂ Sing . To see this, let p ∈ S \ Sing . Then p = F ( t ∗ , z ∗ ) for a unique ( t ∗ , z ∗ ) ∈ A . It follows ρ S ( z ∗ ) ≤ t ∗ < ρ Sing ( z ∗ ) . As ρ S is continuous, ρ S ( z ) < ρ Sing ( z ) holds for all z in a neighborhood of z ∗ in ∂M and, in particular, for some z ∈ ∂M \ Y . This is a contradiction because, for ρ S ( z ) < t < ρ Sing ( z ) , h ( F ( t, z )) = ˜ u ( t (cid:48) , z (cid:48) ) for ( t (cid:48) , z (cid:48) ) (cid:54) = ( t, z ) , and t < ρ Sing ( z ) implies h ( F ( t, z )) = ˜ u ( t (cid:48) , z (cid:48) ) > ˜ u ( t, z ) = u ( F ( t, z )) , forcing z ∈ Y .We deduce S = Sing using lemma 2.2.7 and the fact that
Sing is a splitlocus. (cid:3) ρ S is Lipschitz In this section we study the functions ρ S and λ j defined earlier. The factthat ρ S is Lipschitz will be of great importance later. The definitions andthe general approach in this section follow [ IT00 ], but our proofs are shorter,provide no precise quantitative bounds, use no constructions from Riemannianor Finsler geometry, and work for Finsler manifolds, thus providing a new andshorter proof for the main result in [ LN ]. The proof that λ j are Lipschitzfunctions was new for Finsler manifolds when we published the first versionof the preprint of this paper. Since then, the paper [ CR ] has appeared whichshows that λ is actually semi-concave. Proof of 4.2.5.
It is immediate to see that the functions λ j are continuous,since this is property (R3) of Warner (see [ Wa , pp. 577-578 and Theorem 4.5]).Near a conjugate point x of order k , we can take special coordinates as insection 1.2.2: F ( x , . . . , x n ) = ( x , . . . , x n − k , F n − k +1 , . . . , F n ) Conjugate points near x are the solutions of d ( x , . . . , x n ) = det ( dF ) = (cid:88) σ ( − σ ∂F σ ( n − k +1) ∂x n − k +1 . . . ∂F σ ( n ) ∂x n = 0 From the properties of the special coordinates, we deduce that:(4.4.1) D α d (0) = 0 ∀| α | < k and ∂ k ∂x k d = 1 We can use the preparation theorem of Malgrange (see [ GG ]) to find realvalued functions q and l i in an open neighborhood U of x such that d ( x ) (cid:54) = 0 and: q ( x , . . . , x n ) d ( x , . . . , x n ) = x k + x k − l ( x , . . . , x n ) + · · · + l k ( x , . . . , x n ) and we deduce from (4.4.1) that(4.4.2) D α l i (0) = 0 ∀| α | < i which implies(4.4.3) | l i ( x , . . . , x n ) | < ¯ C max {| x | , . . . , | x n |} i At any conjugate point ( x , . . . , x n ) , we have q ( x ) = 0 , so: − x k = x k − l ( x , . . . , x n ) + · · · + l k ( x , . . . , x n ) and therefore | x | k < | x | k − | l | + · · · + | l k | Combining this and (4.4.3), we get an inequality for | x | at any conjugatepoint ( x , . . . , x n ) , where the constant C ultimately depends on bounds for thefirst few derivatives of F :(4.4.4) | x | k < C max {| x | , . . . , | x n |} k − max {| x | , . . . , | x n |} If | x | > max {| x | , . . . , | x n |} , then | x | k < C | x | k − max {| x | , . . . , | x n |} . Ifthe opposite holds, then | x | k < C max {| x | , . . . , | x n |} k . So we get: | x | < max { C, } max {| x | , . . . , | x n |} This is the statement that all conjugate points near x lie in a cone of fixedwidth containing the hyperplane x = 0 . Thus all functions λ j to λ j + k areLipschitz at ( x . . . , x n ) with a constant independent of x . (cid:3) Remark. A proof of lemma 4.2.5 in the language of section 4.3.1 seems possible:let Λ( M ) be the bundle of Lagrangian submanifolds of the symplectic linearspaces T ∗ p M and let Σ( M ) be the union of the Maslov cycles within each Λ p ( M ) . Define λ : V → Λ( M ) where λ ( x ) is the tangent to Θ at D (Φ( x )) (recall (4.3.2)). The graphs of the functions λ k are the preimage of the Maslovcycle Σ( M ) . The geodesic vector field (transported to T ∗ M ), is transversalto the Maslov cycle. Showing that the angle (in an arbitrary metric) betweenthis vector field and the Maslov cycle at points of intersection can be boundedfrom below is equivalent to showing that the λ k are Lipschitz. Lemma . For any split locus S and point y ∈ ∂M , there are no con-jugate points in the curve t → exp( ty ) for t < ρ S ( y ) . In other words, ρ S ≤ λ . Proof.
Assume there is x with ρ S ( x ) − ε > λ ( x ) . By [ Wa , 3.4], the map F is not injective in any neighborhood of ( x, λ ( x )) . There are points F ( x n , t n ) of S with x n → x and t n < ρ S ( x ) − ε (otherwise S does not split M ). Takinglimits, we see F ( x, t ) is in S for some t < ρ S ( x ) − ε , which contradicts thedefinition of ρ S ( x ) . (cid:3) .4. ρ S IS LIPSCHITZ 35
From now on and for the rest of the paper, S will always be a balancedsplit locus: Lemma . Let E ⊂ ∂M be an open subset whose closure is compactand has a neighborhood where ρ < λ . Then ρ S is Lipschitz in E . Proof.
The map G ( x ) defined in 1.2.1, written as ( F ( x ) , dF x ( r )) is an em-bedding of V into T M . There is a constant c such that for x, y ∈ V :(4.4.5) | F ( x ) − F ( y ) | + | dF x ( r ) − dF y ( r ) | ≥ c min {| x − y | , } Recall the exponential map is a local diffeomorphism before the first con-jugate point. Points p = F (( z, ρ ( z ))) for z ∈ E have a set R p consisting ofthe vector dF ( z,ρ ( z )) ( r ) , and vectors coming from V \ E . Choose one such point p , and a neighborhood U of p . The above inequality shows that there is aconstant m such that: | dF x ( r ) − dF y ( r ) | ≥ m for x = ( z, ρ ( z )) with z ∈ E and y = ( w, ρ ( w )) ∈ Q p with w ∈ V \ E . By thebalanced condition 2.2.8, any unit vector v tangent to S satisfies (cid:92) dF x ( r )( v ) = (cid:92) dF y ( r )( v ) for some such y and so: (cid:92) dF x ( r )( v ) < − ε Thus for any vector w tangent to E both vectors ( w, dρ − ( w )) and ( w, dρ + ( w )) lie in a cone of fixed amplitude around the kernel of (cid:92) dF x ( r ) (the hyperplanetangent to the indicatrix at x ). Application of lemma 4.3.3 shows that ρ isLipschitz. (cid:3) Lemma . Let z ∈ ∂M be a point such that ρ ( z ) = λ ( z ) . Thenthere is a neighborhood E of z and a constant C such that for all z in E with ρ ( z ) < λ ( z ) , ρ is Lipschitz near z with Lipschitz constant C . Proof.
Let O be a compact neighborhood of ( z , λ ( z )) where special coor-dinates apply. Let x = ( z, ρ ( z )) ∈ O be such that ρ ( z ) < λ ( z ) . In particular, d x F is non-singular. We can apply the previous lemma and find ρ is Lipschitznear z . We just need to estimate the Lipschitz constant uniformly. Vectors in R F ( x ) that are of the form dF y ( r ) for y ∈ V \ O , are separated from dF x ( r ) as in the previous lemma and pose no trouble, but now there might be othervectors dF y ( r ) for y ∈ O .Fix the metric (cid:104)·(cid:105) in O whose matrix in special coordinates is the identity.Any tangent vector to S satisfies (cid:92) dF x ( r )( v ) = (cid:92) dF y ( r )( v ) , for some y ∈ Q F ( x ) .A uniform Lipschitz constant for ρ is found if we bound from below the anglein the metric (cid:104)·(cid:105) between r and d x F − ( v ) for any vector v with this property.This is easy to do if y (cid:54)∈ O , so fix a point y ∈ O with F ( x ) = F ( y ) , and let X = dF x ( r ) , Y = dF y ( r ) and α = (cid:98) X − (cid:98) Y . We need to bound from below theangle between r x and the hyperplane ker F ∗ α .This is equivalent to proving that there is ε > independent of x suchthat: F ∗ x α ( r ) (cid:107) F ∗ x α (cid:107) > ε which is equivalent to: (cid:98) X ( X ) − (cid:98) Y ( X ) < ε (cid:107) F ∗ x α (cid:107) (cid:98) Y ( X ) < − ε (cid:107) F ∗ x α (cid:107) in the norm (cid:107) · (cid:107) associated to (cid:104)·(cid:105) .Notice first that X and Y belong to the indicatrix at F ( x ) = F ( y ) , whichis strictly convex. By this and (4.4.5), we see that for some ε > : (cid:98) Y ( X ) < − ε (cid:107) X − Y (cid:107) < − cε (cid:107) x − y (cid:107) So it is sufficient to show that for some C independent of x :(4.4.6) (cid:107) F ∗ x α (cid:107) < C (cid:107) x − y (cid:107) Using a Taylor expansion of ∂ϕ∂x j in the second entry, we see the form F ∗ x α can be written in coordinates (with implicit summation over repeated indices):(4.4.7) F ∗ x α = (cid:16) ∂ϕ∂x j ( p, X ) − ∂ϕ∂x j ( p, Y ) (cid:17) ∂F j ∂x l = ∂ ϕ∂x i x j ( p, X ) ( X i − Y i ) ∂F j ∂x l + O ( (cid:107) X − Y (cid:107) ) = ∂ ϕ∂x i x j ( p, X ) ( X i − Y i ) ∂F j ∂x l + O ( (cid:107) x − y (cid:107) ) Define the bilinear map g ( p, X ) with coordinates ∂ ϕ∂x i ∂x j ( p, X ) . It is sufficientto prove that for some C independent of x : (cid:107) g i,j ( p, X ) ( X i − Y i ) ∂F j ∂x l (cid:107) ≤ C (cid:107) x − y (cid:107) This is equivalent to showing that for every vector v ∈ T V : (cid:107) g i,j ( p, X ) ( X i − Y i ) ∂F j ∂x l v l (cid:107) = (cid:107) g ( p, X )( X − Y, dF ( v )) (cid:107) ≤ C (cid:107) x − y (cid:107) (cid:107) v (cid:107) We can of course restrict to vectors v of norm . The maximum norm isachieved when dF ( v ) is proportional to X − Y . The map d x F is invertible, sofor the vector v = dF − ( X − Y ) (cid:107) dF − ( X − Y ) (cid:107) , we have: sup (cid:107) v (cid:107) =1 (cid:107) g ( p, X )( X − Y, dF ( v )) (cid:107) = (cid:107) g ( p, X )( X − Y, dF ( v )) (cid:107) Thus by (4.4.7) and the convexity of ϕ we have: (cid:107) F ∗ ( z,ρ ( z )) α (cid:107) < C (cid:107) X − Y (cid:107) (cid:107) dF − ( X − Y ) (cid:107) + O ( (cid:107) x − y (cid:107) ) < C (cid:107) x − y (cid:107) (cid:107) dF − ( X − Y ) (cid:107) + O ( (cid:107) x − y (cid:107) ) for constants C and C , and it is enough to show there is ε independent of x and y such that:(4.4.8) (cid:107) dF − ( X − Y ) (cid:107) > ε Let G ( x ) = d x F ( r ) . We have: X − Y = G ( x ) − G ( y ) = dG x ( x − y ) + O ( (cid:107) x − y (cid:107) ) .4. ρ S IS LIPSCHITZ 37 so in order to prove (4.4.6) it is enough to show the following: (cid:107) dF − dG x ( x − y ) (cid:107) > ε for ε independent of x and y .Assume that ( ρ ( z ) , z ) is conjugate of order k , so that ρ ( z ) = λ ( z ) = · · · = λ k ( z ) . Thanks to Lemma 4.2.5 and reducing to a smaller O , we canassume that a = ( λ ( z ) , z ) to a k = ( λ k ( z ) , z ) all lie within O (some of themmay coincide). Let d i = λ i ( z ) − ρ ( z ) be the distance from x to the a i . At each ofthe a i there is a vector w i ∈ ker d a i F such that all the w i span a k -dimensionalsubspace. Recall from section 4.3.1 that we can choose w i forming an almostorthonormal subset for the above metric, in the sense that (cid:104) w i , w j (cid:105) = δ i,j + ε i,j for ε i,j << .The kernel of d y F is contained in K = (cid:104) ∂∂x n − k +1 , . . . , ∂∂x n (cid:105) for all y ∈ O ,and thus K = (cid:104) w , . . . , w k (cid:105) . Write w i = (cid:80) j ≥ n − k +1 w ji ∂∂x j . Then we have ∂∂x ∂∂w i F ( a ) = z i + R i ( a ) , for z i = (cid:80) w ki ∂∂y k , (cid:107) R i ( a ) (cid:107) < ε and a ∈ O . Wededuce ∂∂w i F ( x ) = ∂∂w i F ( a i ) + d i ( z i + v i ) = d i ( z i + v i ) for (cid:107) v i (cid:107) < ε .By the form of the special coordinates, x − y ∈ K . Let x − y = (cid:80) b i w i .Since | w i | is almost , there is an index i such that | b i | > n (cid:107) x − y (cid:107) . We havethe identity: F ( y ) − F ( x ) = d x F ( y − x ) + O ( (cid:107) x − y (cid:107) ) = (cid:88) b i d i ( z i + v i ) + O ( (cid:107) x − y (cid:107) ) Multiplying the above by ± z j , we deduce d j | b j | = − (cid:80) | b i | d j ( ε i,j + v i z j ) + O ( (cid:107) x − y (cid:107) ) , which leads to(4.4.9) (cid:88) | b i | d i < C (cid:107) x − y (cid:107) At the point x , the image by d x F of the unit ball B x V in T x V is containedin a neighborhood of Im ( d a i F ) of radius d i . We use the identity (cid:107) dF − dG x ( x − y (cid:107) x − y (cid:107) ) (cid:107) − = sup { t : tdG x ( x − y (cid:107) x − y (cid:107) ) ∈ d x F ( B x V ) } We can assume the distance between the vectors dG x ( x − y (cid:107) x − y (cid:107) ) and (cid:80) b i (cid:107) x − y (cid:107) z i issmaller than n . In particular, looking at the i coordinate chosen above, wesee that the vector dG x ( x − y (cid:107) x − y (cid:107) ) needs to be rescaled by a number no bigger than nd i in order to fit within the image of the unit ball. In other words, the supabove is smaller than nd i . (cid:107) dF − dG x ( x − y (cid:107) x − y (cid:107) ) (cid:107) > nd i > | b i | nC (cid:107) x − y (cid:107) > ε (cid:107) x − y (cid:107) for ε = n C > , which is the desired inequality. (cid:3) Proof of Lemma 4.2.6.
We prove that ρ is Lipschitz close to a point z . Let E be a neighborhood of z such that λ has Lipschitz constant L , and ρ hasLipschitz constant K for all z ∈ E such that ρ ( z ) < λ ( z ) . Let z , z ∈ E besuch that ρ ( z ) < ρ ( z ) . If ρ ( z ) = λ ( z ) we can compute | ρ ( z ) − ρ ( z ) | = ρ ( z ) − ρ ( z ) < λ ( z ) − λ ( z ) < L | z − z | where L is a Lipschitz constant L for λ in U .Otherwise take a linear path with unit speed ξ : [0 , t ] → ∂M from z to z and let a be the supremum of all s such that ρ ( ξ ( s )) < λ ( ξ ( s )) . Then | ρ ( z ) − ρ ( z ) | < | ρ ( z ) − ρ ( ξ ( a )) | + | ρ ( ξ ( a )) − ρ ( z ) | The second term can be bound: | ρ ( ξ ( a )) − ρ ( z ) | < Ka If ρ ( z ) ≥ ρ ( ξ ( a )) , we can bound the first term as | ρ ( z ) − ρ ( ξ ( a )) | = ρ ( z ) − ρ ( ξ ( a )) < λ ( z ) − λ ( ξ ( a )) < L | t − a | while if ρ ( z ) < ρ ( ξ ( a )) , we have | ρ ( z ) − ρ ( z ) | < | ρ ( ξ ( a )) − ρ ( z ) | so in all cases, the following holds: | ρ ( z ) − ρ ( z ) | < max { L, K } t < max { L, K }| z − z | (cid:3) Take the function h associated to S as in definition 2.2.5. At a cleave point x there are two geodesics arriving from ∂M ; each one yields a value of h byevaluation of ˜ u . The balanced condition implies that (cid:98) X ( v ) = (cid:98) X ( v ) for thespeed vectors X and X of the characteristics reaching x and any vector v tangent to S . But (cid:98) X is dh , so the difference of the values of h from either sideis constant in every connected component of the cleave locus.We define an ( n − -current T in this way: Fix an orientation O in M . Forevery smooth ( n − differential form φ , restrict it to the set of cleave points C (including degenerate cleave points). In every component C j of C computethe following integrals(4.5.1) (cid:90) C j,i h i φ i = 1 , where C j,i is the component C j with the orientation induced by O and theincoming vector V i , and h i for i = 1 , are the limit values of h from each sideof C j .We define the current T ( φ ) to be the sum:(4.5.2) T ( φ ) = (cid:88) j (cid:90) C j, h φ + (cid:90) C j, h φ = (cid:88) j (cid:90) C j, ( h − h ) φ The function h is bounded and the H n − measure of C is finite (thanks tolemma 4.2.6) so that T is a real flat current that represents integrals of testfunctions against the difference between the values of h from both sides.If T = 0 , we can apply lemma 4.3.7 and find u = h . .5. PROOF OF THE MAIN THEOREMS. 39 We will prove later that the boundary of T as a current is zero. Assumefor the moment that ∂T = 0 . It defines an element of the homology space H n − ( M ) of dimension n − with real coefficients. We can study this spaceusing the long exact sequence of homology with real coefficients for the pair ( M, ∂M ) : → H n ( M ) → H n ( M, ∂M ) → H n − ( ∂M ) → H n − ( M ) → H n − ( M, ∂M ) → . . . (4.5.3) We prove that under the hypothesis of4.2.1, the space H n − ( M ) is zero, and then we deduce that T = 0 .As M is open, H n ( M ) ≈ . As M is simply connected, it is orientable, sowe can apply Lefschetz duality with real coefficients ([ Ha , 3.43]) which implies: H n ( M, ∂M ) ≈ H ( M ) and H n − ( M, ∂M ) ≈ H ( M ) = 0 As ∂M is connected, we deduce H n − ( M ) has rank , and T = ∂P forsome n -dimensional flat current P . The flat top-dimensional current P can berepresented by a density f ∈ L n ( M ) (see [ F , p 376, 4.1.18]):(4.5.4) P ( M ) = (cid:90) M f M, , M ∈ Λ n ( M ) We deduce from (4.5.2) that the restriction of P to any open set disjointwith S is closed, so f is a constant in such open set. It follows that the constantis zero because the boundary of P for a constant non-zero function is a currentsupported on ∂M . Assume now that ∂M has k connected com-ponents Γ i . We look at (4.5.3), and recall the map H n − ( ∂M ) → H n − ( M ) isinduced by inclusion. We know by Poincaré duality that H n − ( ∂M ) is isomor-phic to the linear combinations of the fundamental classes of the connectedcomponents of ∂M with real coefficients. We deduce that H n − ( M ) is gener-ated by the fundamental classes of the connected components of ∂M , and thatit is isomorphic to the quotient of all linear combinations by the subspace ofthose linear combinations with equal coefficients. Let R = (cid:88) a i [Γ i ] be the cycle to which T is homologous (the orientation of Γ i is such that,together with the inwards pointing vector, yields the ambient orientation).If we define a ( x ) = a i , ∀ x ∈ Γ i , solve the HJ equations with boundarydata g − a and compute the current (cid:98) T corresponding to that data, we see that (cid:98) T = T − j (cid:93) R , where j is the retraction j of M onto S that fixes points of S and follows characteristics otherwise. The homology class of (cid:98) T is zero, and wecan prove (cid:98) T = 0 as before. It follows that S is the singular set to the solutionof the Hamilton-Jacobi equations with boundary data g − a . For this result we cannot simply use thesequence (4.5.3). We first give a procedure for obtaining balanced split loci in M other than the cut locus.A function a : [ ∂ (cid:102) M ] → R that assigns a real number to each connectedcomponent of ∂ (cid:102) M is equivariant iff for any automorphism of the cover ϕ thereis a real number c ( ϕ ) such that a ◦ ϕ = a + c ( ϕ ) .A function a : [ ∂ (cid:102) M ] → R is compatible iff (cid:101) g − a satisfies the compatibilitycondition ((1.4.3).An equivariant function a yields a group homomorphism from π ( M, ∂M ) into R in this way:(4.5.5) σ → a ( (cid:101) σ (1)) − a ( (cid:101) σ (0)) where σ : [0 , → M is a path with endpoints in ∂M and (cid:101) σ is any lift to (cid:102) M . The result is independent of the lift because a is equivariant. On the otherhand, choosing an arbitrary component [Γ ] of ∂M and a constant a = a ([Γ]) ,the formula:(4.5.6) [Γ] → a ([Γ ]) + l ( π ◦ ˜ σ ) , for any path ˜ σ with ˜ σ (0) ∈ Γ , σ (1) ∈ Γ assigns an equivariant function a to an element l of Hom ( π ( M, ∂M ) , R ) ∼ H ( M, ∂M ) .Up to addition of a global constant, these two maps are inverse of one an-other, so there is a one-to-one correspondence between elements of H ( M, ∂M ) and equivariant functions a (with a + c identified with a for any constant c ).The compatible equivariant functions up to addition of a global constant canbe identified with an open subset of H ( M, ∂M ) that contains the zero coho-mology class.Let (cid:102) M be the universal cover of M . We can lift the Hamiltonian H toa function (cid:101) H defined on T ∗ (cid:102) M and the function g to a function ˜ g defined on ∂ (cid:102) M . The preimage of a balanced split locus for M , H and g is a balancedsplit locus for (cid:102) M , (cid:101) H and ˜ g that is invariant by the automorphism group of thecover, and conversely, a balanced split locus (cid:101) S in (cid:102) M that is invariant by theautomorphism group of the cover descends to a balanced split locus on M .Any function a that is both equivariant and compatible can be used to solvethe Hamilton-Jacobi problem (cid:101) H ( p, du ( p )) = 1 in (cid:102) M and u ( p ) = (cid:101) g ( p ) − a ( p ) .If π ( M ) is not finite, (cid:102) M will not be compact, but this is not a problem (seeremark 5.5 in page 125 of [ L ]). The singular set is a balanced split locus that isinvariant under the action of π ( M ) and hence it yields a balanced split locusin M . We write S [ a ] for this set. It is not hard to see that the map a → S [ a ] is injective.Conversely, a balanced split locus in M lifts to a balanced split locus (cid:101) S in (cid:102) M . The reader may check that the current T (cid:101) S is the lift of T S , and inparticular it is closed. As in the proof of Theorem 4.2.2, we have H ( (cid:102) M ) = 0 ,and we deduce T (cid:101) S = (cid:88) j a j [Λ j ] + ∂P where Λ j are the connected components of ∂ (cid:102) M . .6. PROOF THAT ∂T = 0 This class is the lift of the class of T ∈ H n − ( M ) and thus it is invariantunder the action of the group of automorphisms of the cover. Equivalently,the map defined in (4.5.5) is a homomorphism. Thus a is equivariant. Similararguments as before show that S = S [ a ] .Thus the map a → S [ a ] is also surjective, which completes the proof thatthere is a bijection between equivariant compatible functions a : [ ∂ (cid:102) M ] → R and balanced split loci. ∂T = 0 It is enough to show that ∂T = 0 at all points of M except for a set ofzero ( n − -dimensional Hausdorff measure. This is clear for points not in S .Due to the structure result 3.1.2, we need to show the same at cleave points(including degenerate ones), edge points and crossing points. Along the proof,we will learn more about the structure of S near those kinds of points.Throughout this section, we assume n = dim ( M ) > . This is only tosimplify notation, but the case n = 2 is covered too. We shall comment on thenecessary changes to cover the case n = 2 , but do not bother with the trivialcase n = 1 . . We now take a closer look at pointsof A ( S ) that are also conjugate points of order . Fortunately, because of 3.1.2we do not need to deal with higher order conjugate points. In a neighborhood O of a point x of order , in the special coordinates of section 1.2.2, we have x = 0 and F looks like: F ( x , x , . . . , x n ) = ( x , x , . . . , F n ( x , . . . , x n )) (4.6.1)Let (cid:101) S be the boundary of A ( S ) , but without the points (0 , z ) for z ∈ ∂M .It follows from 4.2.6 that (cid:101) S is a Lipschitz graph on coordinates given by thevector field r and n − transversal coordinates. It is not hard to see that it isalso a Lipschitz graph x = ˜ t ( x , . . . , x n ) in the above coordinates x i , possiblyafter restricting to a smaller open set.Because of Lemma 4.4.1, we know x is a first conjugate point, so we canassume that O is a coordinate cube (cid:81) ( − ε i , ε i ) , and that F is a diffeomorphismwhen restricted to { x = s } for s < − ε / . Definition . A set O ⊂ V is univocal iff for any p ∈ M and x , x ∈ Q p ∩ O we have ˜ u ( x ) = ˜ u ( x ) . Remark.
The most simple case of univocal set is a set O such that F | O isinjective. Lemma . Let x ∈ V be a conjugate point of order 1. Then x has anunivocal neighborhood. Proof.
Let O and U be neighborhoods of x and F ( x ) where the specialcoordinates (4.6.1) hold; let x i be the coordinates in O and y i be those in U .Choose smaller U ⊂ U and O ⊂ F − ( U ) ∩ O so that we can assume thatif a point x (cid:48) ∈ V \ O maps to a point in U , then for the vector Z = dF x (cid:48) ( r ) we have(4.6.2) ˆ Z ( ∂∂y ) < ˆ X ( ∂∂y ) for any X = dF x ( r ) with x ∈ O and also(4.6.3) ˆ Y ( ∂∂y ) > − k for some k > sufficiently small and all Y = dF y (cid:48) ( r ) for y (cid:48) ∈ O .Take x , x ∈ Q q ∩ O for q ∈ U . The hypothesis x , x ∈ Q q implies q = F ( x ) = F ( x ) , and so x j = x j follows for all j < n . Let us write a j = x j = x j for j < n , s = x n and s = x n . Fix a , . . . , a n − and considerthe set H a = { x ∈ O : x i = a i ; i = 2 , . . . , n − } Its image by F is a subset of a plane in the y i coordinates: L a = { y ∈ U : y i = a i , i = 2 , . . . , n − } Points of O not in H a map to other planes. If n = 2 , we keep the samenotation, but the meaning is that H a = O and L a = V .There is ε > such that for t < − ε/ , the line { x = t } ∩ H a mapsdiffeomorphically to { y = t } ∩ L a .Due to the comments at the beginning of this section, (cid:101) S is given as aLipschitz graph x = ˜ t ( x , . . . , x n ) . The identity a = ˜ t ( a , . . . , a n − , s i ) holdsfor i = 1 , because x , x ∈ Q q . We define a curve σ : [ s , s ] → (cid:101) S by σ ( s ) = (˜ t ( a , . . . , a n − , s ) , a , . . . , a n − , s ) . The image of σ by F stays in S ,describing a closed loop based at q ; we will establish the lemma by examiningthe variation of ˜ u along σ .For i = 1 , , let η i : ( − ε i , a ] → H a given by η i ( t ) = ( t, a , . . . , a n − , s i ) bethe segments parallel to the x direction that end at x i , defined from the firstpoint in the segment that is in O . We can assume that the intersection of O with any line parallel to ∂∂x is connected, and that the intersection of U withany line parallel to ∂∂y is connected too. We can also assume ε i < ε .Let D be the closed subset of H a delimited by the Lipschitz curves η , η and σ , and let E be the closed subset of L a delimited by the image of η and η .We claim D is mapped onto E . First, no point in int ( D ) can map to theimage of the two lines, cause this contradicts either ρ ≤ λ , or the fact that ρ ( a , . . . , a n − , s i ) is the first time that the line parallel to the x directionhits (cid:101) S , for either i = 1 or i = 2 . We deduce D is mapped into E .Now assume G = E \ F ( D ) is nonempty, and contains a point p = ( p , ..., p n ) .If Q p contains a point x ∈ O \ F ( D ) , following the curve t → ( t, x , . . . , x n ) backwards from x = ( x , x , . . . , x n ) , we must hit either a point in the imageof η i | ( − ε ,a ) (which is a contradiction with the fact that both ( t, . . . , x n ) for t < x and ( t, a , . . . , a n − , s i ) for t < a are in A ( S ) ; see definition 2.2.1), orthe point q (which contradicts (4.6.3)). Thus for any point p ∈ G , we have Q p ⊂ V \ O .Now take a point p ∈ ∂G , and pick up a sequence approaching it from within G and contained in a line with speed vector ∂∂y . By the above, the set Q forpoints in this sequence is contained in V \ O . We can take a subsequence .6. PROOF THAT ∂T = 0 carrying a convergent sequence of vectors, and thus R p has a vector of theform dF x ∗ ( r ) for x ∗ ∈ D ⊂ O . This violates the balanced condition, becauseof (4.6.2). This implies ∂G = ∅ , thus G = ∅ because E is connected and F ( D ) (cid:54) = ∅ .Finally, we claim there are no vectors coming from V \ O in R p for p ∈ int ( E ) . The argument is as above, but we now approach a point with a vectorfrom V \ O in R p within E = F ( D ) and with speed − ∂∂y . The approachingsequence may be chosen so that it carries a convergent sequence of vectors from F ( D ) , and again (4.6.2) gives a contradiction with the balanced condition.We now compute:(4.6.4) ˜ u ( x ) − ˜ u ( x ) = (cid:90) s s d (˜ u ◦ σ ) ds = (cid:90) σ d ˜ u The curve F ◦ σ runs through points of S . If F ( σ ( s )) is a cleave point,then F ◦ σ is a smooth curve near s . We show that cleave points are the onlycontributors to the above integral. If a point is not cleave, either it is the imageof a conjugate vector, or has more than incoming geodesics. As F ◦ σ mapsinto int ( E ) , all vectors in R F ( σ ( s )) come from O .Let N be the set of s such that σ ( s ) is conjugate. We notice that σ ( s ) is notan A2 point for s ∈ N . This is proposition 3.2.2, and is a standard result forcut loci in Riemannian manifolds. This means that at those points the kernelof dF is contained in the tangent to (cid:101) S . The intersection of (cid:101) S with the plane H a is the image of the curve σ . Thus, for s ∈ N the tangent to the curve λ is the kernel of d σ ( s ) F . If σ is differentiable at a point s we deduce, thanks to4.4.1, that the tangent to the curve λ is the kernel of d σ ( s ) F .We now use a variation of length argument to get a variant of the FinslerGauss lemma. Let c = ( l, w ) be a tangent vector to V ⊂ R × ∂M at the point x = ( t, z ) , and assume d x F ( c ) = 0 . We show that this implies d ˜ u ( c ) = 0 .Let γ s be a variation through geodesics with initial point in z ( s ) ∈ ∂M andthe characteristic vector field at z ( s ) as the initial speed vector, such that ∂∂s z ( s ) = w , and with total length t + sl . By the first variation formula andthe equation for the characteristic vector field at ∂M , the variation of thelength of the curve γ s is ∂ϕ∂v ( p, d x F ( r x )) · d x F ( c ) − ∂ϕ∂v ( p, d z F ( r )) · w = − dg ( w ) ,and by the definition of γ s , it is also l . We deduce l = − dg ( w ) , and thus d ˜ u ( c ) = l + dg ( w ) = 0 .It follows that d σ ( s ) F ( σ (cid:48) ( s )) = 0 at points s ∈ N where σ is differentiable.As σ is Lipschitz, the set of s where it is not differentiable has measure , andwe deduce: (cid:90) N d ˜ u ( σ (cid:48) ) = 0 N is contained in the set of points where d ( F ◦ σ ) vanishes. Thus, by theSard-Federer theorem, the image of N has Hausdorff dimension .Let Σ be the set of points in L a with more than incoming geodesics.From the proof of 3.3.3, we see that the tangent to Σ has dimension andthus Σ has Hausdorff dimension .As F is non-singular at points in [ s , s ] \ N , the set of s in [ s , s ] \ N mapping to a point in Σ ∪ N has measure zero. Altogether, we see that the integral (4.6.4) can be restricted to the set C of s mapping to a cleave point. C is an open set and thus can be expressedas the disjoint union of a countable amount of intervals. Let A be one ofthose intervals. It is mapped by F ◦ σ diffeomorphically onto a smooth curve c of cleave points contained in L a . Points of the form ( t, a , . . . , a n − , s ) for t < ˜ t ( a , . . . , a n − , s ) map through F to a half open ball in E . There mustbe points of D mapping to the other side of c . Because of all the above, c is also the image of other points in [ s , s ] . As c is made of cleave points, itmust be the image of another component of C , which we call B , also mappingdiffeomorphically onto c . Choose a new component A , which is matched toanother component B , different from the above, and so on, till the A i and B i are all the components of C .We can write the integral on B i as an integral on A i (we add a minus sign,because the curve is traversed in opposite directions): (cid:90) A i d ˜ u ( σ (cid:48) ) + (cid:90) B i d ˜ u ( σ (cid:48) ) = (cid:90) A i du l (( F ◦ σ ) (cid:48) ) − du r (( F ◦ σ ) (cid:48) ) where u l and u r are the values of u computed from both sides, evaluated atpoints in U . The balanced condition implies ( F ◦ σ ) (cid:48) ∈ ker( d ˜ u l − d ˜ u r ) , andthus the above integral vanishes. The integral (4.6.4) is absolutely convergentby Lemma 4.2.6, and the proof follows. (cid:3) Remark.
The above proof took some inspiration from [
H87 , 5.2]. The readermay be interested in James Hebda’s tree-like curves . In this sectionwe prove some more results about the structure of a balanced split locus neardegenerate cleave and crossing points. Besides their importance for provingthat ∂T = 0 , we believe they are interesting in their own sake. Lemma . Let p ∈ S be a (possibly degenerate) cleave point, and let Q p = { x , x } .There are disjoint univocal neighborhoods O and O of x and x , and aneighborhood U of p such that for any q ∈ U , Q q is contained in O ∪ O .Furthermore, if we define: A i = { q ∈ U such that Q q ∩ O i (cid:54) = ∅} for i = 1 , , then A ∩ A is the graph of a Lipschitz function, for adequatecoordinates in U . Proof.
The points x and x are at most of first order, so we can take univocalneighborhoods O and O of x and x . By definition of Q p and the compactnessof M , we can achieve the first property, reducing U if necessary.We know (cid:92) d x F ( r ) is different from (cid:92) d x F ( r ) . For fixed arbitrary coordi-nates in U , we can assume that { (cid:92) d x F ( r ) for x ∈ O } can be separated by ahyperplane from { (cid:92) d x F ( r ) for x ∈ O } , after reducing U , O , O if necessary.Therefore, there is a vector Z ∈ T p M and a number δ > such that(4.6.5) (cid:92) d x F ( r )( Z ) < (cid:92) d x (cid:48) F ( r )( Z ) + δ ∀ x ∈ O , x (cid:48) ∈ O .6. PROOF THAT ∂T = 0 for any unit vector Z in a neighborhood G of Z . Let C + = { tZ : t > , Z ∈ G } be a one-sided cone containing Z . We write q + C + for the cone displaced tohave a vertex in q .Choose q ∈ A ∩ A , and Z ∈ G . Let R = { q (cid:48) ∈ U : q (cid:48) = q + tZ, t > } bea ray contained in ( q + C + ) ∩ U . We claim R ⊂ A \ A .For two points q = q + t Z, q = q + t Z ∈ R , we say q < q if and only t < t . If R ∩ A (cid:54) = ∅ , let q be the infimum of all points p > in R ∩ A ,for the above order in R . If q ∈ A (whether q = q or not), we can approach q with a sequence of points q n = F ( x n ) > q carrying vectors d x n F ( r ) with x n ∈ O . The limit point of this sequence is q , and the limit vector is d x F ( r ) for some x ∈ O , but the incoming vector is in − G , which contradicts thebalanced condition by (4.6.5).If q ∈ A \ A , then approaching q with points q < q n = F ( x n ) < q ,we get a new contradiction with the balanced property. The only possibility is R ⊂ A \ A . As the vector Z is arbitrary, we have indeed ( q + C + ) ∩ U ⊂ A \ A .Fix coordinates in U , and let ε = dist ( p, ∂U ) . Let B ε be the ball of radius ε centered at p . By the above, the hypothesis of lemma 4.3.3 are satisfied, for A = A \ A , the cone C + , the number ε , and V = B ε . Thus, we learn fromlemma 4.3.3 that A ∩ A ∩ B ε is the graph of a Lipschitz function along thedirection Z from any hyperplane transversal to Z . (cid:3) The following three lemmas contain more detailed information about thestructure of a balanced split locus near a crossing point. The following is statedfor the case n > , but it holds too if n = 2 , though then L reduces to a singlepoint { a } . Definition . The normal to a subset X ⊂ T ∗ p M is the set of vectors Z in T p M such that M ( Z ) is the same number for all M ∈ X . Lemma . Let p ∈ S be a crossing point. Let B ⊂ T ∗ p M be the affineplane spanned by R ∗ p . Let L be the normal to B , which by hypothesis is a linearspace of dimension n − , and let C be a (double-sided) cone of small amplitudearound L .There are disjoint univocal open sets O , . . . , O N ⊂ V and an open neigh-borhood U of p such that Q q ⊂ ∪ i O i for all q in U .Furthermore, define sets A i as in lemma 4.6.3, and call S = ∪ i,j A i ∩ A j the essential part of S . Define Σ = ∪ i,j,k A i ∩ A j ∩ A k and let C = S \ Σ . (1) At every q ∈ Σ , there is ε > such that Σ ∩ ( q + B ε ) ⊂ q + C . (2) Σ itself is contained in p + C . The next lemma describes the intersection of S with -planes transversalto L . Lemma . Let p ∈ S be a crossing point as above. Let P ⊂ T p M bea -plane intersecting C only at the origin, and let P a = P + a be a -planeparallel to P for a ∈ L . (1) If | a | < ε , the intersection of S , the plane P a , and U is a connectedLipschitz tree. (2) The intersection of S , the plane P a , and the annulus of inner radius c · | a | and outer radius ε : A ( c | a | , ε ) = { q ∈ U : c | a | < | q | < ε } is the union of N Lipschitz arcs separating the sets A i . Figure 4.6.2.
Two possible intersections of a plane P a with S Remark.
We cannot say much about what happens inside P a ∩ B ( P, c | a | ) .The segments in P a ∩ A ( c | a | , ε ) must meet together, yielding a connected tree,but this can happen in several different ways (see figure 4.6.2).Finally, we can describe the connected components of C = S \ Σ within U : Lemma . Under the same hypothesis, for every i = 1 , . . . , N there isa coordinate system in U such that: • The set ∂A i is the graph of a Lipschitz function h i , its domain delimitedby two Lipschitz functions f l and f r , for L ∗ ⊂ L : ∂A i = { ( a, t, h i ( t )) , a ∈ L ∗ , f l ( a ) < t < f r ( a ) }• A connected component C of C contained in ∂A i admits the followingexpression, for Lipschitz functions f and f , for L ⊂ L : C = { ( a, t, h i ( t )) , a ∈ L , f ( a ) < t < f ( a ) } Corollary . H n − (Σ) < ∞ . Proof of corollary.
We apply the general area-coarea formula (see [ F ,3.2.22]), with W = Σ , Z = L , and f the projection from U onto L parallel to P , and m = µ = ν = n − , to learn: (cid:90) Σ ap J f d H n − = (cid:90) L H ( f − ( { z } )) d H n − ( z ) = (cid:90) L H (Σ ∩ P a ) d H n − ( a ) ap J f | Σ is bounded from below, so if we can bound H (Σ ∩ P a ) uniformly, weget a bound for H n − (Σ) .The set C ∩ P a ∩ U is a simplicial complex of dimension , and a standardresult in homology theory states that the number of edges minus the numberof vertices is the same as the difference between the homology numbers of the .6. PROOF THAT ∂T = 0 complex: h − h . The graph is connected and simply connected, so this lastnumber is − . The vertices of C ∩ P a ∩ U consist of N vertices of degree lying at ∂U and the interior vertices having degree at least . The handshakinglemma states that the sum of the degrees of the vertices of a graph is twicethe number of edges, so we get the inequality e ≥ N + 3¯ v for the number e ofedges and the number ¯ v of interior vertices. Adding this to the previous equality e − ( N + ¯ v ) = − , we get ¯ v ≤ N − . We have thus bounded ¯ v = H (Σ ∩ P a ) with a bound valid for all a . (cid:3) Proof of 4.6.5.
This lemma can be proven in a way similar to 4.6.3, but wewill take some extra steps to help us with the proof of the other lemmas.First, recall the map ∆ defined in (4.3.1). Each point x in ∆ − ( R ∗ p ) hasa univocal neighborhood O x . Recall R ∗ p consists only of covectors of norm .Let γ be the curve obtained as intersection of B and the covectors of norm .Instead of taking the neighborhoods O x right away, which would be sufficientfor this lemma, we cover R ∗ p with open sets of the form ∆( O x ) ∩ γ .By standard results in topology, we can extract a finite refinement of thecovering of R ∗ p ⊂ γ by the sets ∆( O x ) ∩ γ consisting of disjoint non-emptyintervals I , . . . , I N . Let ˜ I i be the set of points tx for t ∈ (1 − ε , ε ) and x ∈ I i , and choose a linear space M of dimension n − transversal to B .Define the sets of our covering: O i = ∆ − ( ˜ I i + B ( M , ε )) for the ball of radius ε in M ( ε and ε are arbitrary, and small).We can assume that Q q ⊂ ∪ i O i for all q in U by reducing U and the O i further if necessary, hence we only need to prove the two extra properties toconclude the theorem.The approximate tangent to Σ at a point q ∈ Σ ∩ U is contained in thenormal to R ∗ q (recall the definition 1.1.2 of approximate tangent cone, and useproposition 3.3.3, or merely use the balanced property). If R ∗ q is contained ina sufficiently small neighborhood of γ and contains points from at least threedifferent I i , its normal must be close to L . Thus if we chose ε and ε smallenough, the approximate tangent to Σ ∩ U at a point q ∈ Σ is contained in C .If property (1) did not hold for any ε at a point q , we could find a sequenceof points converging to q whose directions from q would remain outside C ,violating the above property.Finally, the second property holds if we replace U by U ∩ B ε , for the number ε that appears when we apply property (1) to p . (cid:3) Proof of 4.6.6.
Just like in 4.6.3, we can assume that each set { (cid:92) d x F ( r ) for x ∈ O i } can be separated from the others by a hyperplane (e.g., a direction Z i ),such that:(4.6.6) (cid:92) d x F ( r )( Z ) < (cid:92) d x (cid:48) F ( r )( Z ) + δ ∀ x ∈ O i , x (cid:48) ∈ O j , i (cid:54) = j for some δ > and any unit vector Z in a neighborhood G i of Z i . Thanksto the care we took in the proof of the previous lemma, we can assume all Z i belong to the plane P in the statement of this lemma: indeed the intervals I i Figure 4.6.3. S near a crossing pointcan be separated by vectors in any plane transversal to L , and the sets ∆( O i ) are contained in neighborhoods of the I i .Define the one-sided cones C + i = { tZ : t > , Z ∈ G i } . The above impliesthat the intersection of each C + i with P is a nontrivial cone in P that consistsof rays from the vertex.By the same arguments in 4.6.3, we can be sure that whenever q ∈ A i , then ( q + C + i ) ∩ U ⊂ A i . This implies that ∂A i is the graph of a Lipschitz functionalong the direction Z i from any hyperplane transversal to Z i . We notice ∂A i is (Lipschitz) transversal to P , so for any a ∈ L , ∂A i ∩ P is a Lipschitz curve.As the cone C is transversal to P , and the tangent to Σ is contained in C , wesee Σ ∩ P a consists of isolated points.Thus S ∩ P a is a Lipschitz graph and Σ ∩ P a is the set of its vertices. If itwere not a tree, there would be a bounded open subset of P a ∩ U \ S withboundary contained in S . An interior point q belongs to some A k . Then thecone q + C + k is contained in A k , but on the other hand its intersection with P a contains a ray that must necessarily intersect S , which is a contradiction.We notice P a ∩ ( p + C + i ) ⊂ A i . This set is a cone in P a (e.g. a circularsector) with vertex at most a distance c | a | from p + a , where c > dependson the amplitude of the different C i .If a = 0 , the N segments departing from p with speeds Z i belong to each A i respectively. Let us assume that the intervals I i appearing in the last proof aremet in the usual order I , I . . . , I N when we run along γ following a particularorientation, and call P i the region delimited by the rays from p with speeds Z i and Z i +1 (read Z instead of Z N +1 ). If there is a point q ∈ P i ∩ A k ∩ B ( ε ) for sufficiently small ε , then ( q + C + k ) ∩ U would intersect either p + C + i or p + C + i +1 , and yield a contradiction if k is not i or i + 1 . Thus P i ⊂ A i ∪ A i +1 . .6. PROOF THAT ∂T = 0 Clearly there must be some point q in P i ∩ A i ∩ A i +1 , to which we can applylemma 4.6.3. A i ∩ A i +1 is a Lipschitz curve near q transversal to Z i (and to Z i +1 ), and it cannot turn back. The curve does not meet Σ , and it cannotintersect the rays from p with speeds Z i and Z i +1 , so it must continue up to p itself. For any q ∈ A i ∩ A i +1 , the cone q + C + i is contained in A i , and the cone q + C + i +1 is contained in A i +1 . This implies there cannot be any other branchof A i ∩ A i +1 inside P i .This is all we need to describe S ∩ P ∩ B ( ε ) : it consists of N Lipschitzsegments starting at p and finishing in P ∩ ∂B ( ε ) . The only multiple point is p . For small positive | a | , we know by condition (2) of the previous lemma that P a ∩ Σ ⊂ C ∩ P a = B ( c | a | ) ∩ P a for some c > . Similarly as above, defineregions P ia ⊂ P a ∩ A ( c | a | , ε ) delimited by the rays from a with directions Z i and Z i +1 , and the boundary of the ring A ( c | a | , ε ) , for a constant c > max( c , c ) .Take c big enough so that for any q ∈ P ia and any k (cid:54) = i, i + 1 , q + C + k ∩ U ∩ P a intersects either p + C + i or p + C + i +1 . The same argument as above shows that A i ∩ A i +1 ⊂ P ia ⊂ A i ∪ A i +1 . We conclude there must be a Lipschitz curve ofpoints of A i ∩ A i +1 , which starts in the inner boundary of A ( c | a | , ε ) , and endsup in the outer boundary. (cid:3) Figure 4.6.4.
A neighborhood of a crossing point (this view isrotated with respect to figure 4.6.3)
Proof of 4.6.7.
First we assume U has a product form U = L ∗ × P ∗ for opendiscs L ∗ ⊂ L and P ∗ = B ( P, ε ) ⊂ P .Recall ∂A i is the graph of a Lipschitz function along the direction Z i fromany hyperplane transversal to Z i . Let H i = L + W be one such hyperplane that contains the subspace L and the vector line W ⊂ P , and construct coordinates L × W × < Z i > . It follows from the previous lemma that ∂A i ∩ P ∗ a is aconnected Lipschitz curve. In these coordinates ∂A i is the graph of a Lipschitzfunction h i . Its domain, for fixed a , is a connected interval, delimited by twofunctions f l : L ∗ → W and f r : L ∗ → W . Condition (1) of lemma 4.6.5 assuresthey are Lipschitz.A connected component C of C is contained in only one A i ∩ A j . We canexpress it in the coordinates defined above for ∂A i . The intersection of C witheach plane P a is either empty or a connected Lipschitz curve. The second partfollows as before. (cid:3) Using lemma 4.6.2, we show without much effort that ∂T vanishes near edge points. Using the structure results from the previoussection, we show also that it vanishes at cleave points (including degenerateones) and crossing points. Proposition . Let p ∈ S be an edge point. Then the boundary of T vanishes near p . Proof.
Let p be an edge point with Q p = { x } . Let O be a univocal neigh-borhood of x . It follows by a contradiction argument that there is an openneighborhood U of p such that Q q ⊂ O for all q ∈ U . Recall the definition of T : T ( φ ) = (cid:88) j (cid:90) C j, ( h − h ) φ For any cleave point q ∈ U with Q q = { x , x } , h i ( q ) = ˜ u ( x i ) . By the above,both x and x are in O . As O is univocal, we see h = h at q . The integrandof T vanishes near p , and thus ∂T = 0 . (cid:3) Proposition . Let p ∈ S be a (possibly degenerate) cleave point.Then ∂T vanishes near p . Proof.
Use the sets U , A and A of lemma 4.6.3.Whenever φ is a n − differential form with support contained in U , wecan compute: T ( φ ) = (cid:90) A ∩ A ( h − h ) φ The components of cleave points inside either A or A do not contribute to theintegral, for the same reasons as in the previous lemma. Recall the definitionof ∂T , for a differential n − form σ : ∂T ( σ ) = T ( dσ ) = (cid:90) A ∩ A ( h − h ) dσ We can apply a version of Stokes theorem that allows for Lipschitz functions.We will provide references for this later: T ( dσ ) = (cid:90) A ∩ A d ( h − h ) σ .6. PROOF THAT ∂T = 0 The balanced condition imposes that for any vector v tangent to A ∩ A at a non-degenerate cleave point q with Q q = { x , x } . ˆ X ( v ) = ˆ X ( v ) for the incoming vectors X i = d x i F ( r ) . Recall that H n − -almost all points arecleave, and dh i is dual to the incoming vector X i , so T ( dσ ) = 0 . (cid:3) Proposition . Let p ∈ S be a crossing point. Then the boundary ofthe current T (defined in 4.5.2) vanishes near p . Proof.
We use lemma 4.6.7 to describe the structure of connected componentsof C near p . Let Σ T , the set of higher order points , be the set of those pointssuch that R ∗ q spans an affine subspace of T ∗ q M of dimension greater than .Take any connected component C of C contained in ∂A i . ∂ C decomposesinto several parts: • The regular boundary, consisting of two parts D and D : D = { ( a , . . . , a n − , f ( a ) , h i ( f ( a ))) , ∀ a ∈ L ∗ such that f l ( a ) < f ( a ) < f ( a ) } D = { ( a , . . . , a n − , f ( a ) , h i ( f ( a ))) , ∀ a ∈ L ∗ such that f ( a ) < f ( a ) < f r ( a ) }• The points of higher order, or ∂ C ∩ Σ T . • The singular boundary, or those points q = ( a , . . . , a n − , f ( a ) , h i ( f ( a ))) where f ( a ) = f ( a ) and R q is contained in an affine plane. • A subset of ∂U .Using a version of Stokes theorem that allows for Lipschitz functions, wesee that (cid:90) C vdσ = (cid:90) C d ( vσ ) − (cid:90) C ( dv ) σ = (cid:90) D vσ − (cid:90) D vσ − (cid:90) C ( dv ) σ for any function v and n − form σ with compact support inside U . Indeed, thelast coordinate of the parametrization of C is given by a Lipschitz function,so we can rewrite the integral as one over a subset of L × W , and only Gauss-Green theorem is needed. We can apply the version in [ F , 4.5.5], whose onlyhypothesis is that the current H n − (cid:98) ∂ C must be representable by integration.Using [ F , 4.5.15] we find that it is indeed, because its support is contained ina rectifiable set. Here we are assuming that D is oriented as the boundary of C , while D is oriented in the opposite way, to match the orientation of D .Notice we have discarded several parts of ∂ C : • A subset of ∂ C inside ∂U does not contribute to the integral because supp ( σ ) ⊂⊂ U . • ∂ C ∩ Σ T does no contribute because it has Hausdorff dimension at most n − . • The singular boundary does not contribute either, because the normal to (cid:101) C at a point of the singular boundary does not exist (see [ F , 4.5.5]).We now prove that ∂T = 0 . For a form σ of dimension n − and compact support inside U : T ( dσ ) = (cid:88) i (cid:90) C i ( h l − h r ) dσ = (cid:88) i (cid:90) C i d ( h l − h r ) σ + (cid:88) i (cid:32)(cid:90) D i, ( h l − h r ) σ − (cid:90) D i, ( h l − h r ) σ (cid:33) where D i, and D i, are the two parts of the regular boundary of C i .The first summand is zero and the remaining terms can be reordered (thesum is absolutely convergent because h is bounded and H n − (Σ) is finite): (cid:88) i (cid:32)(cid:90) D i, ( h l − h r ) σ − (cid:90) D i, ( h l − h r ) σ (cid:33) = (cid:90) Σ \ Σ T (cid:88) ( i,j ) ∈ I ( q ) ( h i,j,l − h i,j,r ) σdq where every point q ∈ Σ \ Σ T has a set I ( q ) consisting of those i and j = 1 , such that q is in the boundary part D j of the component C i . The integrand atpoint q is then: σ (cid:88) ( i,j ) ∈ I ( q ) ( h i,j,l − h i,j,r ) where h i,j,l is the value of ˜ u ( x ) coming from the side l of component C i andboundary part D j .By the structure lemma 3.1.2, we can restrict the integral to crossing points.Let O , . . . , O N be the disjoint univocal sets that appear when we apply 4.6.5to p . For a crossing point q , I ( q ) is in correspondence with the set of indices k such that O k ∩ Q p (cid:54) = ∅ . Indeed, the intersection of S with the plane P a containing q is a Lipschitz tree, and q is a vertex, and belongs to the regularboundary of the components that intersect P a in an edge. The h i,j,l in the sumappear in pairs: one is the value from the left coming from one component C i and the value from the right of another component C i (cid:48) . Each one comes froma different side, so they carry opposite signs, and they cancel. The integrandat q vanishes altogether, so ∂T = 0 . (cid:3) HAPTER 5
A new proof of the Ambrose conjecture for generic -manifolds We give a proof of the
Ambrose conjecture , a global version of the Car-tan local lemma. The proof is given only for a generic class of Riemannianmanifolds of dimension . In 2010, J. Hebda gave a proof in [ H10 ] of the Am-brose conjecture for a (different) generic class of Riemanian manifolds of anydimension. His proof is also much shorter. However, his proof does not extendto arbitrary metrics and we think that our proof might, even though we havebeen unable to do so to this day. Indeed, the proof presented here extends tosome manifolds that are not covered by the result of J. Hebda, so this is trulya different approach.Finally, some of the techniques presented here, such as the conjugate de-scending flow , or the linking curves might be useful for other problems, ascommented in the chapter 6.
Let ( M , g ) and ( M , g ) be two Riemannian mani-folds of the same dimension , with selected points p ∈ M and p ∈ M . Wewill speak about the pointed manifolds ( M , p ) and ( M , p ). Any linear map L : T p M → T p M induces the map ϕ = exp ◦ L ◦ (exp | O ) − , defined inany domain O ⊂ T p M such that e | O is injective (tipically, O is a normalneighborhood of p ).A classical theorem of E. Cartan [ C ] identifies a situation where this map isan isometry. The following is both a reformulation and a slight generalization: Definition . Let ( M , p ) and ( M , p ) be complete Riemannian man-ifolds of the same dimension with base points, and L : T p M → T p M a linearmap.Let γ and γ be the geodesics defined in the interval [0 , , with γ startingat p with initial speed vectors x ∈ T p M and γ starting at p with initialspeed L ( x ) .For any three vectors v , v , v in T p M , define: • R ( v , v , v ) is the vector of T p M obtained by performing parallel trans-port of v , v , v along γ , computing the Riemann curvature tensor at thepoint γ (1) ∈ M acting on those vectors, and then performing paralleltransport backwards into the point p . • R ( v , v , v ) is the vector of T p M obtained by performing parallel trans-port of L ( v ) , L ( v ) , L ( v ) along γ , computing the Riemann curvaturetensor at the point γ (1) ∈ M acting on those vectors, then performingparallel transport backwards into the point p , and finally applying L − toget a vector in T p M .
534 CHAPTER 5. A NEW PROOF OF THE AMBROSE CONJECTURE FOR GENERIC -MANIFOLDS If R ( v , v , v ) = R ( v , v , v ) ∀ v , v , v ∈ T p M for any two geodesics γ and γ as above, we say that the curvature tensors of ( M , p ) and ( M , p ) are L -related. The usual way to express that M and M are L -related is to say that the parallel traslation of curvature along geodesics on M and M coincides. Definition . We say ( M , p ) and ( M , p ) are L -related iff they havethe same dimension and, whenever exp | O is injective for some domain O ⊂ T p M , then the map ϕ = exp ◦ L ◦ (exp | O ) − is an isometric inmersion. Theorem . If the curvature tensors of ( M , p ) and ( M , p ) are L -related, then ( M , p ) and ( M , p ) are L -related. Proof.
Lemma 1.35 of [ CE ]. (cid:3) In 1956 (see [ A ]), W. Ambrose proved a global version of the above theorem,but with stronger hypothesis: if the parallel traslation of curvature along brokengeodesics on M and M coincide, then there is a global isometry ϕ : M → M whose differential at p is L . It is simple to prove that ϕ can be constructedas above. Ambrose himself showed that is enough if the hypothesis holds forbroken geodesics with only one “ elbow ”. The reader can find more details inthe standard reference [ CE ].However, he conjectured that the same hypothesis should suffice, except forthe obvious counterexample of covering spaces: The
Ambrose conjecture states that if thecurvature tensor of ( M , p ) and ( M , p ) are L -related, and if furthermore M and M are simply connected, there is an isometry ψ : M → M such that ψ ◦ exp = exp ◦ L . Definition . A Riemannian covering is a local isometry that is alsoa covering map.
Conjecture
Ambrose Conjecture ) . Let ( M , p ) and ( M , p )be two L -related pointed Riemannian manifolds.Then there is a Riemannian manifold ( M, p ) (the synthesis of ( M , p ) and ( M , p ) ), linear isometries L i : T p M s → T p i M i , for i = 1 , , and Riemanniancoverings π i : M s → M i for i = 1 , such that π i ◦ exp p = exp p i ◦ L i and L ◦ L = L . T p M exp p (cid:15) (cid:15) L (cid:123) (cid:123) L (cid:35) (cid:35) T p M p (cid:15) (cid:15) L (cid:47) (cid:47) T p M p (cid:15) (cid:15) M π (cid:122) (cid:122) π (cid:36) (cid:36) M M .2. NOTATION AND PRELIMINARIES 55 In particular, if M and M are simply-connected, the maps π i are isome-tries, and π ◦ π − : ( M , p ) → ( M , p ) is an isometry (“ the ” isometry) whosetangent at p is L . The main result of this chapter is:
Theorem . The Ambrose Conjecture 5.1.5 holds if the metric of M belongs to the generic class of metrics G M , as defined in 5.3.7. Remark.
The synthesis manifold that we build is a least common Riemanniancovering (see 5.4.9).
Ambrose was able to prove the conjecture if all thedata is analytic. In [ Hi ], in 1959, the conjecture was generalized to paralleltransport for affine connections, and in [ BH ], in 1987, to Cartan connections.Also in 1987, in the paper [ H87 ], James Hebda proved that the conjecture wastrue for surfaces that satisfy a certain regularity hypothesis, that he was ableto prove true in 1994 in [
H94 ]. J.I. Itoh also proved the regularity hypothesisindependently in [ I ]. The latest advance came in 2010, after we had startedour research on the Ambrose conjecture, when James Hebda proved in [ H10 ]that the conjecture holds if M is a heterogeneous manifold . Such manifoldsare generic. M is an arbitrary Riemannian manifold, p a point of M , ( M , p ) and ( M , p ) are two Riemannian manifolds that are L -related.Throughout this chapter, e stands for exp p and e for exp p ◦ L . T p M has the Riemannian manifold structure induced by the scalar product g p . We denote by R ( v ) = | v | the norm in T p M . Using this name will be usefulwhen we use non-linear coordinates in T p M . The radial vector field at v ∈ T p M is the vector ∂ r = ∂∂r = v | v | . Finally, we also define: B R = { x ∈ T p M : | x | < R } B R ( y ) = { x ∈ T p M : | x − y | < R } The proof of the Ambrose conjecture for surfaces given by James Hebda in[
H87 ] relies on properties of
Cut p , the cut locus of M with respect to p . Letus define also the injectivity set O p ⊂ T p M , consisting of those vectors x in T p M such that d (exp p ( tx ) , p ) = t for all (cid:54) t (cid:54) , and let TCut p = ∂O p bethe tangent cut locus . It is a well known fact that TCut p maps onto Cut p by exp p .In our proof, we will need to use a set bigger than the injectivity set, definedas follows. Recall the functions λ k : S p M → R as the parameter t ∗ for which t · x is the k -th conjugate point along t → tx (counting multiplicities. Weproved in 4.2.5 that these functions are Lipschitz. We define V as the setof tangent vectors such that | x | (cid:54) λ ( x/ | x | ), a set with Lipschitz boundary.Indeed, in [ CR ], it was shown that λ is semiconcave. It is well known that O p ⊂ V .Let AC p ( X ) be the space of absolutely continuous curves in the manifold M starting at p , with the topology defined as in [ H87 ]. We will also use the -MANIFOLDS affine developement Dev p : AC p ( M ) → AC ( T p M ) defined in that reference,or in the standard reference [ KN ].Finally, we introduce tree-formed curves, following James Hebda ([ H87 ]).The model for a tree-formed curve u : [0 , → M is an absolutely continuouscurve that factors through a finite topological tree Γ . In other words, u = ¯ u ◦ T for the quotient map T : [0 , → Γ with T (0) = T (1) . The concept is similarto the tree-like paths of the theory of rough paths. J. Hebda uses a moregeneral definition, allowing for an arbitrary quotient map T : [0 , → Γ , andan absolutely continuous curve u such that: (cid:90) t t ϕ ( s )( u (cid:48) ( s )) ds = 0 for any continuous -form ϕ along u ( ϕ ( s ) ∈ T ∗ u ( s ) M ) that factors through Γ( T ( s ) = T ( s ) implies ϕ ( s ) = ϕ ( s )) , and t , t such that T ( t ) = T ( t ) . Thusif Γ = [0 , and T is the identity, the definition is empty, and we will ratheruse the definition saying that a certain curve u is tree-formed with respect toan identication map with T ( t ) = T ( t ) as a rigorous way to say that u | [ t ,t ] isa tree-like path. In the most common case, T (0) = T (1) , and we say the curveis fully tree-formed . Inthis section we give a sketch of the paper [
H87 ]. The reader can find moredetails in that paper.Theorem 5.1.3 shows that ϕ = exp ◦ L ◦ (exp | U p ) − is an isommetricimmersion from U p = M \ Cut p into M . The starting idea is to prove thatwhenever a point in Cut p is reached by two geodesics γ and γ , meaning that e ( γ (cid:48) (0)) = e ( γ (cid:48) (0)) , then e ( γ (cid:48) (0)) = e ( γ (cid:48) (0)) . Then the formula ϕ ( p ) = e ( x ) , for any x ∈ ( O p ∪ TCut p ) ∩ e − ( p ) gives a well-defined map ϕ : M → M that is an isometry at least on U p .As we know from 3.3.2, the cut locus looks specially simple at the cleavepoints , for which there are exactly two minimizing geodesics from p , and bothare non-conjugate. Near a cleave point, the cut locus is a smooth hyper-surface. The rest of the cut locus is more complicated, but we know that H n − (Cut \ Cleave) = 0 and, indeed, that
Cut \ Cleave has Hausdorff dimen-sion at most n − , for a smooth Riemannian manifold.An isometric inmersion from M \ A into a complete manifold, with H n − ( A ) = 0 , can be extended to an isometric inmersion from M . Thus,it only remains to show that, for a cleave point q = e ( x ) = e ( x ) , we have e ( x ) = e ( x ) .The way to do this is to find for each cleave point q as above, a sequence Y j of curves in T p M such that Y j ( t ) ∈ int( O p ) for all j and t , Y j (0) = e ((1 − /j ) x ) , Y j (1) = e ((1 − /j ) x ) , and Y j converges to a curve Y in TCut p (in the metric space AC( M ) of absolutely continuous curves) such that Y (0) = x , Y (1) = x , and e ◦ Y : [0 , → M is fully tree-formed .Consider the curve u = γ x ∗ ( e ◦ Y ) , the concatenation of the geodesic withinitial speed x with the curve e ◦ Y , defined in the interval [0 , l + l ] , where l i is the length of each of these two segments. If Y is absolutely continuous, this isan absolutely continuous curve in T p M , and so admits an affine developement .3. GENERIC EXPONENTIAL MAPS 57 from p . Composing with L we get a curve in T p M , and the inverse affinedevelopement from p yields a curve v in M .J. Hebda proves that the affine developement and the inverse affine devel-opement of a tree-formed curve that factors through Γ is also tree-formed andfactors through Γ . From e ( x ) = e ( x ) we learn u (1 /
2) = u (1) , so that u factors through some Γ with T (1 /
2) = T (1) , and this shows v (1 /
2) = v (1) .We also know that e ◦ ( e | O p ) − is an isometric immersion from U p into M ,and thus the curves γ (1 − /j ) x ∗ ( e ◦ Y j ) map isometrically to ˜ γ (1 − /j ) x ∗ ( e ◦ Y j ) ,where ˜ γ x is the geodesic in M with initial speed L ( x ) . The affine developementconmutes with an isometry, and we learn that v j = e ◦ ( e | O p ) − ◦ u j , so that e ( x ) = lim j →∞ e ( Y j (0)) = lim j →∞ v j (1 /
2) = v (1 / and similarly, v (1) = e ( x ) .The way to find the curves Y j works only in dimension . Let S p M bethe set of unit vectors in T p M parametrized with a coordinate θ , and define ρ : S p M → R as the first cut point along the ray t → tv for t > (and ρ ( θ ) = ∞ if there is no cut point). Given a cleave point q = e ( x ) = e ( x ) ,with x i = ( ρ ( θ i ) , θ i ), then ρ is finite in at least one the two arcs in S p M that join θ and θ , which we write [ θ , θ ] . Then the curve Y ( θ ) = ( ρ ( θ ) , θ )defined in [ θ , θ ] , together with the curves Y j ( θ ) = ((1 − j ) ρ ( θ ) , θ ) , satisfiesthe previous hypothesis.It is important that Y be absolutely continuous, which follows once it isproved that ρ is. This was shown independently in [ H94 ] and [ I ], and latergeneralized to arbitrary dimension in [ IT00 ]. . Indimension higher than , there is no natural choice for such a curve Y . Indeed,a manifold can be built for which this technique does not work, roughly asfollows:Using the techniques in [ We2 ], we can build a three dimensional manifold M whose cut locus with respect to a point does not contain conjugate points(in other words, any minimizing geodesic segment is non-conjugate). Let q = e ( x ) = e ( x ) be a cleave point and Y be a path joining x and x within thetangent cut locus. Assume for simplicity that the path consists only of cleavepoints and isolated non-cleave points (this is generic in a certain sense, as wewill see later).If e ◦ Y is fully tree-formed, then it has one terminal vertex q = e ( x ) .We can approach this vertex with a sequence of cleave points q j = e ( x j ) = e ( x j ) such that x j → x and x j → x . But then x is conjugate and minimiz-ing, contrary to the hypothesis. A generic perturbation of a Riemannian metric greatly simplifies the typesof singularities that can be found on the exponential map ([ We ],[ K ]) or thecut locus with respect to any point ([ B77 ]). In [ We ], A. Weinstein showedthat for a generic metric, the set of conjugate points in the tangent space near -MANIFOLDS a singularity of order k is given by the equations: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x . . . x k x x k +1 . . . x k − ... ... x k x k − . . . x k ( k +1)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 where x , . . . x n are coordinates in T p M , and k ( k + 1) / (cid:54) n . This is called aconical singularity.In [ B77 ], M. Buchner studied the energy functional on curves starting at p and the endpoint fixed at a different point of the manifold, as a familyof functions parametrized by the endpoint. He proved a multitransversalitystatement about this family of functions that we will comment on later, andthen used this information to provide a description of the cut locus of a genericmetric.It is well known that a exponential map only has lagrangian singularities.In [ K ], Fopke Klok showed that the generic singularities of the exponentialmaps are the generic singularities of lagrangian maps. These singularities are,in turn, described by means of the generalized phase functions of the singular-ities. This is the approach more useful to our purposes. A generalized phase function is a map F : U × R k → R such that D q F = (cid:16) ∂F∂q , . . . , ∂F∂q k (cid:17) : U × R k → R k is transverseto { } ∈ R k . We will use a result that relates generalized phase functionsdefined at U × R k and Lagrangian subspaces of T ∗ U : Proposition . If L ⊂ T ∗ U is a Lagrangian submanifold and p ∈ L ,it is locally given as the graph of φ | C : C → T ∗ U , where C = ( D q F ) − (0) and φ ( x, q ) = ( x, D x F ( x, q )) , for some generalized phase function F .Furthermore, we can assume: • k = corank( L, p ) • F (0 ,
0) = 0 • ∈ R k is a critical point of F (0 , · ) : R k → R • ∂ F∂q i ∂q j = 0 for all i and j in , . . . , k Proof.
This is found in section 1 of [ K ], specifically in proposition 1.2.4 andthe comments in page 320 after proposition 1.2.6. (cid:3) Given a germ of generalized phase function F : R n × R k → R , the lagrangianmap is built in this way: D q F is transverse to { } , and we can assume the last k x -coordinates are such that the derivative of D q F in those coordinates is aninvertible matrix. Let us split the x coordinates in ( y, z ) ∈ R n − k × R k . Ourhypothesis is that D qz F is invertible.The implicit equations D q F = 0 defines functions f j : R n − k × R k → R suchthat, locally near , D qz F ( y, f ( y, q ) , q ) = 0 . Definition . A Lagrangian map λ : L → M is the composition ofa Lagrangian immersion i : L → T ∗ M with the projection π : T ∗ M → M (a Lagrangian immersion is an immersion such that the image of sufficientlysmall open sets are Lagrangian submanifolds). .3. GENERIC EXPONENTIAL MAPS 59 Definition . Two Lagrangian maps λ j =: L j → M j , with correspond-ing immersions i j : L → T ∗ M , j = 1 , , are Lagrangian equivalent iff thereare diffeomorphisms σ : L → L , ν : M → M and τ : T ∗ M → T ∗ M suchthat the following diagram conmutes: L σ (cid:15) (cid:15) i (cid:47) (cid:47) T ∗ M τ (cid:15) (cid:15) π (cid:47) (cid:47) M ν (cid:15) (cid:15) L i (cid:47) (cid:47) T ∗ M π (cid:47) (cid:47) M and τ preserves the symplectic structure. Lagrangian equivalence corresponds to equivalence of generalized phasefunctions (this is proposition 1.2.6 in [ K ]). Two generalized phase functionsare equivalent iff we can get one from the other composing three operations:(1) Add a function g ( x ) to F . This has no effect on the functions f j .(2) Pick up a diffeomorphism G : R n → R n , and replace F ( x, q ) by F ( G ( x ) , q ) .If the map G has the special form G ( x ) = G ( y, z ) = ( g ( y ) , h ( z )) , the effectis to replace the map ( y, q ) → ( y, f ( y, q )) by ( y, q ) → ( y, h − ( f ( g ( y ) , q ))) .(3) Pick up a map H : R n × R k → R k such that D q H is invertible, andreplace F ( x, q ) by F ( x, H ( x, q )) . If the map H does not depend on the z variables, the effect is to replace the map ( y, q ) → ( y, f ( y, q )) by ( y, q ) → ( y, f ( y, H ( y, q ))) Using theorem1.4.1 in [ K ], we get the following result: fix a smooth manifold M , a point p ∈ M . For a residual set of metrics in M the exponential map T p M → M isnonsingular except at a set Sing , which is a smooth stratified manifold with thefollowing strata (we describe the different singularities in some detail below): • A stratum of codimension consisting of folds , or lagrangian singularitiesof type A . • A stratum of codimension consisting of cusps , or lagrangian singularitiesof type A . • Strata of codimension consisting of lagrangian singularities of types A (swallowtail), D − (elliptical umbilic) and D +4 (hyperbolic umbilic). • We do not need to worry about the rest, which consists of strata of codi-mension at least . Definition . We define the sets A , A , etc as the set of all pointsof V that have a singularity of type A , A , etc. We also define C as the set ofconjugate (singular) points and N C as the set of non-conjugate (non-singular)points.
Thus,
Sing is a smooth hypersurface of T p M near a conjugate point oforder (including A , A and A points), and is diffeomorphic to the productof a cone in R with a cube near a conjugate point of order (including D ± ).The A points are characterized as those for which the kernel of the differentialof the exponential map is a vector line transversal to the tangent plane to Sing .Furthermore, the image by exp p of each stratum of canonical singularities isalso smooth. There might be strata of high codimension that are not uniform,in the sense that the exponential map at some points in those strata may not -MANIFOLDS have the same type of singularity (in other words, the singularities are non-determinate ). This only happens in some strata of codimension at least , andis not a problem for our arguments.There are also other generic property that interests us: the image of thedifferent strata intersect “transversally”:Take two different points x , x ∈ T p M mapping to the same point of M ,and assume x and x lie in A ∪ A ∪ A ∪ D . Then the points x and x haveneighborhoods U , U such that exp p ( U ∩ C ) and exp p ( U ∩ C ) are transversal(each pair of strata intersect transversally).This follows from proposition 1 in page 215 of [ B77 ], with p = 2 , so that j k H ( α ) is transversal to the orbit in R × [ J k ( n, where the first jet is oftype T and the second one is of type T . Even though that proposition isstated for manifolds of dimension less or equal than , the proof covers ourstatement for any dimension, because we only need transversality to a fewparticular orbits of low codimension.For any singularity in the above list, we can choose coordinates near x and exp p ( x ) so that exp p is expressed by standard formulas. For example, theformulas near an A point are ( x , . . . , x n − , x n ) → ( x ± x x , x , . . . , x n ) .The coordinates that we will use are derived using generalized phase func-tions (see [ K ] for example). We list the generalized phase functions and thecorresponding coordinates for the exponential function that derives from it forthe singularities A , A , A and D ± : • A : F ( x , ˜ x , x , x , . . . , x n ) = x − ˜ x x exp p : ( x , x , x , . . . , x n ) → ( x , x , x , . . . , x n ) • A : F ( x , ˜ x , x , x , . . . , x n ) = x ± x x − (cid:102) x x exp p : ( x , x , x , . . . , x n ) → ( x ± x x , x , x , . . . , x n ) • A : F ( x , ˜ x , x , x , . . . , x n ) = x + x x + x x − (cid:102) x x exp p : ( x , x , x , . . . , x n ) → ( x + x x + x x , x , x , . . . , x n ) • D − : F ( x , x , ˜ x , (cid:102) x , x , . . . , x n ) = x − x x + x ( x + x ) − (cid:102) x x − (cid:102) x x exp p : ( x , x , x , . . . , x n ) → ( x − x + x x , − x x + x x , x , . . . , x n ) • D +4 : F ( x , x , ˜ x , (cid:102) x , x , . . . , x n ) = x + x + x x x − (cid:102) x x − (cid:102) x x exp p : ( x , x , x , x , . . . , x n ) → ( x + x x , x + x x , x , x , . . . , x n ) Definition . The above expression is the canonical form of theexponential map at the singularity. The canonical form is only defined for thesingularities in the above list.We call adapted coordinates any set of coordinates for which the expres-sion of the exponential map is canonical.
Definition . Let U be a neighborhood of adapted coordinates near aconjugate point x . The lousy metric on U is the metric whose matrix inadapted coordinates is the identity. .3. GENERIC EXPONENTIAL MAPS 61 Remark.
We call this metric lousy because it does not have any geomet-ric meaning, and it depends on the particular choice of adapted coordinates.However, it is useful for doing analysis.However, while the adapted coordinates make the exponential map simple,radial geodesics from p are no longer straight lines, and the spheres of constantradius in T p M are also distorted. We do not know of any result that givesan explicit canonical formula for the exponential map and also keeps radialgeodesics in T p M simple. The results of section 5.4.14 suggest that this mightbe possible to some extent, but the classification that might derive from it mustbe finer than the one above. We will find examples showing that the radialvector can be placed in different, non-equivalent positions.For example, near an A point, C is given by x = x . The radial vector r = ( r , . . . , r n ) at (0 , . . . , is transversal to C , and thus must have r (cid:54) = 0 .There are two possibilities: • A point is A ( I ) iff r > . • A point is A ( II ) iff r < .Even though the exponential map has the same expression in both cases (foradequate coordinates), they differ for example in the following:Let x ∈ A ∩ V (a first conjugate point), and let U be a neighborhood of x of adapted coordinates. Then exp p ( V ∩ U ) is a neighborhood of exp p ( x ) iff x is A ( I ) . A proof for this fact will be trivial after section 5.4.3.In fact, the above can be used as a characterization (for points in A ∩ V )that shows that the definition is independent of the adapted coordinates chosen.We remark that in a neighborhood of an A ( I ) point, there are no A ( II ) points, and viceversa.We will get back to this distinction later, and we will also make a similardistinction with D +4 points. Remark.
Sometimes singularities of real functions of type A are subdividedinto A +3 and A − points. A canonical form for an A ± singularity is F ± ( x , ˜ x , x , x , . . . , x n ) = ± x − x x − (cid:101) x x When F ± are generalized phase functions, each subtype gives equivalent sin-gularities. However, in the work of Buchner, the same singularities appear,now as the energy function in a finite dimensional approximation to the spaceof paths with fixed endpoints. In this second context, it is not equivalent if ageodesic is a local minimum, or a maximum, of the energy functional, and itwould make sense to use the distinction between A +3 and A − , rather than thesimilar-but-not-the-same distinction between A ( I ) and A ( II ) .This can also serve as an illustration that the classification of singularitiesof the exponential map by F. Klok and M. Buchner is not equivalent, eventhough the final result is indeed quite similar. In the classification of F. Klok,the A singularities are not divided into the two subclasses A +3 and A − . Definition . We define G M as the set of Riemannian metrics for thesmooth manifold M such that the singular set of exp p is stratifed by singularitiesof types A , A , A and D ± with the codimensions listed above, plus strata of -MANIFOLDS different types with codimension at least , and such that the images of any twostrata intersect transversally as stated above. Thanks to the work of M. Buchner and F. Klok, we know that this set isopen and dense in the set of all Riemannian metrics for M . -manifold5.4.1. Main idea. For any point x ∈ V , the Cartan lemma provides anisometry from a neighborhood of e ( x ) to one of e ( x ) . We cannot use thisfact to get an isommetric immersion into M from a set much bigger than M \ Cut p , but we can try to collect local mappings to build a covering space,as stated in the main theorem 5.1.6.If e has no singularities, we can pull the metric from M onto T p M andthe desired Riemannian coverings are e and e . In the presence of singularities,the idea is to build the synthesis as a quotient of a subset of V that identifiespairs of points with the same image by both e and e .As mentioned above, as well as NC points for e , there are points of T p M with singularities for e of types A , A , A , D +4 and D − . The A points arefurther divided into A ( I ) and A ( II ) points.Our way to deal with a singularity x of type A ( I ) is to show that it is unequivocal , which means that it can play the same role in the quotient as anon-singular point. Definition . We say that an open set O ⊂ T p M is unequivocal iff e ( O ∩ V ) is open, e ( O ∩ V ) is open and there is an isometry ϕ : e ( O ∩ V ) → e ( O ∩ V ) such that ϕ ◦ e | O ∩ V = e | O ∩ V , for any pointed manifold ( M , p ) that is L -related to ( M , p ) .We say x ∈ V is unequivocal if it has a neighbourhood base consisting ofunequivocal sets. Regarding a singularity x of a different type, we will show that there isa linking curve between x and an unequivocal point y of smaller radius.A linking curve between x and y is a curve α : [0 , t ] → T p M such that α (0) = x , α ( t ) = y , e ◦ α is fully tree-formed and Im( α ) is contained in V .It also satisfies some technical restrictions that we will present later. Linkingcurves play the role of the curve Y in the proof of the conjecture for surfacesby J. Hebda: we will see that if there is a linking curve between two points,they are linked . Definition . Two points x, y ∈ T p M are linked ( x (cid:33) y ) iff either x = y , or: e ( x ) = e ( y ) , e ( x ) = e ( y ) and there are neighborhoods U of x and V of y such that ∀ z ∈ U, w ∈ V : e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) for any pointed manifold ( M , p ) that is L -related to ( M , p ) . Remark.
Note that with the above definition of linked , it may not be anequivalence relation (depending on M ), but in fact, the relation is transitiveunder some conditions that hold in our setting: .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 63 Lemma . Let x, y, z, v ∈ T p M . (1) If x (cid:33) y , z (cid:33) y and y in unequivocal, then x (cid:33) z . (2) Assume that M is a generic manifold.If ˜ x (cid:33) x (cid:33) y (cid:33) ˜ y and ˜ x and ˜ y are unequivocal, then ˜ x (cid:33) ˜ y . Proof.
Let ( M , p ) be a pointed manifold that is L -related to ( M , p ) .The hypothesis of the first part imply that: • ∃ U x , V y : ∀ z ∈ U x , w ∈ V y , e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) • ∃ U y , V z : ∀ z ∈ U y , w ∈ V z , e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) • ∃ W y : e ( W y ) is an open neighborhood of e ( x ) = e ( y ) = e ( z ) Then we take open sets A = e − ( e ( W y ∩ V y ∩ U y )) ∩ U x of x and B = e − ( e ( W y ∩ V y ∩ U y )) ∩ V z of z .Suppose there are z ∈ A, w ∈ B such that e ( z ) = e ( w ) . Then e ( z ) ∈ e ( W y ∩ V y ∩ U y ) , so that there is some v ∈ W y ∩ V y ∩ U y such that e ( v ) = e ( z ) = e ( w ) , and it follows that e ( z ) = e ( w ) .The hypothesis for the second part, in turn, imply that e ( x ) = e ( y ) = e (˜ x ) = e (˜ y ) (we call this point q ), and: • ∃ U ˜ x , V x : ∀ z ∈ U ˜ x , w ∈ V z , e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) • ∃ U x , V y : ∀ z ∈ U x , w ∈ V y , e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) • ∃ U y , V ˜ y : ∀ z ∈ U y , w ∈ V ˜ y , e ( z ) = e ( w ) ⇒ e ( z ) = e ( w ) • ∃ W ˜ x : e ( W ˜ x ) is an open neighborhood of q and there is an isometry ϕ ˜ x : e ( W ˜ x ) → e ( W ˜ x ) such that ϕ ◦ e = e on W ˜ x . • ∃ W ˜ y : e ( W ˜ y ) is an open neighborhood of q and there is an isometry ϕ ˜ y : e ( W ˜ y ) → e ( W ˜ y ) such that ϕ ◦ e = e on W ˜ y .The genericity hypothesis also imply that e ( U x ) ∩ e ( V y ) ∩ e ( U ˜ x ) ∩ e ( V x ) ∩ e ( U y ) ∩ e ( V ˜ y ) ∩ e ( W ˜ x ) ∩ e ( W ˜ y ) is a set with non empty interior. Indeed,for a generic metric, the image by e of a neighborhood of a singular point isa stratifed manifold with non-empty interior, and bounded by hypersurfaces.The image of two such neighborhoods are two transversal stratified manifoldswith at least one point in common, and thus they must share some -cell with q in its boundary, so that the intersection of this cell with the image of theopen sets e ( U ˜ x ∩ W ˜ x ) and e ( V ˜ y ∩ W ˜ y ) also has non-empy interior.Any point q in this set can be expressed as q = e ( x ) = e ( x ) = e ( y ) = e ( y ) for x ∈ W ˜ x ∩ U ˜ x , x ∈ U x ∩ V x , y ∈ V y ∩ U y , y ∈ W ˜ y ∩ U ˜ y , soit follows that e ( x ) = e ( x ) = e ( y ) = e ( y ) , but ( x ) = ϕ ˜ x (( x )) and ( y ) = ϕ ˜ y (( y )) so ϕ ˜ x and ϕ ˜ y are isometries that agree on an open set, sothey must agree at least in the connected component of e ( W ˜ x ) ∩ e ( W ˜ y ) thatcontains q .Thus ˜ x and ˜ y are linked, as we can check by defining U = ( e | W ˜ x ) − ( W ˜ x ∩ W ˜ y ) and V = ( e | W ˜ y ) − ( W ˜ x ∩ W ˜ y ) . (cid:3) Corollary . Let M be a generic manifold such that every point islinked to an unequivocal point.Then the linked relation is transitive. Proof.
Let x, y, z ∈ T p M be such that x (cid:33) y (cid:33) z .Then there are unequivocal points ˜ x, ˜ y, ˜ z ∈ T p M such that x (cid:33) ˜ x , y (cid:33) ˜ y and z (cid:33) ˜ z . -MANIFOLDS By the second part of the above proof, we learn that ˜ x (cid:33) ˜ y (cid:33) ˜ z .Then, by the first part of the above proof, we learn that x (cid:33) ˜ y , then that x (cid:33) ˜ z , and finally x (cid:33) z . x (cid:79) (cid:79) (cid:15) (cid:15) (cid:111) (cid:111) (cid:47) (cid:47) y (cid:79) (cid:79) (cid:15) (cid:15) (cid:111) (cid:111) (cid:47) (cid:47) z (cid:79) (cid:79) (cid:15) (cid:15) ˜ x (cid:111) (cid:111) (cid:47) (cid:47) ˜ y (cid:111) (cid:111) (cid:47) (cid:47) ˜ z (cid:3) In the next section, we build the synthesis manifold M s as a quotient spaceof a subset of V ⊂ T p M , identifying linked points. Define I = ( N C ∪A ( I )) ∩ V and J = ( A ∪ A ( II ) ∪ A ∪ D ± ) ∩ V . The following claim is all we needto use the results in the next section: Theorem . Points in I are unequivocal, and any point in J is linkedto a point in I . We will actually prove the theorem in a simpler situation first:
Definition . A manifold M is easy from p iff the exponential mapfrom p only has singularities of type A and A . Theorem . In an easy manifold, points in I = N C ∪ A ( I ) are un-equivocal, and any point in J = A ∪ A ( II ) is linked to a point in I . In this section, A is an arbitrary topological space, X , X are Riemannian manifolds, and e : A → X , e : A → X are arbitrarycontinuous maps. The concepts of unequivocal point and linked pair of points make sense in this slightly more general setting with the obvious changes. Proposition . Let A be a topological space, X , X Riemannian man-ifolds, e : A → X , e : A → X be continuous maps such that (cid:33) is an equivalence relation and the following property holds:For every x ∈ A , there is some y ∈ A such that: • x is linked to y • y in unequivocal.Then there is a Riemannian manifold X (the synthesis of X and X ), acontinuous map e : A → X and local isometries π : X → X and π : X → X ,such that e i = π i ◦ e , for i = 1 , . A e (cid:6) (cid:6) e (cid:24) (cid:24) e (cid:15) (cid:15) X π (cid:126) (cid:126) π (cid:32) (cid:32) X X Proof.
Define X as a quotient by the linked relation: X = A/ (cid:33) .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 65 Let e : A → X be the projection map. We define maps π i : X → X i by π i ([ x ]) = e i ( x ) . Both maps are clearly well defined. • Topology of X : A basis for the topology of X is given by all [ W ] = { [ x ] , x ∈ W } , for an unequivocal open set W . • e is continuous at every point x ∈ A : There is an unequivocal point z ∈ [ x ] , thus ∃ U z , V x : ∀ v ∈ U z , w ∈ V x , e ( v ) = e ( w ) ⇒ e ( v ) = e ( w ) .Let U = [ W ] be a basis open neighborhood of [ x ] : ∃ W z ⊂ U z ∩ e − ( π ( U )) : e ( W z ) is an open neighborhood of e ( z ) andthere is an isometry ϕ z : e ( W z ) → e ( W z ) such that ϕ ◦ e = e on W z .Then O = U x ∩ e − ( e ( W z )) is an open neighborhood of x . We want toshow that O ⊂ e − ( U ) .We first show O ⊂ e − ([ W z ]) : let v ∈ O . There is some w ∈ W z suchthat e ( w ) = e ( v ) and this implies also that e ( w ) = e ( v ) .For the same reason, the sets O and W z also satisfy the necessary propertyto show that v is linked to w , thus [ v ] ∈ [ W z ] .It remains to show that [ W z ] ⊂ [ W ] . We can assume both W and W z areconnected. The unequivocal sets W z and W have associated isometries ϕ z and ϕ , and they agree on e ( U x ) ∩ e ( V z ) ∩ e ( W z ) , a set with non-emptyinterior, so they agree on e ( W z ) . Finally, for any point ˜ z ∈ W z there isanother ˜ x ∈ W such that e (˜ z ) = e (˜ x ) , and ϕ z = ϕ implies e (˜ z ) = e (˜ x ) .So we conclude as before that ˜ z (cid:33) ˜ x . • For i = 1 , , π i | [ W ] is injective for any basis open set [ W ] : WLOG, take i = 1 , and let [ x ] , [ x ] ∈ [ W ] be such that π ([ x ]) = π ([ x ]) . Wecan assume x , x ∈ W . By the property of W , e ( x ) = e ( x ) implies e ( x ) = e ( x ) , and taking U x = V x = W does the rest of the job ofproving that x (cid:33) x . • For i = 1 , , π i is continuous. WLOG, take i = 1 . We show that π | [ W ] is continuous, for a basis set [ W ] : let U be an open subset of π ([ W ]) = e ( W ) . Then ( π | [ W ] ) − ( U ) = [ W ] ∩ π − ( U ) = [ W ∩ e − ( U )] is open,because W ∩ e − ( U ) ⊂ W , and e ( W ∩ e − ( U )) = U , and thus W ∩ e − ( U ) is also unequivocal. • For a basis open set [ W ] , π i ([ W ]) is open by definition. Hence, π i is openfor i = 1 , . Thus, π i | [ W ] is an homeomorphism onto its image. • Hence, π and π are local homeomorphisms. We can use π to give X the structure of a Riemannian manifold, which trivially makes π alocal isometry. For an unequivocal set W , with e | W = ϕ ◦ e | W , then π ◦ ( π | [ W ] ) − = ϕ is an isometry from π ([ W ]) = e ( W ) to π ([ W ]) = e ( W ) , so π is also a local isometry. (cid:3) Let us mention that the synthesis that we constructed satisfies an universalproperty, and thus is unique up to global isometry:
Lemma . Under the same hypothesis of 5.4.8, the synthesis manifold X constructed in the proof satisfies the following universal property:For any Riemannian manifold X (cid:48) , continuous surjective map e (cid:48) : A → X (cid:48) and local isometries π (cid:48) : X (cid:48) → X and π (cid:48) : X (cid:48) → X , such that e i = π (cid:48) i ◦ e (cid:48) ,for i = 1 , , there is a local isometry π : X (cid:48) → X such that π (cid:48) i = π i ◦ π and π ◦ e (cid:48) = e : -MANIFOLDS A e (cid:6) (cid:6) e (cid:24) (cid:24) e (cid:48) (cid:15) (cid:15) X (cid:48) π (cid:48) (cid:125) (cid:125) π (cid:48) (cid:33) (cid:33) X X X (cid:48) π (cid:48) (cid:6) (cid:6) π (cid:48) (cid:24) (cid:24) π (cid:15) (cid:15) X π (cid:125) (cid:125) π (cid:33) (cid:33) X X Proof.
Define π ( q ) = [ x ] for any x ∈ A such that e (cid:48) ( x ) = q . For any other y such that e (cid:48) ( y ) = q , we have e i ( y ) = π (cid:48) i ( q ) = e i ( x ) . We can also take openneighborhoods U x , V y of x and y contained on e (cid:48)− ( A ) , for an open neighbor-hood A of q such that π | A is an homeomorphism. Then, if e ( z ) = e ( w ) for z ∈ U and w ∈ V , it follows from π (cid:48) ( e (cid:48) ( z )) = π (cid:48) ( e (cid:48) ( w )) that e (cid:48) ( z ) = e (cid:48) ( w ) andthus e ( z ) = e ( w ) . It follows that x (cid:33) y and π is well defined.We also check that π i ( π ( q )) = e i ( x ) = π (cid:48) i ( e (cid:48) ( x )) = π (cid:48) i ( q ) . Any q ∈ X (cid:48) has a neighborhood U (cid:48) ⊂ X (cid:48) such that π (cid:48) i | U (cid:48) is an isometry. There is also U ⊂ X such that π i | U is an isometry. Let V (cid:48) = ( π (cid:48) ) − ( π (cid:48) ( U (cid:48) ) ∩ π ( U )) . Then π | V (cid:48) = ( π − i ◦ π (cid:48) i ) | V (cid:48) , and thus π is a local isometry. (cid:3) Remark.
We have not proved that π and π are coverings maps. It wouldbe enough to show that X is complete, but this is not true in such generality,as the following example shows:Let A (cid:40) M be an open subset of a connected Riemannian manifold, andlet i : A → M be the inclussion. Take X = X = M and e = e = i . Then e = i , π = π = id M and X = A satisfy the thesis of the theorem, and e and e are isometries, but not covering maps.We will prove in section 5.4.11 that X is complete when M are M arecomplete Riemannian manifolds with a generic metric, A is V , e is exp p and e is exp p ◦ L . A ( I ) first conjugate points are unequivocal. Con-sider an A ( I ) point x in the manifold ( M , p ) that is L -related to ( M , p ) ,and use adapted coordinates near x = (0 , , , in an arbitrarily small neigh-borhood O : • Define γ ( x , x ) = x . • Let A be the subset of O given by x < γ ( x , x ) . e maps difeomorphi-cally A onto a big subset of e ( O ) . Only the points with x = 0 , x (cid:62) are missing. x is A ( I ) , so ¯ A ⊂ V , and e ( O ∩ V ) is open. • For any ( x , x ) , the pair of points ( x , x , x ) and ( − x , x , x ) map tothe same point by e , the curve t → ( t, t , x ) , t ∈ [ − x , x ] maps to atree-formed curve. This shows that the two points map to the same pointby e as well. The details go exactly like in two dimensions. • Define a map ϕ : e ( O ) → e ( O ) by ϕ ( p ) = e ( a ) , for any a ∈ ¯ A such that p = e ( a ) . By the above, this is unambigous. • The rest of the proof proceeds as in lemma 2.1 in [
H87 ]: for a pair of linkedpoints x = ( x , x , x , . . . , x n ) and ¯ x = ( − x , x , x , . . . , x n ) , we have twodifferent local isometries from a neighborhood of p = e ( x ) = e (¯ x ) into M , given by e ◦ ( e | O i ) − , for neighborhoods O i of x and ¯ x such that e ( O ) = e ( O ) and we need to show that they agree. They both send .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 67 p to the same point, and we only need to check that their differential isthe same. These are linear isometries, and they agree on the hyperplane x = 0 (tangent to the image of ∂A : x = 0 , x (cid:62) ). It is easy to see thatthey both preserve orientation (for example: there is continuous curve oflocal isometries joining them), so they coincide. • We know that ϕ ◦ e ( x ) = e ( x ) , for x ∈ ¯ A . Let y ∈ O \ ¯ A . There is a uniquepoint x in the radial line through y in ∂A . We know ϕ ◦ e ( x ) = e ( x ) ,and the radial segment from x to y map by both ϕ ◦ e and e to ageodesic segment with the same length, starting point and initial vector.We conclude ϕ ◦ e ( y ) = e ( y ) . Remark.
The only place where we used that the point is A ( I ) is when weassumed that A ⊂ V . We now introduce the main ingredient in the con-struction of the linking curves. The idea in the definition of conjugate flow wasused in [
H82 ] to prove lemma 2.2, but the idea for that proof is attributed toan anonymous referee , and we cannot track the origin of the idea any further.Near a conjugate point of order 1, the set C of conjugate points is a smoothhypersurface. Furthermore, we know ker dF does not contain r by Gauss’lemma. Thus we can define a one dimensional distribution D within the set ofpoints of order by the rule:(5.4.1) D = (ker dF ⊕ < r > ) ∩ T C
Definition . A conjugate descending curve (CDC) is a smoothcurve, consisting only of A points, except possibly at the endpoints, and suchthat the speed vector to the curve is in D and has negative scalar product withthe radial vector r . Therefore, the radius is decreasing along a descending flowline of conjugate points.The canonical parametrization of a CDC γ is the one that makes exp p ◦ γ (cid:48) a unit vector. By Gauss lemma, it is also the one that makes dR ( γ (cid:48) ) = 1 . Definition . Let α : [0 , t ] → T p M be a smooth curve, and x ∈ T p M be a point such that exp p ( x ) = exp p ( α ( t )) . A curve β : [0 , t ] → T p M is a retort of α starting at x iff β ( t − t ) (cid:54) = α ( t ) for any t ∈ [0 , t ) , but exp p ( α ( t )) =exp p ( β ( t − t )) for any t ∈ [0 , t ] , and β ( t ) is NC for any t ∈ (0 , t ) . Whenever β is a retort of α , we say that β replies to α . A partial retort of α is a retortof the restriction of α to a subinterval [ t , t ] , for < t < t . We have seen that near an A point x , there are coordinates near x and exp p ( x ) such that exp p reads ( x , x , . . . , x n ) → ( x , x , . . . , x n ) . The A pointsare given by x = 0 , and no other point y (cid:54) = x maps to exp p ( x ) . Thus, there isa neighborhood of any CDC such that any CDC has no retorts. Lemma . Let x be an A point. Then there is a C ∞ CDC α : [0 , t ) → T p M with α (0) = x . The CDC is unique, up to reparametrization. Further-more: • | α (0) | − | α ( t ) | = length(exp p ◦ α ) James Hebda said “I wish to thank the referee for the simple proof of lemma 2.2”. -MANIFOLDS • If β is a non-trivial retort of α , then of course, length(exp p ◦ α ) = length(exp p ◦ β ) , but | β ( t ) | − | β (0) | < length(exp p ◦ β ) . We say that segments of descending conjugate flow are unbeatable . Proof.
Both A and the distribution D are smooth near x , so the first partis standard.We also compute: length(exp p ◦ α ) = (cid:90) | (exp p ◦ α ) (cid:48) | = (cid:90) | d exp p ( α (cid:48) ) | By definition of D , α (cid:48) = ar + v is a linear combination of a multiple of the radialvector and a vector v ∈ ker( d exp p ) . By the Gauss lemma, | d exp p ( α (cid:48) ) | = a .On the other hand, v is tangent to the spheres of constant radius, so: | α (0) | − | α ( t ) | = (cid:90) ddt | α | = (cid:90) a = length(exp p ◦ α ) For a retort β : [0 , t ] → T p M , we also have β (cid:48) = br + v for a function b : [0 , t ] → R and a vector v ( t ) ∈ T β ( t ) ( T p M ) that is always tangent to the spheres ofconstant radius, and v ( t ) is not identically zero because e ◦ β is not a geodesic.However, β ( s ) is non-conjugate, so | d exp p ( β (cid:48) ) | = (cid:113) b + | d exp p ( v ) | > b . Theresult follows. (cid:3) Remark.
We recall that the plan is to build linking curves, whose compositionwith the exponential is tree formed. If a linking curve contains a CDC, it mustalso contain a retort for that CDC. The “unbeatable” property of CDCs isinteresting, because the radius decreases along a CDC and along the retortit never increases as much as it decreased in the first place. This way, ourprospective linking curve will stay within a sphere of finite radius. A points. As we mentionedin section 5.3, the radial vector field, and the spheres of constant radius of T p M ,that have very simple expressions in standard linear coordinates in T p M , aredistorted in canonical coordinates. Thus, the distribution D and the CDCs donot always have the same expression in adapted coordinates. In this section,we see what we can say about these curves near an A point. We will use thename R : T p M → R for the radius function, and r for the radial vector field,and we assume that our conjugate point is a first conjugate point (it lies in ∂V ).In a neighborhood O of special coordinates of an A point, C is given by x = x . At each A point, the kernel is spanned by ∂∂x . At points in C ,we can define a 2D distribution D , spanned by r and ∂∂x . We extend thisdistribution to all of O in the following way: Definition . For any point x ∈ O , there are y ∈ C and t such that x = φ t ( y ) , where φ t is the radial flow, and y and t are unique. Define D ( x ) as ( φ t ) ∗ ( D ( y )) . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 69 Let P be the integral manifold of D through x = (0 , , . The integralcurve C of D through x is contained in P , and C \ { x } consists of two CDCs.We claim that if the point is A ( I ) , the two CDCs descend into x , but if thepoint is A ( II ) , they start at x and flow out of O . P is also obtained byflowing the CDC with the radial vector field.We can assume that r is close to r ( x ) in O . The tangent T x to the sphereof constant radius { y : R ( y ) = R ( x ) } must contain ∂∂x (the kernel of d exp p )if x ∈ C , by Gauss lemma, and we can assume that the angle between T x and ∂∂x is small if x (cid:54)∈ C . A is transversal to D , so { x } = A ∩ P . CDCs have non-zero speed, so weonly need to show that the two CDCs have greater radius than x . Otherwise,for some x ∈ C ∩ P close to x , the curve { y : R ( y ) = R ( x ) } ∩ P is forced tomake a sharp turn and become “vertical” (parallel to r ), as it cannot intersect { y : R ( y ) = R ( x ) } ∩ P (the dashed line in figure 5.4.1 below). But its tangentis close to ∂∂x , which is a contradiction. Figure 5.4.1.
A neighborhood of (0 , . . . , in the plane P , if the curve { y : R ( y ) = R ( x ) } ∩ P did not lie below C ∩ P .If r > ( A ( I ) points), then R ( x ) ≥ R ( x ) for any x ∈ C , while r < ( A ( II ) points), implies R ( x ) ≤ R ( x ) for any x ∈ C , as required.Thus, A ( I ) points are terminal for the conjugate flow, but A ( II ) pointsare not. This is fortunate, because A ( II ) points are not unequivocal and thuswe hope to link them to an unequivocal point. We have just learned that wecan at least start a CDC at those point. A joins. We can continue a CDC as long as it stays within a stratumof A points. As we have seen, a CDC may enter a different singularity. Themost important situation is when the CDC reaches an A point, because thenwe can start a non-trivial retort right after the CDC. We may not be ableto continue the retort for the whole CDC curve, but we will deal with thatproblem later.The set of conjugate points is a graph over the x , x plane: x = α ( x , x ) =3 x . A CDC is written t → ( t, t , x ( t )) , for t ∈ [ t , , finishing at an A point (0 , , x (0)) , but it cannot be continued further. Fortunately, we can start a -MANIFOLDS retort for this segment of CDC right from the A point. The retort for thisCDC is given explicitely by t → ( − t, t , x ( t )) .In figure 5.4.2 below, we can see a CDC (in solid red, coming from right toleft), and reaching the A point, and a retort for this curve (in green). On theright hand side, we can see the image by the exponential of the concatenationof both curves (a tree formed curve: the image of each curve is the same butrun in opposite directions). The picture is within the plane P of section 5.4.5and the blue lines are no more than vertical lines with their respective images.They are not geodesics, but are included to help interpret the picture. Figure 5.4.2. near an A pointThese curves, composed of a segment of CDC plus the corresponding retort,map to a fully tree-formed map that shows that the point ( t, t , x ) is linkedto ( − t, t , x ) . We say that the CDC and the retort given above are joinedwith an A join . In order to build linking curves, it is sim-pler to replace CDCs with curves that are close to CDC curves, but avoidcertain “obstacles”. The following remark helps in that respect:
A curve that is sufficiently C -close to a CDC is also unbeatable . Actually,we can say more: the greater the angle between r x and ker d x e , the more wecan depart from the CDC. Definition . The slack A x at a first order conjugate point x is theabsolute value of the sine of the angle between D x and ker( d x e ) . Remark.
The slack is positive iff the point is A Lemma . For any positive numbers
R > and a > there areconstants c > and ε > depending on M , R and a such that the followingholds:Any curve α : [ t , t ] → T p M of A points such that: (1) | α (0) | (cid:54) R (2) (cid:104) α (cid:48) , r (cid:105) < (3) α (cid:48) ( t ) is within a cone around D of amplitude cA α ( t ) (4) the slack at all points of α is bounded below by a constant a > .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 71 has the following properties: (1) α is unbeatable: any retort β satisfies | β ( t ) | − | β (0) | < | α (0) | − | α ( t ) | . (2) | β ( t ) | − | β (0) | < | α (0) | − | α ( t ) | − ε . Proof.
We only need to prove the second statement, as any curve of A pointsdefined in a compact interval has slack bounded from below.Fix a neighborhood U of adapted A coordinates that contains the imageof α . We can assume that one such U contains all of the image of α , otherwisewe can split α into parts.Let v ( t ) be the vector at α ( t ) such that d ( t ) = − r ( α ( t )) + v ( t ) belongs to D x . Then the slack A α ( t ) is | r ( α ( t )) ||− r ( α ( t ))+ v ( t ) | = | d ( t ) | .We reparametrize α so that α (cid:48) ( t ) = d ( t ) + p ( t ) , with p ( t ) is orthogonal to d (this is the canonical parametrization). Then | p ( t ) | < c | α (cid:48) ( t ) | , and for c << ,this implies that | p ( t ) | < ( c/ | d ( t ) | = c A α ( t ) < c a .We compute | α (0) | − | α ( t ) | = (cid:90) t ddt | α | = (cid:90) t (cid:104) r, α (cid:48) (cid:105) ≥ t (1 − c a ) An A point only has one preimage in U , so any retort β of α lies outsideof U . As exp p ( α ( t )) = exp p ( β ( t − t )) , we have: | d exp p ( r ( α ( t ))) − d exp p ( r ( β ( t − t ))) | > ε for some ε depending on U . U , in turn, contains a ball around x of radius atleast r , a number which depends on a and R : the differential of the slack isbounded, so if A x = a > , it cannot drop to in a ball of sufficiently smallradius. Thus, we can switch to a smaller ε > that depends only on a and R .Write β (cid:48) ( t ) = b ( t ) r ( β ( t )) + w ( t ) , where w is a vector orthogonal to r ( β ( t )) .It follows from the above that | b ( t ) | < − ε for some ε depending on M and ε .We compute: | β (0) | − | β ( t ) | = (cid:90) t ddt | β | = (cid:90) t b ( t ) ≤ t (1 − ε ) and the result follows. (cid:3) With this lemma, we can perturb a CDC slightly to avoid some points:
Definition . An approximately conjugate descending curve (ACDC)is a C curve α of A points such that α (cid:48) ( t ) is within a cone around D of am-plitude cA α ( t ) , where c is the constant in the previous lemma for R = α (0 ). Using the results so far, it is not hard to prove (seelemma 5.4.26 and 5.4.31) that there is always a ACDC α : [0 , t ] → T p M starting at any point x ∈ J , whose interior consists only of A points andending up in an A point. We also know that we can start a retort (cid:102) α at the A point.We can continue the retort while it remains in the interior of V , where e isa local diffeomorphism and we can lift any curve. However, we might be unable -MANIFOLDS to continue the retort up to x if the returning curve hits the set of conjugatepoints.If we hit an A point y = (cid:102) α ( t ) , we can take a ACDC β : [0 , t ] → V starting at this point and ending in an A point. If β has a retort ˜ β : [0 , t ] → T p M that ends up in a non-conjugate point ˜ β ( t ) , we can continue with theretort (cid:102) α of α | [0 ,t − t ] starting at ˜ β ( t ) . If (cid:102) α can be continued up to x = α (0) ,the concatenation of (cid:102) α , β , ˜ β and (cid:102) α can play the same as the retort of α (seefigure 5.4.3).There are a few things that may go wrong with the above argument: theretort (cid:102) α may meet J \ A , or β may not admit a full retort starting at β ( t ) ,or α | [0 ,t − t ] may not admit a full retort starting at ˜ β ( t ) . The first problemcan be avoided if the ACDCs are built to dodge some small sets, as we will seelater. Then, if we assume that a retort never meets J \ A , we can iterate theabove argument whenever a retort is interrupted upon reaching an A point.We will prove later that the argument only needs to be applied a finite numberof times.This is the motivation for the definition of linking curve: Definition . A linking curve is a continuous curve α : [0 , t ] → T p M that is the concatenation α = α ∗ . . . ∗ α n of ACDCs and non-trivialretorts of those ACDCs, all of them of finite length , such that: • Starting with the tuple ( α , . . . , α n ) consisting of the curves that α is madeof, in the same order, we can reach the empty tuple by iteration of thefollowing rule:Cancel an ACDC α j together with a retort α j +1 of α j that follows inmedi-ately: ( α , . . . , α j , α j +1 , . . . α n ) → ( α , . . . , α j − , α j +2 , . . . α n ) , if α j +1 is a retortof α j . • The extremal points of the α i are called the vertices of α . The verticesof α fall into one of the following categories: − starting point (first point of α ): a point in J . − end point (last point of α n ): a point in I . − A join, as explained in section 5.4.6. − a splitter : a vertex that joins two ACDCs whose concatenation isalso a ACDC. − a hit : a vertex that joins a retort that reaches A ( I ) transversally,and an ACDC starting at the intersection point. − a reprise : a vertex that joins a retort that completes its task ofreplying to a ACDC α j , and the retort for a different ACDC α i (itfollows from the first condition that i < j ). • The preimage of a point of M by e ◦ α falls into one of the followingcategories: − it can be empty. − it can have one point that is an A join. − it can have two points, one A point in the interior of an ACDC andan NC point in the retort of that ACDC. − it can have two points, the first and the last points of α . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 73 − it can consist of three vertices: a splitter, a hit and a reprise, suchthat the six curves α i contiguous to any of these three points mapto a T -shaped curve, with two curves mapping into each segment ofthe T. See figure 5.4.3. We call this combination of three vertices a standard T . Definition . A standard T consists of three vertices: a splitter, ahit and a reprise, such that the six curves α i contiguous to any of these threepoints map to a T -shaped curve, with two curves mapping into each segmentof the T. See figure 5.4.3. IIIIII IV V VI I II III IV VVI1 2 3 4 5 1 2 3 45
Figure 5.4.3. A Standard T : The left hand side displays a curve α in T p M , while the right hand side displays exp p ◦ α . I, II andIV are ACDCs, III is the retort of II, V is the retort of IV, andVI is the retort of I. Vertices 2 and 4 are A joins, vertex 1 is asplitter, vertex 3 is a hit and vertex 5 is a reprise. There can bemore than two segments between a splitter and its matching hit,and between a hit and its matching reprise. Remark.
A linking curve is non-trivial if it contains at least one ACDC.
Lemma . Let α = α ∗ . . . ∗ α n be a non-trivial linking curve: • α is an ACDC and α n is its retort. • Whenever α k is the retort of α j , for < j < k < n , then α j ∗ . . . ∗ α k isa linking curve. Proof.
The proof is simple and is left to the reader. (cid:3)
Proposition . Let α be a linking curve between x, y ∈ T p M , and M and M two L -related Riemannian manifolds. Then: • | x | > | y |• exp p ◦ α is fully tree-formed (in particular, it is continuous) -MANIFOLDS • e ( x ) = e ( y ) and e ( x ) = e ( y ) Proof.
The first part follows trivially from lemma 5.4.12 and its generaliza-tion, lemma 5.4.15. Each pair of a ACDC and its retort adds a negative amountto the radius of x .For the second part, we reparametrize α to the unit interval [0 , . Let T : [0 , → Im(exp ◦ α ) be the identification given by exp ◦ α . Let us show that u = exp ◦ α is tree-formed with respect to T : let t , t such that u ( t ) = u ( t ) ,and ϕ a continuous -form along u ( ϕ ( s ) ∈ T ∗ u ( s ) M ) that factors through Γ .Then we claim that:(5.4.2) (cid:90) t t ϕ ( s )( u (cid:48) ( s )) ds splits as a sum of integrals over the image by exp of an ACDC and the imageof its matching retort. The curves in each such pair have the same image,and the integrals cancel out, as the integral of a -form is independent of theparametrization, and only differs by sign.The claim follows if u − ( u ( t )) consists of two points, because t is in thedomain of an ACDC α i and t lies in the retort α j of α i . We recall it is possibleto reach an empty tuple by cancelling adjacent pairs of an ACDC and its retort.Thus, in order to cancel α i and α j , it must be possible to cancel all the curves α k with i < k < j . These curves can be matched in pairs { ( α n , α m ) } ( n,m ) ∈P of ACDC and retort, with i < n < m < j for each pair ( n, m ) ∈ P . Then wehave: (cid:90) t t ϕ ( s )( u (cid:48) ( s )) ds = (cid:90) t i t ϕ ( s )((exp ◦ α i ) (cid:48) ( s )) ds + (cid:88) ( n,m ) ∈P (cid:16) (cid:90) t n t n ϕ ( s )((exp ◦ α n ) (cid:48) ( s )) ds + (cid:90) t m t m ϕ ( s )((exp ◦ α m ) (cid:48) ( s )) ds (cid:17) + (cid:90) t t j ϕ ( s )((exp ◦ α j ) (cid:48) ( s )) ds The remaining two integrals also cancel out, proving the claim.If t and t are two of the three points of a standard T, we can take points t ∗ and t ∗ as close to t and t as we want, but in an ACDC and its retort, re-spectively, and such that u ( t ∗ ) = u ( t ∗ ) . The result follows because the integral5.4.2 depends continuously on t and t .The last part is similar to lemma 4.1 in [ H87 ]. In the hypothesis, we areassuming that the curve has a specific structure, which makes the proof simpler,but we do not ask for the sequence of curves converging to the linking curvein the hypothesis of that lemma, so we will have to build it ourselves.Let α = α ∗ . . . ∗ α n be a linking curve between x ∈ V ⊂ T p M and y ∈ V ,with each α i either a ACDC, or the retort of one of the previous ACDCs. Wewrite j = ¯ i whenever α j is a retort of α i . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 75 We want to find an open set O ⊂ V such that exp p | O is injective, Im( α ) ⊂ O , and a sequence of curves α k converging to α in AC( M ) such that Im( α k ) ⊂ O .We first construct a set O as the union of neighborhoods O k of the verticesof α , neighborhoods U i of the ACDCs in α and neighborhoods W j of the retortsof those ACDCs.First, we take disjoint neighborhoods O k of the vertices. We can assumethat they are disjoint with the preimages of the images of the other O k , exceptfor the neighborhoods of the three vertices of the same standard T.Next, we take neighborhoods U i ⊂ T p M of (the interior of the imageof) each ACDC α i in α , such that there is no non-trivial retort of α i in U i . By the third property in the definition 5.4.17, we can assume that U i ∩ exp − p (cid:0) exp p (cid:0) U i (cid:1)(cid:1) is empty unless α i and α i are consecutive ACDCs joinedby a “splitter” vertex, in which case the intersection is only the splitter. Also, U i should only intersect exp − p (cid:0) exp p ( O k ) (cid:1) when O k is a neighborhood of oneof the two endpoints of α i . It follows that no retort of any part of α i passesthrough ∪ U i .And last, the neighborhood W j of (the interior of the image of) the retort α j of α i has to be chosen so that: • W j is disjoint with ∪ j (cid:54) = j exp − p (cid:0) exp p ( W j ) (cid:1) and ∪ i (cid:54) = i exp − p (cid:0) exp p ( U i ) (cid:1) , orconsists of just one vertex if the curves are consecutive. • W j should only intersect exp − p (cid:0) exp p ( O k ) (cid:1) when O k is a neighborhood ofone of the two endpoints of α j .We still have to build the set O . The neighborhood U i of an ACDC α i maps 2:1 to a half ball by exp p . Its intersection with V is a set U i thatmaps 1:1 onto the same image. The neighborhood W j of its retort α j maps1:1 to a tubular neighborhood of Im (cid:0) exp p ◦ α i (cid:1) = Im (cid:0) exp p ◦ α j (cid:1) . We take W j = W j \ e − (exp p ( U i )) as the neighborhood of α j .We next describe how to build neighborhoods for each type of vertex, sothat they are compatible with the neighborhoods U i and W j for the curves. An A join: We take the neighborhood O k defined (in special coordinates)by { x < x ; x ≤ } ∪ { x < x ; x > } (we assume that the CDC inthe join comes from the “left” side x < ). As shown in section 5.4.6, theboundary of O k consists of a half surface foliated by CDCs and anotherhalf surface foliated by the retorts of those curves. A splitter (in a standard T):
For this type of point we proceed as if thetwo ACDCs that join at the split point were one only ACDC. So wetake the neighborhood O k of the splitter point, and intersect it with V : O k = O k ∩ V . A hit (in the same standard T):
For this point (which is A ) we in-tersect its neighborhood O l with V , and also remove the preimage ofthe image of the neighborhood O k of the accompanying splitter: O l =( O l ∩ V ) \ e − ( e ( O k )) . A reprise (in the same standard T):
This is a non-conjugate point, andwe remove from its neighborhood O m the preimage of the images of the -MANIFOLDS neighborhoods O k and O l of the accompanying splitter and hit: O m = O m \ ( e − ( e ( O l )) ∪ e − ( e ( O k ))) .The reader can check that for O = (cid:83) k O k ∪ (cid:83) i O i ∪ (cid:83) j O j , exp p | O isinjective and Im( α ) ⊂ O .Let us build the k -th approximation to α . This will be a curve α k with Im( α k ) ⊂ O , consisting of n + 1 parts: one for each curve α i in α and onefor each vertex v k . The part corresponding to the curve α i is a curve in a /k neighborhood of Im( α i ) . The part corresponding to the vertex v k will be a C curve that has length bounded by Ck for some universal constant and joinsthe approximations to the curves α i adjacent to v k . The argument proceedsnow as sketched in section 5.2.1, or proved in detail in lemma 4.1 of [ H87 ]:Let Y be the concatenation of the radial line in T p M that ends up in α (0) with α , with both curves rescaled so that Y (1 /
2) = α (0) , and Y (1) = α (1) .Let u = e ◦ Y , and let v be the curve obtained by affine developement of u followed by inverse affine developement onto T p M . The curve u is tree-formedwith respect to an identification T with T (1 /
2) = T (1) , and as we have seenit follows that v also has that property. In particular, v (1 /
2) = v (1) Let Y k be the concatenation of the radial line in T p M that ends up in α k (0) with α k , with both curves rescaled so that Y k (1 /
2) = α k (0) , and Y k (1) = α k (1) .Let u k = e ◦ Y k , and let v k be the curve obtained by affine developement of u k followed by inverse affine developement onto T p M . The curves u k arecontained in O , where e ◦ e − is an isometry, so v k = e ◦ e − ◦ u k .It follows at last that v (1 /
2) = lim k v k (1 /
2) = lim k v k (1 /
2) = e lim k Y k (0) = e ( x ) and v (1 /
2) = lim k v k (1) = lim k v k (1) = e lim k Y k (1) = e ( y ) , and thus e ( x ) = e ( y ) . (cid:3) As we promised, the following is also true:
Proposition . Let α be a linking curve between x, y ∈ T p M , and M and M two L -related Riemannian manifolds.Then x and y are linked. but we defer the proof until 5.4.12. We now begin the proof of theorem 5.4.7. The only places where we assumethat M is easy from p is in lemma 5.4.26 and theorem 5.4.28.The goal of this section is to prove the existence of a linking curve starting atan arbitrary point x ∈ J . The set { y : | y | < | x | , exp p ( y ) = exp p ( x )) } = { y j } isfinite. This follows because { y : | y | (cid:54) | x |} can be covered with a finite amountof neighborhoods of adapted coordinates, and in any of them the preimage ofany point is a finite set. At least one y j realizes the minimum distance from p to q = exp p ( x ) , and must be either A ( I ) or NC (in other words, y ∈ I ). Wewill show that there is a linking curve joining x and one y j ∈ I , though it maynot be the one with minimal radius. Theorem . For any x ∈ J , there is a linking curve that joins x tosome y ∈ I . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 77 We start with a generalization of a linking curve that we can describeinformally as a linking curve under construction : Definition . An aspirant curve is a continuous curve α : [0 , t ] → T p M that is the concatenation α = α ∗ . . . ∗ α n of ACDCs and non-trivial retortsof those ACDCs, such that: • Starting with the tuple ( α , . . . , α n ) consisting of the curves that α is madeof in order, we can reach a tuple with no retorts , by iteration of thefollowing rule: Cancel an ACDC together with a retort of that ACDC that follows rightafter it: ( α , . . . , α j − , α j , α j +1 , α j +2 , . . . α n ) → ( α , . . . , α j − , α j +2 , . . . α n ) ,if α j +1 is a retort of α j . • In all other regards, an aspirant curve satisfies the same conditions as alinking curve.The loose
ACDCs in α = α ∗ . . . ∗ α k are the ACDC curves α j for which thereis no retort in α .The tip of alpha is its endpoint α ( t ) . Definition . We define some important sets: S R = B R ∩ exp − p (exp p ( A ∩ B R )) V = { x ∈ V : exp − p (exp p ( x )) ∩ B | x | ⊂ N C ∪ A }SA = { x ∈ A : ∃ y ∈ A , exp p ( y ) = exp p ( x ) , | y | < | x |} In other words, V consists of those points x ∈ V such that all preimagesof exp p ( x ) with radius smaller than | x | are N C or A . Definition . Let F ⊂ C be a finite set. A GACDC with respect to F , or GACDC when F is implicit (G is for generic ) is an ACDC α such that • Im( α ) is contained in ( C ∩ V ) \ F . • for y ∈ B | α ( t ) | ∩A such that exp p ( α ( t )) = exp p ( y ) , exp p ◦ α is transversalto exp p ( A ∩ B ε ( y )) at t , for some ε > . The motivation for the definition of GACDC is to find curves starting atpoints x ∈ I so that any possible retort avoids all singularities that are not A . The GACDC will also “avoid itself”: this is indeed the finite set F that itmust avoid, as we will see later. Lemma . For any
R > there is L > such that any GACDCstarting at x ∈ A ∩ B R has length at most L , and can be extended until itreaches an A point ( F can be any finite set). Proof.
First, we prove local existence (and thus, continuation) of GACDC.Let x ∈ A .Let y ∈ C ∩ exp − p ( x ) . Let U ⊂ M be a neighborhood of q = exp p ( x ) =exp p ( y ) , W and W disjoint neighborhoods of x and y mapping into U . C ∩ W is a smooth hypersurface containing x . C ∩ W may have conical singularities,but is a stratified manifold in any case. The transversality result mentioned atthe end of section 5.3 implies that exp p ( C ∩ W ) is transversal to each stratumof exp p ( C ∩ W ) . -MANIFOLDS The CDCs foliate
C ∩ W , so the set of points of C whose CDC sinksinto a stratum of exp − p (exp p ( C ∩ W )) with singularities other than A haspositive codimension in C ∩ W . Replacing one small subcurve of the CDC withan ACDC we can move from one CDC to a neighbouring one, thus avoidingthose singularities. We might not be able to avoid that our ACDC meets exp − p (exp p ( A )) , but we can take our ACDC so that it intersects that settransversally.We can continue the ACDC within a patch U of adapted coordinates. Thereis some L > such that any ACDC within U can be extended by a curve oflength at most L that may end up in an A point, or reach the boundary of U .There is a smaller neighborhood V ⊂ U such that any GACDC starting at x ∈ V is continued within U up to an A point, or up to a point in ∂U withsmaller radius that any point in V . Thus V is transient, in the sense that anACDC that passes through V will either finish or leave U and never returnto V . It is simple to choose such a set V ; it will be clear how to do it after weprove claim 5.4.28.We have shown that there is an GACDC with bounded length that exists V ,but indeed, the length of any ACDC in V is also bounded, because in theplane A , any ACDC is a C graph over any CDC.The radius decreases along an ACDC, and thus an ACDC starting at x never leaves { v ∈ T p M : | v | < | x |} . Take a finite cover of this set by tran-sient sets. An ACDC that starts at x will run through a finite amount oftransient sets. Each transient set only contributes a finite length to the totallength of the GACDC that started at x . (cid:3) Diagram 5.4.4 shows the algorithm that we follow in order to find the linkingcurves, starting with the trivial aspirant curve { x } .The linking curve is built step by step, starting with the trivial curve α = { x } , and adding segments to the aspirant curve α = α ∗ . . . ∗ α k following theserules: Descent:
If the end of α k is a point in J , let γ be a GACDC containedin V that starts at x . The curve γ must also avoid the finite set F =exp − p (Im(exp p ◦ α ) ∩ B | α (0) | ) ∩ C . We also know that γ intersects SA in afinite set and, for convenience, we split γ into r GACDCs α k +1 , . . . , α k + r such that each of these curves intersects SA only at its extrema. Thenew curve α ∗ α k +1 ∗ · · · ∗ α k + r ends up in an A point. The next step isa retort. Retort: If α k : [0 , T ] → V is a ACDC ending up in an A point, add theretort α k +1 of α k that starts at the A join. This is always possible, since α k does not intersect SA . The new tip of α ∗ α k +1 will be N C , A or A ,but the latter can only happen if α ∗ α k +1 is a linking curve. Reprise:
If the tip of α is N C and α is not a linking curve, let α j be thelatest loose curve in α . We add the retort α k +1 of α j starting at the tipof α . This is always possible, since α j does not intersect SA . The newtip of α ∗ α k +1 will be N C , A or A , but the latter can only happen if α ∗ α k +1 is a linking curve. Success!: If α is a linking curve, we report success and stop the algorithm.For completeness, the algorithm also reports success if α = { x } , for x ∈ I . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 79 The algorithm can also be presented in a recursive fashion. We start withsome definitions: • T ip ( α ) = α ( T ) , for any curve α defined in an interval [0 , α ] . • Ret ( α, y ) is the retort of α starting at y , for any curve α contained in V \ SA , and a point y ∈ V such that exp p ( y ) = exp p ( T ip ( α )) .Then for any x ∈ V , we define an aspirant curve L ( x ) by the followingrules: • If x ∈ J , then L ( x ) = { x }• If x ∈ I , then compute the GACDC curve γ = γ ∗ · · · ∗ γ r , as above.Then L ( x ) = γ ∗ L ( T ip ( γ )) ∗ Ret ( γ , T ip ( L ( T ip ( γ )))) Remark.
The reader have probably noticed that γ to γ r are discarded, andonly γ is kept (the ACDC up to the first A point). This causes a smallproblem with the recursive definition because of the non-deterministic descentstep. We have shown that there is a GACDC starting at any point in I , andthis curve intersects SA in finitely may points, but if we only keep the firstsegment of the GACDC up to the first intersection with SA and repeat theprocess, we have not shown that an A point will be reached in finitely manysteps. This can be solved in one blow by an application of the axiom of choice.We might also come back to 5.4.26 and refine it as needed. But the easiestsolution is to use the iterative version of the algorithm.In order to satisfy the last technical condition in the definition of linkingcurve, we have added to the “Descent” section the condition that exp p ◦ α k +1 does not intersect the image of exp p ◦ α .We recall that we can ask that α k +1 avoids a finite set F ⊂ C . The image of exp p ◦ α is the same as the image by exp p of Im( α ) ∩ A , or in other words, theimage by exp p of only the ACDCs in α . Each ACDC α j in α was built so that itdid not intersect exp − p (exp p ( S )) , for any strata S of singularities with smallerradius than α , except for strata of A singularities, which it would intersecttransversally. The ACDC α k +1 is contained in a strata S of A points, andthus S ∩ exp − p ( α j ) is a finite set.Thus, lemma 5.4.26 guarantees that we can always perform the “ descent ”step in the diagram. We have already shown why the other steps can alwaysbe performed.We conclude that it is always possible to perform one more step of thealgorithm, if it hasn’t reported “success!” yet. However, the algorithm may gethooked up in an infinite sequence of GACDC, retorts and reprises. We devotethe rest of the section to prove that this is not the case, for a generic metric. Definition . A pair (
S, O ) of open subsets of T p M with ¯ S ⊂ O , is transient iff for any point x in S ∩ J , a finite number of iterations of thealgorithm starting at { x } gives an aspirant curve that extends outside of O (orreports success!), and then any curve obtained by any number of iterations ofthe algorithm never has its endpoint in S .The gain of a transient pair ( S, O ) is the infimum of all | x | − | y | , for all x ∈ S , y ∈ V \ O such that there is an aspirant curve starting at x and endingat y .A transient pair is positive if it has positive gain. -MANIFOLDS Theorem . For any point x of type NC, A or A there is a positivetransient pair ( S, O ) , with x ∈ S . It follows from this theorem that there is a linking curve starting at anypoint.Define: R = sup (cid:26) R : ∀ x ∈ B R , the algorithm starting at x reportssucess! after a finite amount of iterations (cid:27) We will assume that R is finite and derive a contradiction, thus showingthe existence of linking curves for all points in A . Take a covering of B R bya finite number of neighborhoods { S i } Ni =1 , where ( S i , O i ) are transient pairs.Then B R + ε is also covered by ∪ S i for some ε > . Let ε be the minimum of ε , and all the gains of the N pairs.Take a point x ∈ B R + ε and assume x ∈ S . Iterate the algorithm until itreports success! or builds an aspirant curve α with endpoint y outside of O .Thanks to the way we have chosen ε , we can assume | y | < R , and byhypothesis there is a linking curve that joins y to some point z . Append thatlinking curve to α to achieve an aspirant curve starting at x and ending at z .For this aspirant curve to become a linking curve, it remains to reply to all theloose ACDCs in α . Each of them, except possibly its endpoint, is contained in V \SA . If, after replying to one of them, we hit an A point y , then y ∈ B R ,and thus we can append a linking curve that joins y to some z ∈ N C ∩ B | y | .Then we can continue to reply to the remaining loose ACDCs, and the processfinishes in a finite number of steps. This is the desired contradiction. It onlyremains to prove theorem 5.4.28. Let x ∈ V be a point and O be a cubical neighborhood of adapted coordinatesaround it. S will be a “small enough” subset of O : N C : The algorithm reports success! in one step for any non-conjugate point,so any S ⊂ O , such that O has no conjugate points, satisfies the claim.The gain is the infimum of the empty set, + ∞ , so the pair is positive. A : The CDC α starting at x that reaches ∂O has a length ε > . For x in a sufficiently small neighborhood S of x , there is a GACDC α thatreaches y ∈ ∂O and has length at least ε/ (for any finite set F ).If there is an aspirant curve that starts with α , and later has a retort of α , starting at a point z , then | z | < | y | , because the restriction of the curvefrom y to z is a linking curve.Further, α is unbeatable, so that any non-trivial retort of this short curvewill increase the radius at most | x | − | y | − δ for some δ > . The inequalitystill holds with δ/ if instead of α we have a GACDC starting at some x in a small enough neighborhood V of x .So if we take S as the intersection of V and a ball of radius δ/ , then ( S, O ) is transient, and the gain is at least δ/ . A : We recall that the set of singular points C near an A point is an hy-persurface, and the stratum of A points is a smooth curve. An ACDCstarting at any A point will flow either into the stratum of A pointstransversally (within C ), or into the boundary of O . .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 81 For points in a smaller neighborhood V ⊂ O , one of the following thingshappen: • If an ACDC starting at x ∈ V ∩ A flows into an A point, then itcan be replied in one step, and the algorithm stops. The algorithmalso stops if x ∈ A . • If the ACDC starting at x ∈ V ∩ A flows into y ∈ ∂O , the argumentis the same as that for an A point.This concludes the proof of claim 5.4.5, and thus we can apply proposition5.4.8 to build the synthesis manifold, for easy manifolds.We have chosen to defer the proof for the existence of linking curves forgeneric manifolds to section 5.4.13. The next section does not require the easyhypothesis, so the reader is presented with a full argument that works for somemanifolds for which the Ambrose conjecture was yet unknown. π and π are covering maps. We still have to provethat the synthesis manifold M given by theorem 5.4.8 is a covering space of M and M . We start with a general lemma: Lemma . Let exp p : T p M → M be the exponential map from a point p in a Riemannian manifold M . Then for any absolutely continuous path x : [0 , t ] → T p M , the total variation of t → | x ( t ) | is no longer than the lengthof t → exp p ( x ( t )) . In particular: | x ( t ) | − | x (0) | < length(exp p ◦ x ) Proof.
For an absolutely continuous path x : length(exp p ◦ x ) = (cid:90) | (exp p ◦ x ) (cid:48) | = (cid:90) | d exp p ( x (cid:48) ) | The speed vector x (cid:48) = ar + v is a linear combination of a multiple of theradial vector and a vector v perpendicular to the radial direction. By theGauss lemma, | d exp p ( x (cid:48) ) | = (cid:113) a + | d exp p ( v ) | ≥ | a | . On the other hand, v istangent to the spheres of constant radius, so: V t ( | x | ) = (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ddt | x | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) | a | ≤ length(exp p ◦ x ) (cid:3) Define d : M → R by: d ( q ) = inf x ∈ e − ( q ) {| x |} If we could prove that e is the exponential map of the Riemannian manifold M at the point p = e (0) , it would follow that d is the distance to p , and thefollowing proposition would be trivial. Proposition . d is distance-decreasing. In other words: | d ( q ) − d ( q ) | ≤ d M ( q , q ) -MANIFOLDS Proof.
We can assume that q and q both lie in the same basic open set [ O ] .Otherwise, take a smooth path joining q and q of length at most d M ( q , q )+ ε and place enough intermediate points q i . If we prove that | d ( q i ) − d ( q i +1 ) |
As we mentioned in section 5.3,the radial vector field, and the spheres of constant radius of T p M , which havevery simple expressions in standard linear coordinates in T p M , are distortedin canonical coordinates. Thus, the distribution D and the CDCs do notalways have the same expression in adapted coordinates. In this section, westudy them qualitatively. We will use the name R : T p M → R for the radiusfunction, and r for the radial vector field, and we assume that our conjugatepoint is a first conjugate point (it lies in ∂V ).(5.4.14.1). A points. In a neighborhood O of an A point, T p M can be strat-ifed as an isolated A point, inside a stratum of dimension of A points, insidea smooth surface consisting otherwise on A points. The conjugate points aregiven by x + 2 x x + x = 0 , and the A points are given by the additionalequation x + 2 x = 0 . The kernel is generated by the vector ∂∂x at anyconjugate point and we can assume that D is close to ∂∂x in O ∩ C .We do not know precisely where the radial vector is, but the distribution D is a smooth line distribution and its integral curves are smooth. Thus, the A point belongs to exactly one integral curve of D .As we saw, A ( I ) (resp A ( II ) ) points have neighborhoods without A ( II ) (resp A ( I ) ) points. The A point splits A into two branches, and it canbe shown easily that they must be of different types. Composing with thecoordinate change ( x , x , x ) → ( − x , x , x ) if necessary, we can assume thatthe CDCs travel in the directions shown in figure 5.4.5.(5.4.14.2). D − points. In a neighborhood O of adapted coordinates near a D − point, C is a cone given by the equations − x − x + x . The kernelof de at the origin is the plane x = 0 , which intersects this cone only at( , . . . , ). Three generatrices of the cone consist of A points (they are givenby the equations x = 0 , x − x = 0 and x + x = 0 , plus the equation ofthe cone), and the rest of the points are A .The radial vector field ( r , r , r ) at the origin must lie within the solidcone − r − r + r > , because the number of conjugate points (countingmultiplicities) in a radial line through a point close to ( , , ), must be . Inparticular, | r | > . Composing with the coordinate change ( x , x , x ) → ( − x , − x , − x ) to the left and ( x , x , x ) → ( x , x , − x ) to the right, ifnecessary, we can assume that r > .The kernel at the origin is contained in the tangent to the hypersurface T = { R ( y ) = R (0) } , and the radius always decreases along a CDC. Thus aCDC starting at a first conjugate point moves away from the origin and may -MANIFOLDS either hit an A point, or leave the neighborhood. Thus these points are notsinks of CDCs starting at points in V .We now claim that there are three CDCs that start at any D − point andflow out of O , and three CDCs that flow into any D − point, but the latter onesare contained in the set of second conjugate points.Recall that the D − point is the origin. We write the radial vector as itsvalue at the origin plus a first order perturbation: r = r + P ( x ) with | P ( x ) | < C | x | for some constant C .We will consider angles and norms in O measured in the adapted coordi-nates in order to derive some qualitative behaviour, even though these quanti-ties do not have any intrinsic meaning.We can measure the angle between a generatrix G and D by the determinantof a vector in the direction of G , the radial vector r and the kernel k of e :the determinant is zero if and only if the angle is zero. The angle between k and r in this coordinate system is bounded from below, and the norm of r isbounded close to . Thus if we use unit vectors that span G and k , we get anumber d ( x ) that is comparable to the sine of the angle between G and theplane spanned by r and k . Thus c | d ( x ) | is a bound from below to | sin( α ) | ,where α is the angle between G and D , for some c > .The kernel is spanned by ( − x + x , x , if − x + x (cid:54) = 0 . The generatrixof C at a point ( x , x , x ) ∈ C is the line through ( x , x , x ) and the origin.So d is computed as follows: d ( x ) = 1 x + x + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x − x + x x r r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Let us look for the roots of the lower order ( -th order) approximation: d ( x ) = 1 x + x + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x − x + x x r r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where ( r , r , r ) are the coordinates of r .The equation (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x − x + x x r r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 is homogeneous in the variables x , x and x , so we can make the substitution − x + x = 1 in order to study itssolutions. We only miss the direction λ (1 , , , where D is not aligned with G because it consists of A points.Points in C now satisfy x − x = 0 , and d ( x ) becomes p ( x ) = − ( a − x + bx − ( a + 3) x + b , for a = r r and b = r r (recall r > ).The lines of A points correspond to x = − √ , x = √ , and the third line lies at ∞ . We prove that p has three different roots, one in each interval: ( −∞ , − √ ),( − √ , √ ), ( √ , ∞ ). This follows inmediately if we prove lim x →−∞ p ( x ) = −∞ , p ( − √ ) > , p ( √ ) < and lim x →∞ p ( x ) = ∞ for all a and b such that a + b < . The first and last one are obvious, so let us look at the second .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 85 one. The minimum of p ( − √ √ a + 23 b + 4 √ in the circle a + b (cid:54) can be found using Lagrange multipliers: it is exactly and is attained only at the boundary a + b = 1 . The third inequality isanalogous.Thus, there is exactly one direction where D is aligned with G en eachsector between two lines of A points. Take polar coordinates ( φ, r ) in C ∩ V .The roots of d are transversal, and thus if φ corresponds to a root of d ,then at a line in direction φ close to φ , the angle between D and G is at least c ( φ − φ ) + η ( φ, r ), for c > and η ( φ, r ) = o ( r ). If, at a point in the linewith angle φ , and sufficiently small r > , we move upwards in the directionof D (in the direction of increasing radius), we hit the line of A points, notthe center. There are two CDCs starting at each side of every A point. Acontinuity argument shows that there must be one CDC in each sector thatstarts at the origin (see figure 5.4.6).Reversing the argument, we see that there are three CDCs that descendinto the elliptic umbilic point, one in each sector, all contained in the the setof second conjugate points.(5.4.14.3). D +4 points. The conjugate points in a neighborhood of adapted co-ordinates lie in the cone C given by x x − x = ( x + x ) − ( x − x ) − x .This time, the kernel of d exp p at the origin intersects this cone in two linesthrough the origin, and the inside of the cone x x − x > is split into twoparts. There is one line of A points, the generatrix of the cone with parametricequations: t → ( t, t, t ) .The radial vector at r = ( r , r , r ) must lie within the solid cone r r − r > , for the same reason as above. Composing with the coordinate change ( x , x , x ) → ( − x , − x , − x ) to the left and ( x , x , x ) → ( x , x , − x ) tothe right, if necessary, we can assume that r > and r > .We write the radial vector as its value at the origin plus a first order per-turbation: r = r + P ( x ) with | P ( x ) | < C | x | for some constant C .As before, the radius decreases along a CDC, but this time, a CDC startingat a first conjugate point might end up at the origin. Let F be the half cone offirst conjugate points (given by the equations x x = x and ( x + x ) < ).Let F + be the points of F with radius greater than the origin. Its tangent coneat the origin is F ∩ { x < } or F ∩ { x > } , depending on the sign of thethird coordinate of r .As in the previous case, we can measure the angle between a generatrix G and D by the determinant of a vector in the direction of G , the radial vector r and the kernel k of e . This time, the kernel is spanned by ( − x , x , in thechart x (cid:54) = 0 . d ( x ) = 1 x + x + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x − x x r r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) -MANIFOLDS Again, we look for the roots of the lower order ( -th order) approximation,which is equivalent to looking for the zeros of: ˜ d ( x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x − x x a b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in the cone C , for a = r r and b = r r . We can make the substitution x = − inorder to study the zeros of the polynomial (we choose x < because we areinterested in the half cone of first conjugate points). This implies x = − x fora point in C , and we are left with p ( x ) = − x − bx + ax +1 = 0 . If b +3 a > , p has two critical points − b ±√ b +3 a , otherwise it is monotone decreasing. Buteven when p has two critical points, the local maximum may be negative, orthe local minimum positive, with one real root.The vector r must satisfy r (cid:54) = 0 and x x − x > , or ab > . There aretwo chambers for r : r > and r < . We will say that a D +4 point suchthat r > (resp, r < ) is of type I (resp, type II).If r > (or a, b > ), then r and L ∩ F lie at opposite sides of thekernel of de at the origin. The cubic polynomial p has limit ∓∞ at ±∞ , and p (0) > . The line of A points intersects x = − at x = − . We check that p ( x = −
1) = 2 − a − b is always negative in the region a > , b > , ab > .Thus there is exactly one positive root, and two negative ones, one at eachside of the line of A points. This correponds to the top right picture in figure5.4.7, where the x axis is vertical, and the CDCs descend, because r > .The positive root gives a direction that is tangent to a CDC that entersinto the D +4 point, but moving to a nearby point we find CDCs that missthe origin, and approach either of the two CDCs that depart from the origin,corresponding to the negative roots of p .However, if r < (type II), p may have one or three roots. We revert thedirection of the CDC taking p ( z ) = p ( − x ) . We note that p (0) = 1 > , and p (cid:48) ( z ) > for z > , a < and b < , so there cannot be any positive root. ACDC starting at a point in F flows away from the stratum of A points andout of the neighborhood (see the bottom pictures at figure 5.4.7). It can bechecked by example that both possiblities do occur.We want to remark that if there are three roots, the D +4 point is the end-point of the CDCs starting at any point in a set of positive H measure. For-tunately, all these points are second conjugate points. This is the main reasonwhy we build the synthesis as a quotient of V rather than all of T p M but moreimportant: this is a hint of the kind of complications we might find in arbitrarydimension, or for an arbitrary metric, where we cannot list the normal formsand study each possible singularity separately. Remark.
In order to find out the number of real roots of p , for any value of a and b , we used Sturm’s method. However, once we found out the results, wefound alternative proofs and did not need to mention Sturm’s method in theproof. The precise boundary between the sets of a, b such that p has one orthree real roots is found by Sturm method. It is given by: p = − a b − a − b − ab + 243 = 0 .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 87 A CDC starting at J singularities . We have shown that there is a CDCstarting at an A point and a D +4 of type II, while there are two CDCs startingat a D +4 point of type I, three CDCs starting at a D − point, and at least onestarting at a D +4 point of type II. This will imply, ultimately, that points ofthose kinds are also linked to a point in I . Let x ∈ V be a point and O be a cubicalneighborhood of adapted coordinates around it. S will be a “small enough”subset of O : A : Near an A point, C is a smooth hypersurface and A is a smooth curvesitting inside C . The A point is isolated and splits the curve A into twoparts. One of them, which we call Branch I, consists of A ( I ) points, andthe other branch consists of A ( II ) points. The conjugate distribution D coincides with the kernel of exp p at the A point, and is contained in thetangent to the manifold of A points.As we saw before, a CDC that ends up in the A point can be perturbedso that it either hits an A point, or leaves the neighborhood.Let H be the set of points such that the CDC starting at that point flowsinto the A point. H is a smooth curve, and splits U into two parts. Oneof them, U , contains only A points, while the other, U , contains all the A points.Look at figure 5.4.8: a CDC starting at a point y ∈ U flows into theboundary of U without meeting any obstacle. A CDC α starting at apoint x ∈ U , however, flows into the branch I of A . We can start aretort β at that point, but it will get interrupted when exp ◦ β reaches thestratum of the queue d’aronde that is the image of two strata of A pointsmeeting transversally. The retort cannot go any further because only thepoints “above exp( C ) ” (the side of ) have a preimage, and points in themain sheet of exp( C ) have only one preimage, that is A . When he hitthe stratum of A points, we follow a CDC to get a curve that leaves theneighborhood in a similar way as the curve starting at y did. D : Any CDC starting at any point in a neighborhood of a D − , or D +4 oftype I point leaves the neighborhood without meeting other singularities.A nearby GACDC will also do. We only have to worry about the one CDCthat flows into the D +4 of type II, but we always take a nearby GACDCthat avoids the center.This concludes the proof of claim 5.4.5 for generic metric, and thus we canapply proposition 5.4.8 to build the synthesis manifold. The results in section5.4.11 can be applied without changes, and thus the theorem is proved forgeneric manifolds. -MANIFOLDS StartStart with the trivial curve {x}Is x unequivocal? Tip is A Tip is A or NC:Is there anyremainingloose ACDC? DescentDescend along a GACDC upto an A pointYes RetortAdd a retort tothe last ACDC.Which type of point is the tip?Success!NoRepriseReply the loose ACDCthat appears last.Which type of pointis the new tip? YesNo Figure 5.4.4.
Flow diagram for building linking curves .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 89 A Branchof A (I) Branchof A (II) Figure 5.4.5.
The distribution D and the CDCs at the conju-gate points near an A point. A A A Figure 5.4.6.
CDCs in the half-cone of first conjugate pointsnear an elliptic umbilic point, using the chart ( x , x ) → ( x , x , − (cid:112) x + x ) , for r = (0 , , . The distribution D makeshalf turn as we make a full turn around x + x = 1 , spinning inthe opposite direction. -MANIFOLDS Figure 5.4.7.
An hyperbolic umbilic point.Explanation of figure 5.4.7. In the
TopLeft corner, the cone C appearsin blue, the line of A points in green, the radial vector at the originin red, and the CDCs in red.The other pictures show the CDCs in the parametrization of the halfcone of first conjugate points, obtained by projecting onto the planespanned by (1 , − , and (0 , , . The red dots indicate the directionswhere D is parallel to the generatrix of the cone. The A points lie inthe half vertical line with x < . TopRight: a > , b > . BottomLeft: a < , b < , p has only one real root. BottomRight: a < , b < , p has three distinct real roots. .4. PROOF OF THE CONJECTURE FOR GENERIC -MANIFOLD 91 U A B r a n c h I o f A B r a n c h II o f A x y e x p p ( x ) e x p p ( y ) U Figure 5.4.8.
This picture shows a neighborhood of an A pointin T p M , together with the linking curves that start at x and y (to the left) and the image of the whole sketch by exp p (to theright).HAPTER 6 Further questions
In this chapter we collect open questions that are suggested by the previouswork. At some points, we comment on our attempts to prove these conjectures.The author did spend quite some time working in some of them, specially thelast one, but very little time in some of the others, and thinks that some ofthem are suitable for a student, specially the conjectures about magician’s hats. k conjugate cut points In the exponential map of a Finsler manifold (recall we only need to considerthe exponential from the boundary), the image of the focal points of order k can have Hausdorff dimension n − k .However, theorem 3.1.1 and the classical result 3.2.3 suggest the followingconjecture: Conjecture . Let M be a Finsler manifold with boundary of dimen-sion n . The set of points p in M such that there is a minimizing geodesic oforder k from ∂M to p has Hausdorff dimension at most n − k − . Inspection of the proofs of those two results actually suggests to divide theset C k conjugate points of order k into two subsets: C k : Points such that the kernel of the exponential is contained in the tangentcone to the set of conjugate points. C k : The complementary set in C k .We can venture a further conjecture that would imply the previous onewith little effort. Recall the definition of CDCs in 5.4.10. Conjecture . If, at a conjugate point x ∈ V of order k , the kernelof the exponential is not contained in the tangent cone to the set of conjugatepoints, then there is a CDC starting at x . The reason why this conjecture is enough to prove 6.1.1 is that the imageunder the exponential of a CDC is never a minimizing curve, as shown in lemma2.2 of [
H82 ].While thinking about this conjecture, we came out with a similar one, thatdoes not follow from or implies the previous one in a direct way, but we believeis interesting in its own right:
Conjecture . Let γ : ( − ε, → C be a C curve of conjugate points.Then γ can be extended to a C curve of conjugate points defined in ( − ε, ε ) for some ε > . This conjecture states that the set of conjugate points, which can be acomplicated subset of the tangent space (or V in the general setting 1.2), does
934 CHAPTER 6. FURTHER QUESTIONS not have edges or pointed tips. The conjecture holds for generic manifolds, asfollows trivially from the first structure result for generic manifolds, [ We ]. Theconjecture can probably be proved by approximating the metric with genericones. The techniques in chapter 4 could in principle be applied to other firstorder PDEs, or systems of PDEs. The idea of using the balanced propertywas actually inspired by the Rankine-Hugoniot conditions for the shock insolutions to equations of conservation laws. We think that the main appeal ofthe balanced condition is that, unlike the Rankine-Hugoniot conditions, it doesnot assume any a priori regularity on the shock. So we think it is interestingto study whether the balanced property, and at least some of the structureresults, can be carried over to other equations.In particular, we believe our proofs of 4.2.5 and 4.2.6 are more easily ex-tensible to other settings than the previous ones in the literature. This maysimplify the task of proving that the singular locus for other PDEs have locallyfinite n − Hausdorff measure.However, this may not be possible for conservation laws, as the Rankine-Hugoniot conditions are really very different to the balanced property, in thatthey only prescribe one tangent vector that must be contained in the tangentplane to the shock, while the balanced condition prescribes all the tangentdirections. Hamilton-Jacobi Cauchy problems are also different in nature, andthe results presented here may not apply. We would like to mention that L.C. Evans has recently made big improvements in the understanding of the HJCauchy problems for non-convex hamiltonian (see [
E10 ] and [ E ]). It wouldbe very interesting if non-convex hamiltonians were also better understood forBVPs. HJBVP problems with a Hamiltonian dependent on u seems a morefeasible target.Finally, Philippe Delanoë and others have suggested that it would be in-teresting to try this approach in sub-riemannian geometry. The Ambrose conjecture is related to the Poincaré conjecture in severalways. We mention one link between them that haunted the author for sometime:The proof of the Ambrose conjecture for surfaces by James Hebda worksin two dimensions because the cut locus is a tree. In [
H87 ], he mentions thatit works also in those manifolds whose cut locus is triangulable and descendssimplicially to a point . In a compact, simply connected manifold, the cut locusis homotopic to a point, but even if the cut locus is triangulable, it may notbe possible to collapse simplices one by one until the whole cut locus becomestrivial. For example, the -sphere admits a metric such that the cut locus withrespect to a certain point is the house with two rooms (see figure 6.3.1).The first motivation for studying split loci was that they may help overcomethis difficulty, if only we could find a split loci that does collapse simpliciallyto a point. But if that strategy worked for any manifold, it would also provide .3. POINCARÉ CONJECTURE 95 Figure 6.3.1.
The “house with two rooms”a proof of the Poincaré conjecture. The reason is that in a compact simply-connected manifold with such a collapsible split locus from a point p we can finda vector field with only one source p , and only one sink, using the deformationretract that collapses the split locus onto one point. It is well known that thisproves that the original manifold is homeomorphic to the sphere. Thus thefollowing conjecture is stronger than the Poincaré conjecture: Conjecture . Any compact simply-connected -manifold admits ametric and a split locus that collapses simplicially to a point. Let us explain the motivation behind the conjecture:Let M be a compact simply-connected -manifold. Any manifold admits ametric whose geodesic flow in T M is ergodic.If the manifold did not have any pair of conjugate points, the exponentialmap from any point would be a covering map, but this is incompatible withthe hypothesis. So we can assume that there is one point x ∈ T M such thatthe geodesic γ ⊂ T M starting at x meets a point y = γ ( T ) such that y isconjugate to x along γ . The continuity of λ shows that the same happens forany point in a neighborhood U ⊂ T M of x .The geodesic flow being ergodic, any geodesic eventually enters U , anddevelops a conjugate point. This means that λ is finite.Any metric close to the ergodic one will also have finite λ . We can chooseone such metric such that the exponential map is generic in the sense of Klok,Buchner or both.The compact set (cid:110) x : λ (cid:16) x | x | (cid:17) ≤ | x | (cid:111) ⊂ T M maps many-to-one to M , sothat each point has several preimages. The genericity hypothesis allows todecompose M into finitely many chambers. Every point in each chamber hasthe same number of preimages. That decomposition induces another one in T M , in which the chambers map diffeomorphically into chambers of M , butdifferent chambers of T M may map into the same chamber of M . The goal isthen to select only one of the preimages of each chamber in M , in such a waythat the union of all the chambers in T M is star-shaped with respect to theorigin. This is equivalent to selecting a split locus that is composed of imagesof conjugate points and geodesics that cut the radial geodesics so that eachpoint in M has only one preimage before the split locus is reached.There is no guarantee that any of these split loci collapses to a point. Infact, some of these selections of split loci might be quite similar to the cut locus.Our idea was to use linking curves to make some of the required choices, in orderto make a partial selection of chambers in a way such that the correspondingsplit locus is tree-like.Faced with the multiplicity of chambers, we can pick up one generic A first conjugate point x and find a linking curve starting at that point. This linking curve can also provide a linking curve for any point in the same -cellto which x belongs. In fact, the linking curve will pass through several -cells,and this could help us select the right chambers, whose boundary would mapto a -dimensional complex full of tree-formed curves. The main problem isthat in three dimensions, a generic linking curve can intersect twice the sameradial line, so this procedure does not help to select chambers.Thus we conjecture that in three dimensions, the conjecture 6.3.1 is false,but in higher dimensions, we do not know. Indeed, in higher dimensions, evenif the conjecture is false, there is still hope that this argument can be useful insome way... We arrived at the following definition when we were working on 5.4.11.We wanted to be able to lift paths in M to the synthesis manifold M , and a pre-compactness result for the exponential map just seemed natural. Definition . A magician’s hat with respect to p ∈ M , is an openset U in a Riemannian manifold M such that its preimage by the exponentialfrom p has an unbounded connected component. Conjecture . (1) Let p be a point in a Riemannian manifold M . Any point q ∈ M has a“sufficiently small” neighborhood that is not a magician’s hat with respectto p . (2) For any “bound on curvature” K , there is a “diameter” d such that onany manifold M with curvature bounded by K , any set of diameter lessthan or equal to d is not a magician’s hat with respect to any point. Weintentionally leave open the question of what are the appropriate notionsof “bound on curvature”. The reason for choosing such a name is that a curve contained in the preim-age of a magician’s hat U that goes to infinity corresponds to a family ofgeodesics with starting point p and endpoint in U that get longer and longer.This, to the author, would be similar to a magician pulling a long handkerchiefout of his hat. The Ambrose conjecture involves both topological and analytical challenges.We already mentioned in section 6.3 one topological difficulty. The Poincaréconjecture was an open, and very hot conjecture, for many years. Many re-knowned topologists and geometers failed at finding a proof using an arsenalof algebraic topology, knot theory, hyperbolic geometry, and what not. If amathematician working in the Ambrose conjecture does not feel like giving atopological proof of the Poincaré conjecture in the way, some strategies are notvery promising. However, we thought that the idea of building a synthesis as in5.4.2 would allow to cast away the topological difficulties, allowing us to proveAmbrose conjecture, but not Poincaré’s. Tree formed curves, now refurbishedinto linking curves, would help make the necessary identifications. .5. PROOF FOR A 3-MANIFOLD WITH AN ARBITRARY METRIC 97
Using the cut locus for a synthesis would not work, because there is nocanonical way to find linking curves. An arbitrary split locus is no better, ofcourse, if it does not have neither simpler topology, nor a canonical way to findlinking curves within it. The conjugate descending curves seemed like the onlysensible choice.But these curves are very tricky. The first obvious problem is that thesingularities of the exponential map are complicated. A first step towardsdealing with that problem is to use our theorem 3.1.1, building a synthesis of most points of the manifold, and then extending the construction to a bona-fidesynthesis by completion of the metric.
In an arbitrary metric, the possible singularities of theexponential map no longer belong to a finite family of canonical forms. Inorder to prove the Ambrose conjecture, we will need to find a wider categoryof conjugate points that are unequivocal, and a wider category of conjugatepoints that are linked to the unequivocal points. It is also convenient to workwith a remainder of conjugate points about which we know very little, butsuch that the set of such points has sufficiently small Hausdorff dimension sothat we can ignore them in our arguments.
Definition . The cousins of x ∈ V are the preimages of its image bythe exponential map. C ( x ) = e − ( e ( x )) The younger cousins of x ∈ V are the cousins of smaller radius: YC( x ) = e − ( e ( x )) ∩ B (0 , | x | ) We defer the definitions of terminal points of order 1 and the types ofconjugate points of order for later sections. Definition . We define some categories of points in T p M (recall N C are the non-conjugate points): − R = N C ∪ { terminal points of order }− S = { non terminal points of order }∪ { conjugate points of order and type I }− T = { conjugate points of order and type II }∪ { conjugate points of order }− I = R− J = { x ∈ S : YC( x ) ⊂ R ∪ S}− K = T ∪ { x ∈ S : YC( x ) (cid:42) R ∪ S}
We call the following the
IJK conjecture : Conjecture . • Points in I are unequivocal. • Points in J are linked to a point in I• e ( K ) has Haussdorf dimension at most n − Remark.
The last part follows directly from 3.1.1.
Using 6.5.3, we can build a synthesis U of U = M \ e ( K ) ⊂ M and U = M \ e ( K ) ⊂ M using theorem 5.4.8. Proposition . The maps π and π in the synthesis are coveringmaps. Proof.
Proposition 5.4.30 follows as in section 5.4.11, once we notice that forany R :(1) X R = { x ∈ A ∩ B (0 , R ) , Y C ( x ) ⊂ N C ∪ A } is ( n − -rectifiable, withfinite H n − measure(2) H n − ( e ( V \ ( N C ∪ A ))) = 0 It follows that a generic path of finite length intersects e ( T p M \ N C ) onlyat a finite number of points in X R . However, U and U are not complete, sowe will prove directly that π , for example, has the path lifting property.Let p : [0 , b ] → U be a smooth, unit speed path, and q ∈ π − ( p (0)) . As π is a local homeomorphism, a path can always be lifted for a short time. Let q : [0 , t ∗ ) → U be a lift of p starting at q for a maximal time t ∗ . The Lipschitzproperty of d shows that d ( q ( t )) < d ( q ) + t for any t < t ∗ .First we prove that q can be extended to the compact interval [0 , t ∗ ] : Forall t < t ∗ , ∃ x ( t ) such that e ( x ( t )) = q ( t ) and | x ( t ) | (cid:54) | x (0) | + t . There isa sequence t n (cid:37) t ∗ such that x ( t n ) converges to some x ∗ ∈ T p M such that | x ∗ | (cid:54) | x (0) | + t ∗ . Also, e ( x ∗ ) = p ( t ∗ ) , so x ∗ ∈ I ∪ J because p ( t ∗ ) ∈ U . Thisway we extend q by setting q ( t ∗ ) = e ( x ∗ ) , and it holds that q ( t ) → q ( t ∗ ) .Finally, assume t ∗ < b . As we have mentioned already, we can extend q toa path defined up to time t ∗ + ε . This completes the proof that π is a coveringmap. (cid:3) The subsets U of M and U of M are big enough so that the constructionextends to provide a synthesis of M and M : Proposition . Let π : Y → X be a Riemannian covering of Riemann-ian manifolds. Assume X ⊂ ¯ X where ¯ X is a complete Riemannian manifoldand H n − ( ¯ X \ X ) = 0 .Then there is a unique Riemannian covering from the completion ¯ Y of Y into ¯ X that restricts to π . In particular, ¯ Y is a Riemannian manifold. The proposition follows from the following general topology lemma:
Lemma . Let π : Y → X be a covering map of locally simply-connectedspaces. Assume X ⊂ ¯ X where ¯ X is a locally simply-connected space such thatthe intersection of any non-empty simply-connected open set V ⊂ ¯ X with X isnon-empty and simply-connected.Then there is a locally simply-connected space ¯ Y and maps i : Y → ¯ Y , ¯ π : ¯ Y → ¯ X such that: • ¯ π is a covering map Project a set S ⊂ R n such that H n − ( S ) < ∞ onto the orthogonal hyperplane to thesegment [ q , q ] . Using the co-area formula, we see that almost sure, a line parallel to [ q , q ] intersects S at a finite number of points. .5. PROOF FOR A 3-MANIFOLD WITH AN ARBITRARY METRIC 99 • ¯ π ◦ i = π • For any simply-connected non-empty set O ⊂ ¯ Y , Y ∩ i − ( O ) is non-emptyand simply-connectedMoreover, the space ¯ Y that we construct has the following universal property:Let ˜ Y , ˜ π, ˜ i satisfy the above properties. Then ˜ Y is a covering space of ¯ Y ,with a covering map ρ : ˜ Y → ¯ Y such that ¯ π ◦ ρ = ˜ π and ρ ◦ ˜ i = i .Thus ¯ Y is characterized by the above properties up to isomorphism. Proof.
The space is built as equivalence classes of pairs ( x, V, A ) , where x ∈ ¯ X , V ⊂ ¯ X is a simply-connected neighborhood of x , A ⊂ Y and π | A : A → U = V ∩ X is a homeomorphism.Two pairs ( x, V, A ) and ( x (cid:48) , V (cid:48) , A (cid:48) ) are equivalent iff x = x (cid:48) and there is anopen simply connected set V (cid:48)(cid:48) ⊂ V ∩ V (cid:48) such that A ∩ A (cid:48) ∩ π − ( V (cid:48)(cid:48) ) (cid:54) = ∅ .The basis open sets of Y are the sets O V,A = { [( x, V, A )] , x ∈ V } , forany V ⊂ ¯ X open simply-connected and A ⊂ Y open simply-connected suchthat π ( A ) = V ∩ X . It follows that A is one of the connected components of π − ( V ∩ X ) .The map i is defined by i ( y ) = [( π ( y ) , π ( A ) , A )] , where A ⊂ Y is anysimply-connected open neighborhood of y .The map ¯ π is given by ¯ π ([( x, V, A )]) = x .Let V ⊂ ¯ X be an open simply-connected set. Its preimage by π consistsof all the classes [( x, V, A i )] , where x ∈ V and A i is one of the connectedcomponents of π − ( V ∩ X ) (each of which is homeomorphic to V ∩ X , becauseit is simply-connected). There are no more classes: let [( x (cid:48) , V (cid:48) , A (cid:48) )] be a classwith x (cid:48) ∈ V . Then V ∩ V (cid:48) is a neighborhood of x (cid:48) which contains a simplyconnected neighborhood V (cid:48)(cid:48) of x (cid:48) .As π | A (cid:48) is a homeomorphism, there is a point y ∈ A (cid:48) ∩ π − ( V (cid:48)(cid:48) ) (cid:54) = ∅ . As π ( y ) ∈ V , y belongs to one of the A i and thus [( x (cid:48) , V (cid:48) , A (cid:48) )] = [( x (cid:48) , V, A i )] .The sets O V,A i = { [( x, V, A i )] x ∈ V } , where V is fixed and A i are theconnected components of π − ( V ) , therefore they are open and disjoint. Themap ¯ π restricts to an homeomorphism from each O V,A i onto V (an open set O contained in O V,A i is of the form O V (cid:48) ,A i ∩ π − ( V (cid:48) ∩ X ) for V (cid:48) = π ( O ) ⊂ V ). Inparticular, each O V,A i is connected, and thus π − ( V ) = (cid:116) O V,A i satisfies thestack property.The third property follows because ¯ π is an homeomorphism when restrictedto a simply-connected set.In order to prove the universal property, let ˜ Y , ˜ π, ˜ i satisfy the stated prop-erties. For a point y ∈ ˜ Y , we define ρ ( y ) = [(˜ π ( y ) , ˜ π ( ˜ V ) , ˜ i − ( ˜ V ))] , where ˜ V isa simply-connected neighborhood of y .We check that ρ ◦ ˜ i = i : i ( y ) = [( π ( y ) , π ( A ) , A )] for a simply connectedneighborhood A of y , and ρ (˜ i ( y )) = [(˜ π (˜ i ( y )) , ˜ π ( ˜ V ) , ˜ i − ( ˜ V ))] for a simplyconnected neighborhood ˜ V of ˜ i ( y ) . The two points are the same because y ∈ A ∩ ˜ i − ( ˜ V ) , which a non-empty open set which contains a simply-connectedneighborhood of y . It is trivial to check that ¯ π ◦ ρ = ˜ π .Let ˜ V ⊂ ˜ Y be simply-connected. It follows from ¯ π ◦ ρ = ˜ π that ρ is anhomeomorphism when restricted to ˜ V . (cid:3)
00 CHAPTER 6. FURTHER QUESTIONS
Proof of 6.5.5: The topological spaces X and Y satisfy the hypothesis ofthe lemma by standard results of dimension theory (see [ HW ]). Remark.
We asked for suggestions about the general topology lemma 6.5.6on the algebraic topology list ALGTOP-L.Ben Wieland suggested that the lemma is also true, and more natural, if thehypothesis is that ¯ X admits a basis B of simply-connected open sets such thatthe intersection of any basis set with X is non-empty and simply-connected. So it remainsto prove the IJK conjecture 6.5.3. The first task, of course, is to define terminalpoints precisely:
Definition . A point x is terminal if there is no CDC starting at thatpoint. We list some conjectures related to conjecture 6.5.3:
Conjecture . All terminal points of order are unequivocal. Conjecture . All terminal points are unequivocal.
Conjecture . A point x of order is terminal iff it has a neighbor-hood U of special coordinates such that e ( U ) is a neighborhood of e ( x ) . Conjecture . A point x is terminal iff it has a neighborhood U ofspecial coordinates such that e ( U ) is a neighborhood of e ( x ) . Conjecture . The image by the exponential of all the conjugateterminal points of order k has Hausdorff dimension at most n − k − Conjecture . All non-terminal points of order are linked to apoint of smaller radius. Conjecture . All non-terminal points are linked to a point of smallerradius.
If we plan to use linking curves, its definition should be appropriately gen-eralized, otherwise it is clear that linking curves, with finitely many segments,will not exist in arbitrary Riemannian manifolds. We propose the followingdefinition:
Definition . A linking curve between x and y is a curve α :[0 , T ] → T p M such that [0 , T ] is split into two subsets A and B , so that: • B is closed, and the Hausdorff dimension of exp p ( B ) is . • A is open, a countable union of open intervals I n , n ∈ I and J m , m ∈ I ,so that I n is an ACDC curve and J n is a retort. The image by the exponential of such a curve would be fully tree-formed,and the same proof we used in 5.4.20 would do, but it would be more technicallychallenging to prove that its extremae are linked without using new ideas.
Conclusiones
En el capítulo 2 introdujimos algunos resultados útiles. El teorema 2.1.6,por ejemplo, da mucha más potencia a los resultados de estructura de [ LN ],pues permite eliminar la restricción, importante para las aplicaciones a prob-lemas de frontera, de que el dato de frontera sea nulo.Los resultados del capítulo 3 hacen parecer razonable la conjetura 6.1.1.Además, hemos mostrado aplicaciones concretas para el resultado de estructura3.1.1, y sugerido otras, como las posibles aplicaciones al movimiento brownianoen variedades al final de la sección III.Los resultados del capítulo 4 son a nuestro entender bastante completos:trabajamos con hipótesis bastante habituales, como la convexidad del Hamil-toniano, sin las cuales la misma definición de solución de viscosidad no estánclaras. Ni siquiera es habitual rebajar las condiciones de regularidad: en elpaper [ LN ] se trabaja con abiertos C , , pero el resto de datos son C ∞ . Nopensamos que rebajar la regularidad hubiera producido resultados cualitativa-mente distintos, en este contexto. Sin embargo, nos parece muy interesantehaber evitado limitarnos a abiertos del plano simplemente conexos, pues éstonos hubiera cerrado los ojos al bello resultado 4.2.4, que dice mucho sobre lanaturaleza de los balanced split loci.Respecto a la conjetura de Ambrose, hemos dado una demostración paramétricas genéricas susceptible de ser generalizada a métricas arbitrarias. Esteúltimo paso es técnicamente muy complicado, ya que supone un mejor en-tendimiento de las singularidades de la aplicación exponencial que permitanseguir un campo de vectores sobre una superficie singular que luego debe ser“respondido" con curvas que deben en lo posible mantenerse alejadas de lassingularidades. Sin embargo, creemos que nuestro enfoque es original, queintroduce algunas ideas nuevas e interesantes, como el enunciado, a nuestroentender muy natural, del lema 5.4.29, o las conjeturas 6.4.2, y que usa deformas nuevas ideas poco conocidas, como las curvas de flujo conjugado de-scendiente, implícitas en el trabajo [ H82 ], o la síntesis de dos variedades, queaparece ya en [ O ].En definitiva, además, creemos que el problema no era fácil, como comen-tamos en las secciones 6.3 o 6.5. ibliography [A] W. Ambrose: Parallel translation of riemannian curvature . Ann. of Math. (2) (1956), 337–363.[APS] W. Ambrose, R. S. Palais and I. M. Singer: Sprays . An. Acad. Brasil. Ci. (1960),163–178.[AG1] P. Angulo Ardoy and L. Guijarro: Cut and singular loci up to codimension . Ann. Inst. Fourier (Grenoble) (2011), no. 4, 1655–1681. Preprint available at arxiv.org/abs/0806.2229 (2009).[AG2] P. Angulo Ardoy and L. Guijarro: Balanced split sets and Hamilton-Jacobi equa-tions . Calc. Var. Partial Differential Equations (2011), no. 1-2, 223–252. Preprintavailable at arxiv.org/abs/0807.2046 (2008-2009).[AAC] G. Alberti, L. Ambrosio and P. Cannarsa: On the singularities of convex functions . Manuscripta Math. (1992), no. 3-4, 421–435.[BL] D. Barden and H. Le: Some consequences of the nature of the distance function onthe cut locus in a Riemannian manifold . J. London Math. Soc. (2) (1997), no. 2,369–383.[BH] R. A. Blumenthal and J. J. Hebda: The generalized Cartan-Ambrose-Hicks theorem . C. R. Acad. Sci. Paris Sér. I Math (1987), no. 14, 647–651.[B] M. A. Buchner:
The structure of the cut locus in dimension less than or equal to six . Compositio Math. (1978), no. 1, 103–119.[B77] M. A. Buchner: Stability of the cut locus in dimensions less than or equal to 6 . Invent.Math. (1977), no. 3, 199–231.[B77II] M. A. Buchner: Simplicial structure of the real analytic cut locus . Proc. Amer. Math.Soc. (1977), no. 1, 118–121.[C] É. Cartan: Leçons sur la géométrie des espaces de Riemann. Les Grands ClassiquesGauthier-Villars. Éditions Jacques Gabay, Sceaux, 1988.[CS] P. Cannarsa and C. Sinestrari: Semiconcave functions, Hamilton-Jacobi equations,and optimal control. Progress in Nonlinear Differential Equations and Their Applica-tions, . Birkhäuser Boston, Inc., Boston, MA, 2004.[CR] M. Castelpietra and L. Rifford: Regularity properties of the distance functions toconjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and appli-cations in Riemannian geometry
ESAIM Control Optim. Calc. Var. (2010), no. 3,695–718. Preprint available at arXiv:0812.4107v1 (2008).[CE] J. Cheeger and D. G. Ebin: Comparison theorems in Riemannian geometry. Revisedreprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008.[dC] M. P. do Carmo: Riemannian geometry. Mathematics: Theory & Applications.Birkhäuser Boston, Inc., Boston, MA, 1992.[D] J. J. Duistermaat: Fourier integral operators. Progress in Mathematics, .Birkhäuser Boston, Inc., Boston, MA, 1996.[E10] L. C. Evans: Adjoint and compensated compactness methods for Hamilton-JacobiPDE . Arch. Ration. Mech. Anal. (2010), no. 3, 1053–1088.[E] L. C. Evans:
Envelopes and nonconvex Hamilton-Jacobi equations . Available at math.berkeley.edu/ ∼ evans/envelopes.HJ.pdf [F] H. Federer: Geometric measure theory. Die Grundlehren der mathematischen Wis-senschaften, . Springer-Verlag New York Inc., New York 1969. Bibliography [GG] M. Golubitsky and V. Guillemin: Stable mappings and their singularities. GraduateTexts in Mathematics, . Springer-Verlag, New York-Heidelberg, 1973.[GS] H. Gluck and D. Singer: Scattering of geodesic fields. I . Ann. of Math. (2) (1978),no. 2, 347–372.[GSII] H. Gluck and D. Singer:
Scattering of geodesic fields. II . Ann. of Math. (2) (1979), no. 2, 205–225.[Ha] A. Hatcher: Algebraic topology. Cambridge University Press, Cambridge, 2002. Avail-able at ∼ hatcher [Hi] N. Hicks: A theorem on affine connexions . Illinois J. Math. (1959), 242–254.[H82] J. J. Hebda: Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem . IndianaUniv. Math. J. (1982), no. 1, 17–26.[H83] J. J. Hebda: The local homology of cut loci in Riemannian manifolds . Tôhoku Math.J. (2) (1983), no. 1, 45–52.[H87] J. J. Hebda: Parallel translation of curvature along geodesics . Trans. Amer. Math.Soc. (1987), no. 2, 559–572.[H94] J. J. Hebda:
Metric structure of cut loci in surfaces and Ambrose’s problem . J. Differ-ential Geom. (1994), no. 3, 621–642.[H10] J. J. Hebda: Heterogeneous Riemannian manifolds . Int. J. Math. Math. Sci. 2010 ,Article ID 187232, 7 pp.[HW] W. Hurewicz and H. Wallman: Dimension theory. Princeton Mathematical Series,v. 4. Princeton University Press, Princeton, N. J., 1941.[I] Jin-ichi Itoh: J. Itoh:
The length of a cut locus on a surface and Ambrose’s problem . J. Differential Geom. (1996), no. 3, 642–651.[IT98] J. Itoh and M. Tanaka: The dimension of a cut locus on a smooth Riemannianmanifold . Tôhoku Math. J. (2) (1998), no. 4, 571–575.[IT00] J. Itoh and M. Tanaka: The Lipschitz continuity of the distance function to the cutlocus . Trans. Amer. Math. Soc. (2001), no. 1, 21–40.[KN] Sh. Kobayashi and K. Nomizu: Foundations of differential geometry. I. IntersciencePublishers, a division of John Wiley & Sons, New York-London, 1963.[K] F. Klok:
Generic singularities of the exponential map on Riemannian manifolds . Geom. Dedicata (1983), no. 4, 317–342.[LN] Y. Li and L. Nirenberg: The distance function to the boundary, Finsler geometry,and the singular set of viscosity solutions of some Hamilton-Jacobi equations . Comm.Pure Appl. Math. (2005), no. 1, 85–146.[L] P. L. Lions: Generalized solutions of Hamilton-Jacobi equations. Research Notes inMathematics, . Pitman, Boston, MA, 1982.[M] J. Milnor: Morse theory. Annals of Mathematics Studies, . Princeton UniversityPress, Princeton, N. J., 1963.[Me] A. C. Mennucci: Regularity and variationality of solutions to Hamilton-Jacobi equa-tions. I. Regularity (2nd Edition) . ESAIM Control Optim. Calc. Var. (2007), no. 2,413–417.[MM] C. Mantegazza and A. C. Mennucci: Hamilton-Jacobi equations and distance functionson Riemannian manifolds . Appl. Math. Optim. (2003), no. 1, 1–25.[O] B. O’Neill: Construction of Riemannian coverings . Proc. Amer. Math. Soc. (1968),1278–1282.[Oz] V. Ozols: Cut loci in Riemannian manifolds . Tôhoku Math. J. (2) (1974), 219–227.[Wa] F. W. Warner: The conjugate locus of a Riemannian manifold . Amer. J. Math. (1965), 575–604.[We] A. Weinstein: The generic conjugate locus . In Global Analysis (Proc. Sympos. PureMath., Vol. XV, Berkeley, Calif., 1968), 299–301. Amer. Math. Soc., Providence, R. I.,1970.[We2] A. Weinstein:
The cut locus and conjugate locus of a riemannian manifold . Ann. ofMath. (2)87