Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds
LLINKING CURVES, SUTURED MANIFOLDS AND THEAMBROSE CONJECTURE FOR GENERIC 3-MANIFOLDS
PABLO ANGULO ARDOY
Abstract.
We present a new strategy for proving the
Ambrose conjecture ,a global version of the Cartan local lemma. We introduce the concepts oflinking curves, unequivocal sets and sutured manifolds, and show that anysutured manifold satisfies the Ambrose conjecture. We then prove that the setof sutured Riemannian manifolds contains a residual set of the metrics on agiven smooth manifold of dimension . Introduction
Let ( M , g ) and ( M , g ) be two complete Riemannian manifolds of the same di-mension , with selected points p ∈ M and p ∈ M . Any linear map L : T p M → T p M induces a map between the pointed manifolds ( M , p ) and ( M , p ): ϕ =exp p ◦ L ◦ (exp p | O ) − defined in ϕ ( O ) , for any domain O ⊂ T p M such that e | O is injective (for example, if exp p O is a normal neighborhood of p ).A classical theorem of E. Cartan [C] identifies a situation where this map is anisometry.For x ∈ T p M , let γ be the geodesic on M defined in the interval [0 , , startingat p with initial speed vector x and γ be the geodesic on M starting at p withinitial speed L ( x ) .Let P γ : T p i M i → T γ i (1) M i denote parallel transport along a curve γ . Definition 1.1.
The curvature tensors of ( M , p ) and ( M , p ) are L -related ifand only if for any x ∈ T p M : (1.1) P ∗ γ ( R γ (1) ) = L ∗ P ∗ γ ( R γ (1) ) In the definition, P ∗ γ i ( R γ i (1) ) is the pull back of the (0 , curvature tensor at γ i (1) ∈ M i by the linear isometry P γ i , for i = 1 , P ∗ γ i ( R γ i (1) )( v , v , v , v ) = R γ i (1) (cid:0) P γ i ( v ) , P γ i ( v ) , P γ i ( v ) , P γ i ( v ) (cid:1) for any four vectors v , v , v , v in T p i M i , and L ∗ is used to carry the tensor P ∗ γ ( R γ (1) ) from p ∈ M to p ∈ M .The usual way to express that the curvature tensors of ( M , p ) and ( M , p ) are L -related is to say that the parallel translation of curvature along correspondinggeodesics on M and M coincides. This certainly holds if L is the differential of aglobal isometry between M and M . Theorem 1.2.
If the curvature tensors of ( M , p ) and ( M , p ) are L -related, and exp | O is injective for some domain O ⊂ T p M , then ϕ = exp ◦ L ◦ (exp | O ) − is an isometric immersion.Proof. The proof of lemma 1.35 in [CE] works for any domain O such that exp | O is injective. (cid:3) The author was partially supported by research grant ERC 301179, and by INEM. a r X i v : . [ m a t h . DG ] A p r ablo Angulo-ArdoyIn 1956 (see [A]), W. Ambrose proved a global version of the above theorem, butwith stronger hypothesis.A broken geodesic is the concatenation of a finite amount of geodesic segments.The Ambrose’s theorem states that if the parallel translation of curvature along broken geodesics on M and M coincide, and both manifolds are simply connected ,then the above construction gives a global isometry ϕ : M → M whose differentialat p is L . It is enough if the hypothesis holds for broken geodesics with only one“ elbow ” (the reader can find more details in [CE]).In [Hi], in 1959, the result of W. Ambrose was generalized to parallel transportfor affine connections; in [BH], in 1987, to Cartan connections; and in [PT], in 2002,to manifolds of different dimensions.The Ambrose’s theorem has found applications to inverse problems (see [HHILU]and [KLU]).Ambrose also posed the following conjecture: Conjecture 1.3 ( Ambrose Conjecture ) . Let ( M , p ) and ( M , p ) be twosimply-connected Riemannian manifolds with L -related curvature.Then there is a global isometry whose tangent at p is L . Ambrose himself was able to prove the conjecture if all the data is analytic. In1987, in the paper [H87], James Hebda proved that the conjecture was true forsurfaces that satisfy a certain regularity hypothesis, that he was able to prove truein 1994 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 the Ambroseconjecture, when James Hebda proved in [H10] that the conjecture holds if M isa heterogeneous manifold . Such manifolds are generic.The strategy of James Hebda in [H87] can be rephrased in the following terms:in any Riemannian surface, for any cleave point q , there is always a cut locus linkingcurve (see definition 3.4) that joins the two minimizing geodesics that reach q . Weprove in Theorem 3.5 that this strategy does not carry over to higher dimension,and present a new strategy towards a proof for the Ambrose conjecture in dimensiongreater than .We refer the reader to section 5.1 for the terms used in the following definition: Definition 1.4.
A pointed manifold ( M , p ) is sutured (resp. strongly sutured )if and only if for any x ∈ T p M , there is an unequivocal y ∈ T p M with (cid:107) y (cid:107) ≤ (cid:107) x (cid:107) that is linked to x (resp. strongly linked ). MAIN THEOREM A.
The Ambrose conjecture holds if ( M , p ) is a suturedmanifold. Our conjecture is that all manifolds are strongly sutured, but in this paper weonly prove it for manifolds whose exponential map has some generic transversalityproperties (see definition 6.7):
MAIN THEOREM B.
The set of strongly sutured
Riemannian metrics on a dimensional differentiable manifold M contains a residual set of metrics. The proof of Main Theorem B involves several technical difficulties but it isalso quite constructive: we build linking curves using the linking curve algorithm (although there is a non-deterministic step at which we have to choose one curvethat avoids some obstacles). In theorem 7.19, we prove that the algorithm alwaysproduces a special type of linking curve, starting on any conjugate point.The proof of the Ambrose conjecture in [H10] certainly works for a generic classof Riemannian manifolds of any dimension, and is shorter than the proof presentedhere. However, that proof does not seem to be extendable to arbitrary metrics.2inking curves, sutured manifolds and the Ambrose conjectureIndeed, the class of Riemannian manifolds for which we prove the Ambrose conjec-ture is not contained in the corresponding class in [H10], so this is truly a differentapproach. In the last section, we show how the ideas in this paper could be usedto complete the proof of the conjecture for all Riemannian manifolds.For the proof of these results we have introduced some new concepts that webelieve are interesting in their own sake, such as linking curves and synthesis man-ifolds in section 5 or conjugate descending flow in section 7.2.The outline of the paper is as follows: In section 3 we interpret the proof in [H82]in our own terms and show why it only works in dimension 2. In section 4 we studytree-formed curves and prove lemma 4.3 about the affine development of curves inmanifolds with L -related curvature. In section 5 we define quasi-continuous linkingcurves, unequivocal sets and synthesis manifolds, and prove our Main Theorem A.In section 6 we collect useful results about the exponential map for a generic metric.In section 7 we define conjugate descending curves, prove that they are unbeatable ,define finite conjugate linking curves (FCLCs), and prove that they can be builtfor a generic metric using the linking curve algorithm .The results in this paper are mostly included in the author’s thesis [A], but theyhave been reorganized to make it more clear and more general, and a few short butpowerful results have been added. We warn the reader of that document that somedefinitions have changed with respect to that document.1.1. Acknowledgments.
We thank Yanyan Li and Biao Yin, who introduced theauthor to the Ambrose conjecture. We also thank Luis Guijarro and James Hebdafor their support and suggestions. 2.
Notation M is an arbitrary Riemannian manifold, p a point of M , ( M , p ) and ( M , p ) are two Riemannian manifolds with L -related curvature.Let e stand for exp p and e for exp p ◦ L .We denote by Cut p , the cut locus of M with respect to p (see chapter 5 of[CE] for definitions and basic properties). Let us 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 forall (cid:54) t (cid:54) , and let TCut p = ∂O p be the tangent cut locus . The set TCut p mapsonto Cut p by exp p .In our proof, we will make heavy use of a subset of t p M bigger than the injectivityset, defined as follows. We define the functions λ k : S p M → R , where λ k ( x ) isthe parameter t ∗ for which t ∗ · x is the k -th conjugate point along t → exp p ( tx ) (counting multiplicities). These functions were shown to be Lipschitz in [IT00]. In[CR], it was shown that λ is semiconcave. Together with L.Guijarro, the authorproved in [AGII] that these functions are also Lipschitz in Finsler manifolds. Wedefine V as the set of tangent vectors x such that | x | (cid:54) λ ( x/ | x | ), a set withLipschitz boundary. It is well known that O p ⊂ V .3. James Hebda’s tree formed curves
Tree formed curves.
Let AC p ( X ) be the space of absolutely continuouscurves in the manifold M starting at p , with the topology defined as in [H87].Affine development Dev p : AC p ( M ) → AC ( T p M ) for absolutely continuouscurves is also defined in that reference, extending the common definition in [KN]. Tree-formed curves are similar to the tree-like paths of the theory of rough paths(see [HL]), but we will stick to the original definition in [H87]. The model for atree-formed curve u : [0 , → M is an absolutely continuous curve that factorsthrough a finite topological tree Γ . In other words, u = ¯ u ◦ T for some continuous3ablo Angulo-Ardoymap ¯ u : Γ → M and a map T : [0 , → Γ with runs through each edge of the treeexactly twice, in opposite directions. The tree Γ is the topological quotient of theunit interval by the map T .The definition also allows for a “partial identification”. Definition 3.1.
Let T : [0 , → Γ be a quotient map, and u an absolutely contin-uous curve. Then u is tree formed with respect to T if and only if (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 (thismeans that T ( s ) = T ( s ) implies ϕ ( s ) = ϕ ( s ) ), and for any t , t such that T ( t ) = T ( t ) .If T (0) = T (1) , we say the curve is fully tree-formed . If 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 to anidentification map with T ( t ) = T ( t ) as another way to say that u | [ t ,t ] is a fullytree-formed curve. Theorem 3.2 ([H87, Theorem 3.3]) . Tree formedness is preserved by affine devel-opment: • If u ∈ AC p ( M ) is tree formed for an identification T , then Dev p ( u ) ∈ AC ( T p M ) is also tree formed for the same T . • If v ∈ AC ( T p M ) is tree formed for an identification T , then Dev − p ( v ) ∈ AC p ( M ) is also tree formed for the same T . The proof of the Ambrose conjecture for surfaces by James Hebda.
In this section we give a sketch of the paper [H87], which is important for latersections. The reader can find more details in that paper.Theorem 1.2 shows that ϕ = exp ◦ L ◦ (exp | U p ) − is an isometric immersionfrom U p = M \ Cut p into M . The starting idea is to prove that whenever apoint 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 anisometry at least on U p .It is a well-known fact that the cut locus looks specially simple at the cleavepoints , for which there are exactly two minimizing geodesics from p , and bothare non-conjugate (see [Oz], for example). Near a cleave point, the cut locus isa smooth hypersurface. The rest of the cut locus is more complicated, but weknow that H n − (Cut \ Cleave) = 0 and, indeed, that
Cut \ Cleave has Hausdorffdimension at most n − , for a smooth Riemannian manifold.An isometric immersion from M \ A into a complete manifold, for any set A such that H n − ( A ) = 0 , can be extended to an isometric immersion 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 curve Y whoseimage is contained in TCut p (in the metric space AC( T p M ) of absolutely contin-uous curves) such that Y (0) = x , Y (1) = x , and e ◦ Y : [0 , → M is fullytree-formed .James Hebda proves in lemma 4.1 of [H87] that this implies that e ( x ) = e ( x ) .We extend that lemma in our lemma 4.4, so that it is simpler to use, and moregeneral. This is an important concept for this paper: Definition 3.3. A linking curve is an absolutely continuous curve Y : [0 , l ] → T p M such that e ◦ Y is a fully tree formed curve. Definition 3.4. A cut locus linking curve is a linking curve Y whose image iscontained in the tangent cut locus, so that e ◦ Y is a fully tree formed curve withimage contained in the cut locus. J. Hebda’s way to find the cut locus linking curves works only in dimension .Let S p M be the 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 on the ray. The tangent cut locus is parametrizedby θ → ( ρ ( θ ) , θ ) , defined on the subset of S p M where ρ is finite. Given a cleavepoint q = e ( x ) = e ( x ) , with x i = ( ρ ( θ i ) , θ i ), then ρ is finite in at least one ofthe 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 ] , satisfies the previous hypothesis.It is important that Y be absolutely continuous, which follows once it is provedthat ρ is. This was shown independently in [H94] and [I], and later generalized toarbitrary dimension in [IT00].3.3. Difficulties to extend the proof to dimension higher than . In dimen-sion higher than , there is no natural choice for a cut locus linking curve joiningthe speed vectors of the two minimizing geodesics that reach a cleave point. Indeed,we can show that for some manifolds it is impossible to do so: Theorem 3.5.
Let M be a smooth manifold of dimension , and p a point in M .There is an open subset of the set of smooth Riemannian metrics such that for anycleave point q = exp p ( x ) = exp p ( x ) from p , there is not CutLC whose extremaare x and x .Proof. Using the main theorem in [We2], there is a metric g on M whose tangentcut locus from p does not contain conjugate points (in other words, all segmentswith P as one endpoint are non-conjugate). Any metric sufficiently close to g willalso have disjoint cut and conjugate locus.Let q = exp p ( x ) = exp p ( x ) be a cleave point and Y : [0 , L ] → T p M be aCutLC joining x and x , where T : [0 , → Γ is the identification map of exp p ◦ Y .We can change the parameter t : [0 , l ] → [0 , L ] so that ( u ◦ t )( s ) has unit speed(the identification is reparametrized accordingly ( T ◦ t )( s ) ). We simply assume thatthe speed vector of u has norm one wherever it is defined and keep the letter t forthe parameter.Let t ∗ = L/ The following possibilities may occur:(1) There is some t (cid:54) = t ∗ such that T ( t ) = T ( t ∗ ) .(2) There is some ε > such that any point in [ t ∗ − ε, t ∗ + ε ] is not identifiedto other points by T .(3) There is a sequence t n → t ∗ and a sequence r n → t ∗ such that r n (cid:54) = t n and T ( r n ) = T ( t n ) (4) There is some ε > , a sequence t n → t ∗ and a sequence r n such that | r n − t ∗ | > ε and T ( r n ) = T ( t n ) .The second option is in contradiction with the hypothesis. The reason is thatfor any continuous -form ϕ along u | [ t ∗ − ε,t ∗ + ε ] , we must have (cid:90) ϕ ( s )( u (cid:48) ( s )) ds = (cid:90) t ∗ + εt ∗ − ε ϕ ( s )( u (cid:48) ( s )) ds = 0 but since T does not identify points in [ t ∗ − ε, t ∗ + ε ] to other points, we can choosethe continuous -form ϕ | [ t ∗ − ε/ ,t ∗ + ε/ freely, and this implies that u (cid:48) is null on thatinterval, which is in contradiction with having unit speed.The fourth option implies the first, since a subsequence of the r n will convergeto some r which is not in ( t ∗ − ε, t ∗ + ε ) and so is different from t ∗ .5ablo Angulo-ArdoyIf the third option holds, since T ( r n ) = T ( t n ) implies u ( r n ) = u ( t n ) , or exp p ( Y ( t n )) =exp p ( Y ( r n )) , any neighborhood of Y ( t ∗ ) contains a pair of different points with thesame image, which implies that exp p is not a local diffeomorphism at Y ( t ∗ ) , incontradiction with the fact that the image of Y is contained in the tangent cutlocus, which does not contain conjugate points.Only the first option remains. In this case, it follows from definition 3.1 that thecurve Y | [ t ,t ∗ ] is tree formed, for the identification T | [ t ,t ∗ ] (if t ∗ < t , we restrict to [ t ∗ , t ] ). The length of [ t , t ∗ ] is smaller than L/ and Y | [ t ,t ∗ ] is also a CutLC. Wecan iterate the argument to get a sequence of nested closed intervals whose lengthdecreases to . The point in the intersection of that sequence is a conjugate point,by a similar argument as in the third option above, and this is again a contradiction. (cid:3) Affine development and tree formed curves
In this section we extend the main results of sections 3 and 4 in [H87].For this whole section, let ( M , p ) and ( M , p ) be two manifolds with L -relatedcurvature. Definition 4.1.
The local linear isometry induced by x ∈ T p M is defined by I x = P γ ◦ L ◦ P − γ where γ is the geodesic on M with γ (0) = x , γ is the geodesic on M with γ (0) = L ( x ) and P α is the parallel transport along the curve α . Remark.
Since parallel transport along γ ∈ AC p ( M ) depends continuously on γ (see [H87, 6.1,6.3]), the map x → I x is continuous. Lemma 4.2.
Let x ∈ T p M be a regular point of e , and O any neighborhood of x in T p M such that e | O is injective. Let f x be the local isometry e ◦ L ◦ ( e | O ) − from e ( O ) in M to e ( O ) in M . Then I x = d e ( x ) f x Proof.
See lemma 1.35 in [CE] (cid:3)
We define
Dev i : AC( M i ) → AC( T p i M i ) as the affine development of absolutelycontinuous curves in M i based at p i , for i = 1 , . Lemma 4.3.
Let Y : [0 , l ] → V be an absolutely continuous curve such that Y (0) = 0 . Then: (1) I Y ( l ) = P v ◦ L ◦ P − u . (2) At any point t where u (cid:48) ( t ) and v (cid:48) ( t ) are defined, I Y ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) . (3) v = (Dev ) − ◦ L ◦ Dev ( u ) .Proof. We first assume that the image of the curve Y is contained in the interiorof V . Notice that if Y is a radial line, the first statement is just the definition of I Y ( l ) . Define: J = (cid:110) t : L ◦ P − u | [0 ,t ] = P − v | [0 ,t ] ◦ I Y ( t ) (cid:111) We will prove that J = [0 , l ] by proving it is open and close.If [0 , t ) ⊂ J , we take a sequence t j (cid:37) t to find by continuity of parallel transportand L ◦ P − u [0 ,tj ] = P − v [0 ,tj ] ◦ I Y ( t j ) that L ◦ P − u [0 ,t ] = P − v [0 ,t ] ◦ I Y ( t ) , so closednessfollows.Assume now [0 , t ] ⊂ J . Y ( t ) is in the interior of V by hypothesis, so there is ε > and a neighborhood O of Y | [ t − ε,t + ε ] , and an isometry ϕ : e ( O ) → e ( O ) with ϕ ◦ e | O = e | O . Then for any t < t < t + εP − u | [0 ,t = P − u | [0 ,t ] ◦ P − u | [ t,t v . By hypothesis, L ◦ P − u | [0 ,t ] = P − v | [0 ,t ] ◦ I Y ( t ) . We have ϕ ◦ e | O = e | O so, as parallel transport commutes with isometries, wehave P − v | [ t,t ◦ I Y ( t ) = P − v | [ t,t ◦ d u ( t ) ϕ = d u ( t ) ϕ ◦ P − u | [ t,t = I Y ( t ) ◦ P − u | [ t,t It follows that [ t, t + ε ) ⊂ J , so J is open and the first item follows when the imageof Y is contained in the interior of V .We next prove that I Y ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) for any t ∈ [0 , l ] such that Y (cid:48) ( t ) is defined.This is clear when Y ( t ) is in the interior of V , because, for an isometry f definedin a neighborhood O of Y ( t ) where e | O is injective and Y ([ t − ε, t + ε ]) ⊂ O , wehave v | [ t − ε,t + ε ] = f ◦ u | [ t − ε,t + ε ] and I Y ( t ) = d u ( t ) f .We now deal with curves whose image intersects the boundary of V . Such acurve Y can be approximated in AC ( T p M ) by curves Y k ( t ) = (1 − k ) Y ( t ) thatstay in int ( V ) , so that | Y k − Y | AC ( T p M ) → . Taking limits as k goes to infinite,the first item follows by continuity of parallel transport, the second because x → I x is continuous and by a standard use of the chain rule.The third claim is equivalent to Dev ( v ) = L ◦ Dev ( u ) , and this follows byintegration if we prove(4.1) L ( P − u | [0 ,t ] ( u (cid:48) ( t ))) = P − v | [0 ,t ] ( v (cid:48) ( t )) for almost every t ∈ [0 , But this clearly follows from the two earlier items. (cid:3)
Remark.
We will only need the above lemma, but it is worth mentioning that theabove also holds for a more general path Y : [0 , → T p M . There are (at least)two ways to do it:(1) As the set of singular points of e is a Lipschitz multi-graph (see TheoremA of [IT00]), we can approximate Y by paths that are transverse to the setof conjugate points. The proof that J is open at an intersection point t consists of gluing two intervals ( t − ε, t ) and ( t , t + ε ) where e is notsingular, and continuity of I X makes the gluing possible.(2) The above approach is straightforward but poses some technical difficulties.An alternative approach is to approximate the metric by a generic one and Y by a generic path in T p M . The manifold M with the new metric willno longer have curvature L -related to that of M , but the local maps I X can still be defined as a continuous family of linear isomorphisms. Thepath Y will cross the set of conjugate points transversally, and only at A singularities, which will simplify the proof. Lemma 4.4.
Let Y : [0 , l ] → T p M be a linking curve whose image is containedin V . Then: • e ( Y (0)) = e ( Y ( l )) . • I Y (0) = I Y ( l ) Proof.
Let r : [0 , → V be the radial path from to Y (0) . Define u = e ( r ∗ α ) and v = e ( r ∗ α ) , which are absolutely continuous curves defined on the interval [0 , l + 1] . Then v = Dev − ◦ L ◦ Dev ( u ) by the previous lemma.By its definition, u is tree-formed for an identification map T with T (1) = T ( l + 1) , so it follows that v also is, by theorem 3.2. It follows that e ( Y (0)) = v (1) = v ( l + 1) = e ( Y ( l )) . 7ablo Angulo-ArdoyFor the second claim, observe that: I Y ( l ) = P e ◦ ( r ∗ Y ) ◦ L ◦ P − e ◦ ( r ∗ Y ) = P e ◦ r ◦ P e ◦ Y ◦ L ◦ P − e ◦ Y ◦ P − e ◦ r We can simplify this expression, since both e ◦ Y and e ◦ Y are fully tree formed: I Y ( l ) = P e ◦ r ◦ L ◦ P − e ◦ r and that is the definition of I Y (0) . (cid:3) Synthesis
For any point x ∈ int ( V ) , the Cartan lemma provides an isometry from aneighborhood of e ( x ) to one of e ( x ) . Our plan is to collect those local mappingsto build a covering space. Definition 5.1.
A Riemannian covering is a local isometry that is also a coveringmap (see [O] for a motivation).
Definition 5.2.
Let A be a topological manifold, X , X are Riemannian mani-folds, and e : A → X , e : A → X are continuous surjective maps.A synthesis of X and X through e and e is a Riemannian manifold X ,together with a continuous map e : A → X , and Riemannian coverings π i : X → X i for i = 1 , such that π i ◦ e = e i . 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 If π i are only local isometries, then X is called a weak synthesis . We will use this extension of the Ambrose conjecture in terms of synthesis man-ifolds (see section 3 in [O]).
Conjecture 5.3 ( Ambrose Conjecture following O’Neill ) . Let ( M , p )and ( M , p ) be two Riemannian manifolds with L -related curvature. Define e =exp p and e = L ◦ exp p .Then there is a synthesis M of M and M through e and e , and a point p ∈ M such that π i ( p ) = e i (0) for i = 1 , .In particular, if M and M are simply-connected, then π ◦ π − : ( M , p ) → ( M , p ) is the unique isometry whose tangent at p is L . If e has no singularities, we can pull the metric from M onto T p M andthe desired Riemannian coverings are π = e and π = e . In the presence ofsingularities, the idea is to build the synthesis as a quotient of A = V that identifiespairs of points with the same image by both e and e .5.1. Unequivocal points and linked points.Definition 5.4.
We say that an open set O ⊂ V is unequivocal if and only if e ( O ) is an open set, and there is an isometry ϕ O : e ( O ) → e ( O ) such that ϕ O ◦ e | O = e | O . Definition 5.5.
We say x ∈ V is unequivocal if there is a sequence of unequivocalsets W n such that e ( W n ) is a neighborhood base of e ( x ) . Remark.
The above definition allows for points x ∈ V that are not isolated in e − ( e ( x )) . This is important if we want a definition of sutured manifold that mayhold for all Riemannian manifolds.We plan to identify points in T p M that are joined by a linking curve. However, inorder to build a quotient space, we need some kind of openness as in Lemma 5.11.In order to define a relaxed version of the above relation for which Lemma 5.11holds, we need to allow curves with some sort of “controlled” discontinuities. Definition 5.6. A quasi-continuous linking curve is a bounded curve Y : [0 , l ] → V such that: (1) The composition e ◦ Y is an absolutely continuous tree formed curved. (2) For every point t , there is an ε > such that either Y | [ t − ε,t + ε ] is abso-lutely continuous, or its image is contained in an unequivocal set W . Definition 5.7.
Two points x, y ∈ V are strongly linked (by the curve Y ) iffthere is a linking curve Y : [0 , l ] → V such that Y (0) = x and Y ( l ) = y .Two points x, y ∈ V are linked ( x (cid:33) y ) if and only if there is a quasi-continuous linking curve Y : [0 , l ] → V such that x is the limit of Y ( t j ) for somesequence t j (cid:38) and y is the limit of Y ( t j ) for some sequence t j (cid:37) l . Main properties of unequivocal sets and linked points.
In this sectionwe extend some results from section 4.
Lemma 5.8.
Let W be any unequivocal neighborhood of x ∈ T p M . Let ϕ : e ( W ) → e ( W ) be the local isometry such that ϕ ◦ e = e . Then I x = d e ( x ) ϕ In particular, it depends only on e ( x ) .Proof. If x is a regular point of e , we know from lemma 4.2 that I x = d e ( x ) f x .Both f x and ϕ are isometries that agree on the open set e ( U ) , for an open set U ⊂ W such that e is injective when restricted to U . Thus f x and ϕ agree on U and the result follows.For a conjugate point x , we take limits of a sequence of regular points, since I z = d e ( z ) ϕ for any regular point z ∈ W , and z → I z is continuous. (cid:3) Lemma 5.9.
Let Y : [0 , l ] → V be a bounded curve such that: • Y (0) = 0 . • u = e ◦ Y and v = e ◦ Y are absolutely continuous. • For every point t , there is an ε > such that either Y | [ t − ε,t + ε ] is abso-lutely continuous, or its image is contained in an unequivocal set W .Then: (1) I Y ( l ) = P v ◦ L ◦ P − u . (2) At any point t where u (cid:48) ( t ) and v (cid:48) ( t ) are defined, I Y ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) . (3) v = (Dev ) − ◦ L ◦ Dev ( u ) .Proof. Define J as in lemma 4.3: J = (cid:110) t : L ◦ P − u | [0 ,t ] = P − v | [0 ,t ] ◦ I Y ( t ) (cid:111) We know that I Y ( t ) is continuous at every t where Y is continuous. By the lasthypothesis and the previous lemma, I Y ( t ) is also continuous at points where Y isdiscontinuous. It follows that J is closed and it remains to prove that it is open.Let t ∈ J .If Y | [ t − ε,t + ε ] is absolutely continuous and its image is contained in int ( V ) , weprove that [ t , t + ε ) ⊂ J as we did in Lemma 4.3. If Y | [ t − ε,t + ε ] is containedin an unequivocal set W , there is an isometry ϕ : e ( W ) → e ( W ) such that9ablo Angulo-Ardoy ϕ ◦ u | [ t − ε,t + ε ] = v | [ t − ε,t + ε ] so, as parallel transport commutes with isometries,and using Lemma 5.8, we have, for t < t < t + εP − v | [ t ,t ◦ I Y ( t ) = P − v | [ t ,t ◦ d u ( t ) ϕ = d u ( t ) ϕ ◦ P − u | [ t ,t = I Y ( t ) ◦ P − u | [ t ,t and we deduce that [ t , t + ε ] ⊂ J as in Lemma 4.3.Finally, if Y | [ t − ε,t + ε ] is absolutely continuous but its image is not contained in int ( V ) , we define for every k a modified curve: Y t ,ε,k ( t ) = Y ( t ) t ≤ t (cid:0) − εk ( t − t ) (cid:1) Y ( t ) t < t ≤ t + ε (cid:0) − k (cid:1) Y ( t ) t + ε < t Since Y t ,ε,k | [ t − ε,t + ε ] is absolutely continuous and its image is contained in int ( V ) ,we learn that for any t < t + εL ◦ P − u k | [0 ,t ] = P − v k | [0 ,t ] ◦ I Y t ,ε,k ( t ) and since Y k converges to Y in AC ( T p M ) , we have proven that [ t , t + ε ) ⊂ J .We now turn to the proof that I Y ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) for almost every t ∈ [0 , . Wehave already shown this if Y | [ t − ε,t + ε ] is absolutely continuous and Y ( t ) belongs to int ( V ) . If Y | [ t − ε,t + ε ] is absolutely continuous but Y ( t ) does not belong to int ( V ) ,we construct the same curves Y t ,ε,k : we know that for every t ∈ [0 , for which u (cid:48) ( t ) and v (cid:48) ( t ) are defined, we have I Y t ,ε,k ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) . Since I Y t ,ε,k ( t ) converges to I Y ( t ) as k goes to infinity, it follows that I Y ( t ) ( u (cid:48) ( t )) = v (cid:48) ( t ) .Finally, if Y | [ t − ε,t + ε ] is contained in an unequivocal set W , let ϕ : e ( W ) → e ( W ) be the isometry in the definition of unequivocal set. By lemma 5.8 I Y ( t ) ( u (cid:48) ( t )) = ( d u ( t ) ϕ )( u (cid:48) ( t )) = ( ϕ ◦ u ) (cid:48) ( t ) = v (cid:48) ( t ) The third item follows from the first and the second as in Lemma 4.3. (cid:3)
Lemma 5.10.
Let x, y ∈ V be linked points. Then (1) e ( x ) = e ( y ) (2) e ( x ) = e ( y ) (3) I x = I y Proof.
Let Y be a quasi-continuous linking curve that links x and y .The first part is obvious from the definition because e ( x ) and e ( y ) are theextrema of the fully tree formed curve e ◦ Y .The second and third parts follow as in 4.4, because the curve r ∗ Y satisfies thehypothesis of lemma 5.9. (cid:3) Lemma 5.11.
Let x ∈ V be linked to some z ∈ W for an unequivocal set W . Thenthere is a neighborhood U ⊂ V of V that contains x and such that every y ∈ U islinked to some w ∈ W .Proof. We define U as the connected component of e − ( e ( W )) ∩ V that contains x . For y ∈ U , we want to prove that y is linked to some w ∈ W .Let Z : [0 , l ] be a quasi-continuous linking curve that joins x and z . We want tofind curves A : [0 , a ] → U and B : [0 , a ] → W such that e ◦ ( A ∗ Z ∗ B ) is fullytree formed, A (0) = y . This holds if we choose an arbitrary absolutely continuouspath A with A (0) = x and A ( a ) = y , and B ( t ) so that e ( B ( t )) = e ( A ( a − t )) . Wemay not be able to choose an absolutely continuous path B , but since its image iscontained in W , Y is a quasi-continuous linking curve. (cid:3) Remark.
Such a choice of B is not very elegant, and requires using the axiom ofchoice. The interested reader can find a more constructive alternative in the proofof Proposition 5.15.5.3. Construction of the synthesis.Theorem 5.12.
Let M , M be Riemannian manifolds with L -related curvature,such that for every x ∈ V is linked to some unequivocal point y ∈ V .Then there is a weak synthesis of M and M through e and e .Proof. Define a set M as a quotient by the equivalence relation: M = ( A/ (cid:33) ) Let e : A → M be the projection map. We define maps π i : M → M i by π i ([ x ]) = e i ( x ) . Both maps are well defined by lemma 5.10.We give M a topology where the basic open sets are [ W ] = { [ x ] , x ∈ W } , forunequivocal open set W . • By hypothesis, every point belongs to some open set, so this is a good basisfor a topology. • e is continuous at every point x ∈ A : Let [ W ] be a basis open neighborhoodof [ x ] , for W unequivocal. There is z ∈ W such that x (cid:33) z , by a quasi-continuous linking curve ρ . By lemma 5.11, there is an open neighborhood U such that any point in U is linked to some point in W . Thus U iscontained in e − ([ W ]) . • π | [ W ] is injective for any basis open set [ W ] : Let [ x ] , [ x ] ∈ [ W ] be suchthat π ([ x ]) = π ([ x ]) . This means e ( x ) = e ( x ) . We can assume x , x ∈ W , which implies [ x ] = [ x ] (using a curve Y that only takes thevalues x and x ). • π | [ W ] is injective for any basis open set [ W ] : If π ([ x ]) = π ([ x ]) for x , x ∈ W , it follows that e ( x ) = e ( x ) , which implies ϕ W ( e ( x )) = ϕ W ( e ( x )) , for the isometry ϕ W in the definition of unequivocal set, whichimplies e ( x ) = e ( x ) and [ x ] = [ x ] . • 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 an open set, because e ( W ∩ e − ( U )) = e ( W ) ∩ U is an open set and W ∩ e − ( U ) ⊂ W , so W ∩ e − ( U ) is unequivocal. • π | [ W ] is continuous, for a basis set [ W ] : let U be an open subset of π ([ W ]) = e ( W ) , and let ϕ W : e ( W ) → e ( W ) be the isometry asso-ciated with W . Then ( π | [ W ] ) − ( U ) = [ W ] ∩ π − ( U ) = [ W ∩ e − ( U )] .This is an open set, because e ( W ∩ e − ( U )) = ϕ − W ( e ( W ∩ e − ( U )) = ϕ − W ( e ( W ) ∩ U ) = ϕ − W ( ϕ W ( e ( W )) ∩ U ) is an open set and W ∩ e − ( U ) ⊂ W , so W ∩ e − ( U ) is unequivocal. • For a basis open set [ W ] , π ([ W ]) = e ( W ) is open by hypothesis, and π ([ W ]) = e ( W ) = ϕ ( e ( W )) is also open. Hence, π i is open for i = 1 , .Thus, π i | [ W ] is an homeomorphism onto its image. • Since π and π are local homeomorphisms, we can use π , for instance,to give M the structure of a smooth Riemannian manifold, which triviallymakes π a local 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 ) onto π ([ W ]) = e ( W ) , so π is also a local isometry. (cid:3) Compactness.
In order to prove Theorem A, we still have to prove that π and π given by theorem 5.12 are covering maps. This requires some sort of“compactness” result, and using the extra hypothesis in the definition of a suturedmanifold. We start with a general lemma: Lemma 5.13.
Let exp p : T p M → M be the exponential map from a point p in aRiemannian manifold M . Then for any absolutely continuous path x : [0 , t ] → T p M , the total variation of t → | x ( t ) | is not greater than the length of 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) t | (exp p ◦ x ) (cid:48) | = (cid:90) t | d exp p ( x (cid:48) ) | The speed vector x (cid:48) = ar + v is a linear combination of a multiple of the radialvector and a vector v perpendicular to the radial direction. By the Gauss lemma, | d exp p ( x (cid:48) ) | = (cid:113) a + | d exp p ( v ) | ≥ | a | . On the other hand, v is tangent to thespheres of constant radius, so: T V t ( | x | ) = (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) ddt | x | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) t | a | ≤ length(exp p ◦ x ) (cid:3) Let us now come back to our hypothesis.
Definition 5.14.
Let ( M , p ) is sutured, and ( M , p ) a manifold with L -relatedcurvature. Let M be the weak synthesis obtained by application of Theorem 5.12and p = e (0) ∈ M .Then the synthesis-distance to p is the function d : M → R given 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 the followingproposition would be trivial. Proposition 5.15.
The synthesis-distance d is a -Lipschitz function on M : | d ( q ) − d ( q ) | ≤ d M ( q , q ) Proof.
Claim.
Given q , q ∈ M and ε > , there is a family of absolutely contin-uous paths β k : [0 , l k ] → V , for k integer, with the following properties: • the curves β k are parametrized so that e ◦ β k has unit speed. This isequivalent to asking that e ◦ β k has unit speed, since π is a local isometry.In particular, l k = length( e ◦ β k ) = length( e ◦ β k ) . • β (0) ∈ e − ( q ) and | β (0) | < d ( q ) + ε/ . • e ( β k ( l k )) → q . • for each k : | β k +1 (0) | ≤ | β k ( l k ) | . • (cid:80) ∞ k =1 l k < d M ( q , q ) + ε/ .The family of curves may be finite or infinite. We will assume the latter, since theformer is strictly simpler. 12inking curves, sutured manifolds and the Ambrose conjectureFrom this and Lemma 5.13 it follows that(5.1) | β N ( l N ) | = | β (0) | + (cid:80) N − k =1 ( | β k +1 (0) | − | β k (0) | ) + ( | β N ( l N ) | − | β N (0) | ) < | β (0) | + (cid:80) Nk =1 ( | β k ( l k ) | − | β k (0) | ) < | β (0) | + (cid:80) Nk =1 l k < d ( q ) + d M ( q , q ) + ε Thus the points β N ( l N ) are bounded and a subsequence of them converge to some x ∈ V which belongs to e − ( q ) and satisfies the same bound. This proves theresult, and it remains to prove the claim.By [H87, 1.1], e ( Sing \A ) has null H n − measure. We define N = e ( Sing \A ) .It follows that N has null H n − measure because N ⊂ π − ( e ( Sing \ A )) , and theimage of a H n − -null set by a local isometry is also H n − -null.Let R = d ( q ) + d M ( q , q ) + ε , and let B ( R ) ⊂ T p M be the open ball ofradius R . The set A ∩ B ( R ) is a smooth n − -manifold (non-compact withoutboundary). Any x ∈ A has a neighborhood U such that e ( U ∩ A ∩ B ( R )) and e ( U ∩ A ∩ B ( R )) are (isometric) smooth n − -manifolds.We start the construction of β choosing a starting point β (0) ∈ e − ( q ) suchthat | β (0) | < d ( q ) + ε/ . The point β (0) may be singular (that is, β (0) ∈ ∂V ),in which case we start β with a short straight path that reaches a new β ( ε/ ∈ int ( V ) ∩ B ( R ) \ e − ( e ( Sing ) ∩ B ( R )) . We can choose β to that its derivativemakes a positive angle with the kernel of the exponential, and this is all we need tochange the parameter so that e ◦ β has unit speed. So we assume that the lengthof e ◦ β is ε/ .By [H82, 4.3], if q / ∈ N , we can find a curve c disjoint from N joining e ( β ( ε/ with q whose length is not greater than d M ( e ( β ( ε/ , q )+ ε/ < d M ( q , q )+ ε/ .We remark that [H82, 4.3] requires that M is complete, something that we have notproved yet. However, the proof of [H82, 4.3] is valid also without this hypothesiswith minor modifications: • let v be a path in M joining e ( β ( ε/ and q , of length smaller than d M ( e ( β ( ε/ , q ) + ε/ • find a finite partition t < t < · · · < t N of the domain of the curve sothat two consecutive points v ( t i ) , v ( t i +1 ) lie in a strongly convex open set. • choose points c ( t ) = v ( t ) , c ( t N ) = v ( N ) and c ( t i ) using [H82, 4.2] (for K = N ) so that the length of c | [ t i ,t i +1 ] is smaller than the length of v | [ t i ,t i +1 ] + ε N .The resulting curve c does not intersect N and has length smaller than length ( v ) + ε/ < d M ( e ( β ( ε/ , q ) + ε/ .If q ∈ N , we can use a similar procedure: take a curve v : [ ε/ , → M from anearby point v ( ε/ into q of length smaller than ε/ and split it by intervals oflength ε/ k +1 (we start from k = 3 for convenience). We can then replace v by abroken geodesic c that avoids N such that the length of the segment c | [ ε/ k ,ε/ k +1 ] is no more than ε/ k . In this way we find a continuous curve c of length smallerthan ε/ that joins q to a point not in N .We also want a curve that is transverse to e ( A ∩ B ( R )) . This is equivalent tobeing transverse to each of the countably many smooth manifolds e ( U ∩A ∩ B ( R )) that we mentioned before. Since transversality to a smooth manifold is a residualproperty [H, 3.2.1], and a countable intersection of residual sets is residual, and inparticular dense, we can find a curve u : [0 , l ] → M joining β ( ε/ and c ( ε/ thatis close to c in the C ([0 , l ] , M ) topology, so that, in particular, the length of u isnot greater than d M ( q , q ) + ε/ , that is transverse to e ( A ∩ B ( R )) and does notintersect N except possibly at the final point.13ablo Angulo-ArdoyAssume that the intersection points of u and e ( A ∩ B ( R )) cluster at a point u ( t ∗ ) (cid:54) = u ( l ) . Then there is a sequence of points x j ∈ A ∩ B ( R ) and times t j → t ∗ such that e ( x j ) = u ( t j ) converge to u ( t ∗ ) . Since the x j are bounded, there is asubsequence converging to x ∗ ∈ Sing ∩ B ( R ) . If x ∗ ∈ A , since u ( t ∗ ) = e ( x ∗ ) , and u is transverse to e ( U ∩ A ) at t ∗ , there is δ > such that u | [ t ∗ − δ,t ∗ + δ ] does notintersect e ( U ∩ A ) , which is a contradiction with the fact that a subsequence ofthe x j converge to x ∗ . If x ∗ ∈ Sing \ A , the contradiction is with the fact thatthe image of u does not intersect N . Note however that it is perfectly possible thatthe intersection points of u and e ( A ∩ B ( R )) cluster at the final point u ( l ) .We have shown that the set of intersection points of u and e ( A ∩ B ( R )) isdiscrete < t < · · · < t j < . . . except at the limit j → ∞ , and is bounded by l .Since β ( ε/ is in int ( V ) , we can start a lift of u from that point. Using theargument of equation 5.1, we see that the curve β will stay in B ( R ) . Thus, thelift will stay in int ( V ) up to t since u | [0 ,t ) does not intersect e ( Sing ) ∩ B ( R ) , sowe get a curve β : [0 , l ] → V that may end in a point in A ∪ int ( V ) .If β ( l ) is in A , we can find a new unequivocal point β (0) that is linked to β ( l ) and with | β (0) | < | β ( l ) | . Since the point e ( β (0)) = e ( β ( l )) belongs tothe image of u and is unequivocal, it can only be a nonsingular point, so we canstart a new lift of u | [ t ,t ] , and so on. The claim follows easily. (cid:3) Proof of Main Theorem A.
We only need to prove that the weak synthesis M builtusing Theorem 5.12 is complete when ( M , p ) is sutured. It is well known thata local isometry is a covering map when the domain is complete (see for examplecorollary 2 in [G]).As mentioned in conjecture 5.3, this implies the original Ambrose conjecturewhen both manifolds are simply connected.Let q n be a Cauchy sequence in M . Then there is R > such that d M ( q n , q ) A generic perturbation of a Riemannian metric greatly simplifies the types ofsingularities that can be found on the exponential map ([We],[K]) or the cut locuswith respect to any point ([B77]). In [We], A. Weinstein showed that for a genericmetric, the set of conjugate points in the tangent space near 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 andthe endpoint fixed at a different point of the manifold, as a family of functionsparametrized by the endpoint. The singularities of the exponential map can be de-tected as degenerate degenerate critical points of the energy functional with bothendpoints fixed, so his results also apply to our setting. He also proved a multi-transversality statement about this family of functions that we will comment on14inking curves, sutured manifolds and the Ambrose conjecturelater, and then used this information to provide a description of the cut locus of ageneric metric.It is well known that a exponential map only has Lagrangian singularities. In[K], Fopke Klok showed that the generic singularities of the exponential maps arethe generic singularities of Lagrangian maps. These singularities are, in turn, de-scribed by means of the generalized phase functions of the singularities. This is theapproach most useful to our purposes. We also wish to mention [JM] for a differentapproach and generalizations of some of these results.6.1. Generalized phase functions. 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 transverse to { } ∈ R k . We will use a result that relates generalized phase functions defined at U × R k and Lagrangian subspaces of T ∗ U : Proposition 6.1. If L ⊂ T ∗ U is a Lagrangian submanifold and p ∈ L , it is locallygiven 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 and thecomments 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 an invertiblematrix. Let us split the x coordinates in ( y, z ) ∈ R n − k × R k . Our hypothesis isthat D qz F is invertible.The implicit equations D q F = 0 defines functions f j : R n − k × R k → R such that,locally near , D qz F ( y, f ( y, q ) , q ) = 0 . Definition 6.2. A Lagrangian map λ : L → M is the composition of a Lagrangianimmersion i : L → T ∗ M with the projection π : T ∗ M → M (a Lagrangian im-mersion is an immersion such that the image of sufficiently small open sets areLagrangian submanifolds). Definition 6.3. Two Lagrangian maps λ j =: L j → M j , with corresponding im-mersions i j : L → T ∗ M , j = 1 , , are Lagrangian equivalent if and only if thereare diffeomorphisms σ : L → L , ν : M → M and τ : T ∗ M → T ∗ M such thatthe following diagram commutes: 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 phase functions(this is proposition 1.2.6 in [K]). Two generalized phase functions are equivalent ifand only if 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 .15ablo Angulo-Ardoy(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 ))) The singularities of a generic exponential map. Using theorem 1.4.1 in[K], we get the following result: fix a smooth manifold M , a point p ∈ M . For aresidual set of metrics in M the exponential map T p M → M is nonsingular exceptat a set Sing , which is a smooth stratified manifold with the following strata (wedescribe 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 6.4. We define the sets A , A , etc as the set of all points of V thathave a singularity of type A , A , etc. We also define C as the set of conjugate(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 of order (including A , A and A points), and is diffeomorphic to the product of a cone in R with a cube near a conjugate point of order (including D ± ). The A pointsare characterized as those for which the kernel of the differential of the exponentialmap is a vector line transverse to the tangent plane to Sing .Furthermore, the image by exp p of each stratum of canonical singularities is alsosmooth. There might be strata of high codimension that are not uniform, in thesense that the exponential map at some points in those strata may not have thesame type of singularity (in other words, the singularities are non-determinate ).This only happens in some strata of codimension at least , and is not a problemfor our arguments.There are also other generic property that interests us: the image of the differentstrata intersect “transversally”:Take two different points x , x ∈ T p M mapping to the same point of M , andassume 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 transverse (eachpair of strata intersect transversally). This follows from proposition 1 in page 215of [B77], with p = 2 , so that j k H ( α ) is transverse to the orbit in R × [ J k ( n, where the first jet is of type T and the second one is of type T .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, the formulasnear 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 functions.We list the generalized phase functions and the corresponding coordinates for theexponential function that derives from it for the 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 6.5. The above expression is the canonical form of the exponentialmap at the singularity. The canonical form is only defined for the singularities inthe above list.We call adapted coordinates any set of coordinates on U ⊂ T p M and V ⊃ exp p ( U ) for which the expression of the exponential map is canonical. Definition 6.6. Let U be a neighborhood of adapted coordinates near a conjugatepoint x . The lousy metric on U is the metric whose matrix in adapted coordinatesis the identity. Remark. We call this metric “lousy” because it does not have any geometricmeaning, and it depends on the particular choice of adapted coordinates. However,it is useful for doing analysis.Although the adapted coordinates make the exponential map simple, radialgeodesics from p are no longer straight lines, and the spheres of constant radiusin T p M are also distorted. We do not know of any result that gives an explicitcanonical formula for the exponential map and also keeps radial geodesics in T p M simple. The results of section 7.8 suggest that this might be possible to some extent,but the classification that might derive from it must be finer than the one above.We will find examples showing that the radial vector 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 transverse to C , and thus must have r (cid:54) = 0 . Thereare two possibilities: • A point is A ( I ) if and only if r > . • A point is A ( II ) if and only if r < .Even though the exponential map has the same expression in both cases (for ade-quate coordinates), they differ for example in the following:Let x ∈ A ∩ V (a first conjugate point), and let U be a neighborhood of x ofadapted coordinates. Then exp p ( V ∩ U ) is a neighborhood of exp p ( x ) if and onlyif x is A ( I ) . A proof for this fact will be trivial after section 7.1.In fact, the above can be used as a characterization (for points in A ∩ V ) thatshows that the definition is independent of the adapted coordinates chosen. Weremark that in a sufficiently small 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 similar dis-tinction with D +4 points. Remark. In the literature, it is common to see singularities of real functions oftype A further subdivided into A +3 and A − points. A canonical form for an A ± F ± ( x , ˜ x , x , x , . . . , x n ) = ± x − x x − (cid:102) x x When F ± are generalized phase functions, both subtypes give equivalent singular-ities. However, in the work of Buchner, the same singularities appear, now as theenergy function in a finite dimensional approximation to the space of paths withfixed endpoints. In this second context, it is not equivalent if a geodesic is a localminimum, or a maximum, of the energy functional, and it would make sense touse the distinction between A +3 and A − , rather than the similar-but-not-the-samedistinction between A ( I ) and A ( II ) .This can also serve as an illustration that the classifications of singularities ofthe exponential map by F. Klok and M. Buchner are not equivalent, even thoughthe 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 6.7. We define H M as the set of Riemannian metrics for the smoothmanifold M such that the singular set of exp p is stratified by singularities of types A , A , A and D ± with the codimensions listed above, plus strata of different typeswith codimension at least , and such that the images of any two strata intersecttransversally as described above. Theorem 6.8. H M is residual in the set of all Riemannian metrics on M .Proof. This is the work of M. Buchner and F. Klok, as we have shown in thissection. (cid:3) Proof of Theorem B In the previous section we have classified the points of T p M for a generic Rie-mannian manifold according to the singularity of the exponential map at that point.We use that classification to split T p M into two sets, according to the role that theyplay when proving that the manifold is sutured: Definition 7.1. I = ( N C ∪ A ( I )) ∩ V J = ( A ∪ A ( II ) ∪ A ∪ D ± ) ∩ V Theorem 7.2. Points in I are unequivocal, and any point in J is strongly linkedto a point in I of smaller radius.Proof of Main Theorem B. By definition, V = I ∪ J for a metric in H M . By The-orem 7.2, all metrics on H M are sutured. Main Theorem B follows by applicationof Theorem 6.8. (cid:3) A ( I ) first conjugate points are unequivocal.Lemma 7.3. Any x ∈ V of type A ( I ) is unequivocal.Proof. Consider an A ( I ) point x in a manifold ( M , p ) whose curvature is L -related to the curvature of ( M , p ) , and use adapted coordinates near x = (0 , , ,in an arbitrarily small neighborhood O : • Define γ ( x , x ) = x . • Let A be the subset of O given by x < γ ( x , x ) . e maps diffeomorphically A onto a big subset of e ( O ) . Only the points with x = 0 , x (cid:62) aremissing. x is A ( I ) , so ¯ A ⊂ V , and e ( O ∩ V ) is open.18inking curves, sutured manifolds and the Ambrose conjecture • 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. • 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 unambiguous. • For a pair of linked points x = ( x , x , x , . . . , x n ) and ¯ x = ( − x , x , x , . . . , x n ) ,we have two different 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 suchthat e ( O ) = e ( O ) and we need to show that they agree. They both send p to the same point, and we only need to check that their differential at p is the same. These are the linear isometries I x and I ¯ x , and they agree by4.4. • We know that ϕ ◦ e ( x ) = e ( x ) , for x ∈ ¯ A . Let y ∈ O \ ¯ A . There is aunique point 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 toa geodesic segment with the same length, starting point and initial vector.We conclude ϕ ◦ e ( y ) = e ( y ) . (cid:3) Remark. The only place where we used that the point is A ( I ) is when we assumedthat A ⊂ V .7.2. Conjugate flow. We now introduce the main ingredient in the constructionof the linking curves. The idea in the definition of conjugate flow was used in lemma2.2 of [H82] for a different purpose.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 of points oforder by the rule:(7.1) D = (ker dF ⊕ < r > ) ∩ T C Definition 7.4. A conjugate descending curve (CDC) is a smooth curve, con-sisting only of A points, except possibly at the endpoints, and such that the speedvector to the curve is in D and has negative scalar product with the radial vector r .Therefore, the radius is decreasing along a CDC.The canonical parametrization of a CDC γ is the one that makes d exp p ( γ (cid:48) ) a unit vector. By Gauss lemma, it is also the one that makes dR ( γ (cid:48) ) = 1 . Definition 7.5. Let α : [0 , t ] → T p M be a smooth curve, and x ∈ T p M be a pointsuch that exp p ( x ) = exp p ( α ( t )) . A curve β : [0 , t ] → T p M is a retort of α startingat x if and only if α ( t ) (cid:54) = β ( t − 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 retort of 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 points are given by x = 0 , and no other point y (cid:54) = x maps to exp p ( x ) . Thus, there is a neighborhood U of any CDC such that any CDC contained in U has no non-trivial retorts containedin U . Lemma 7.6. 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. Furthermore: • | α (0) | − | α ( t ) | = length(exp p ◦ α ) • 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 part isstandard.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 theother 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 of constantradius, 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 . The result 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 must alsocontain a retort for that CDC. The “unbeatable” property of CDCs is interesting,because the radius decreases along a CDC and along the retort it never increasesas much as it decreased in the first place.7.3. CDCs in adapted coordinates near A points. As we mentioned in sec-tion 6, the radial vector field, and the spheres of constant radius of T p M , thathave very simple expressions in standard linear coordinates in T p M , are distortedin canonical coordinates. Thus, the distribution D and the CDCs do not alwayshave the same expression in adapted coordinates. In this section, we study CDCsnear an A point. We will use the name 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 firstconjugate 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 2Ddistribution D , spanned by r and ∂∂x . We extend this distribution to all of O inthe following way: Definition 7.7. 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 )) . The reader may check that D is integrable. Let P be the integral manifoldof D through x = (0 , , . We can assume P is a graph over the x , x plane: x = p ( x , x ) . A is transverse to D , so { x } = A ∩ P . The integral curve C of D through x is contained in P , and C \ { x } consists of two CDCs. We claimthat if the point is A ( I ) , the two CDCs descend into x , but if the point is A ( II ) ,they start at x and flow out of O . P is also obtained by flowing the CDC with theradial vector field.We can assume that r is close to r ( x ) in O . The tangent T x to the sphere ofconstant radius { y : R ( y ) = R ( x ) } must contain ∂∂x (the kernel of d exp p ) if x ∈ C ,20inking curves, sutured manifolds and the Ambrose conjectureby Gauss lemma, and we can assume that the angle between T x and ∂∂x is small if x (cid:54)∈ C .The curves { R ( x ) = R } ∩ P , for any R , are all smooth graphs over the x axis.We claim that the curve { R ( x ) = R ( x ) } ∩ P may not intersect x < x . Assumethat R ( y ) = R ( x ) for some y = ( y , y ) with y < and y > y . Then there isa curve { R ( x ) = R ( x ) − ε } ∩ P , for some < ε << , must intersect C at a point ( x , x , p ( x , x )) with x < , and the tangent to { R ( x ) = R } ∩ P must be ∂∂x .Taking coordinates, x , x in P , we see it is not possible that a graph ( x , t ( x )) over the x axis has t (0) < , t ( y ) > y , and intersect the curve C ∩ P only withhorizontal speed.It follows that x is a local maximum, or minimum, of R , within C . If r > ( A ( I ) points), then R ( x ) ≥ R ( x ) for x ∈ C , while r < ( A ( II ) points), implies R ( x ) ≤ R ( x ) for x ∈ C .Thus, A ( I ) points are terminal for the conjugate flow, but A ( II ) points arenot. We have proved the following: Lemma 7.8. In a neighborhood O of adapted coordinates near an A point x : • C is foliated by integral curves of D . • A is foliated by CDCs. • If x is A ( I ) , exactly two CDCs in O flow into each A point. If x is A ( II ) , exactly two CDCs in O flow out of each A point. • If x is A ( I ) , every CDC in O flow into some A point. If x is A ( II ) ,every CDC in O flow out of some A point. A joins. We can continue a CDC as long as it stays within a stratum of A points. As we have seen, a CDC may enter a different singularity. The mostimportant situation is when the CDC reaches an A point, because then we canstart a non-trivial retort right after the CDC.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)) . We can start a retort for this segment of CDC starting at the A point. The retort for this CDC is given explicitly by t → ( − t, t , x ( t )) .These 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 linked to ( − t, t , x ) . We say that the CDC and the retort given above are joined with an A join .7.5. Avoiding some obstacles. In order to build linking curves, it is simpler toreplace CDCs with curves that are close to CDC curves, but avoid certain “obsta-cles”. The following remark helps in that respect: A curve that is sufficiently C -close to a CDC is also unbeatable . Actually, wecan say more: the greater the angle between r x and ker d x e , the more we candepart from the CDC. Definition 7.9. The slack A x at a first order conjugate point x is the absolutevalue of the sine of the angle between D x and ker( d x e ) . Remark. The slack is positive if and only if the point is A Lemma 7.10. For any positive numbers R > and a > there are constants c > and ε > depending on M , R and a such that the following holds:If a smooth curve α : [0 , T ] → T p M of A points satisfies the following proper-ties: (1) | α (0) | (cid:54) R (2) (cid:104) α (cid:48) ( t ) , r α ( t ) (cid:105) < for all t ∈ [0 , T ] A α ( t ) > a for all t ∈ [0 , T ] (4) α (cid:48) ( t ) is within a cone around D of amplitude c for all t ∈ [0 , T ] then it holds that: • α is ε -unbeatable: any retort β satisfies | β ( T ) | − | β (0) | < ( | α (0) | − | α ( T ) | ) (1 − ε ) Proof. Fix a neighborhood U of adapted A coordinates that contains the image of α . We assume that one such U contains all of the image of α , otherwise we 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 assume that α has the canonical parametrization, so that α (cid:48) ( t ) = d ( t ) + p ( t ) ,with p ( t ) is orthogonal to d . Then | p ( t ) | < c | d ( t ) | = cA α ( t ) < ca .We compute | α ( T ) | − | α (0) | = (cid:90) T ddt | α | = (cid:90) T (cid:104) r, α (cid:48) (cid:105) ≤ T ( − ca ) An A point only has one preimage in U , so any retort β of α lies outside of 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 contains a ball around x of radius at least r , anumber which depends on a , R , and an upper bound on the differential of the slackin the ball of radius R . Thus, we can switch to a smaller ε > that depends onlyon 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 )) . Itfollows from the above that | b ( t ) | < − ε for some ε depending on ε and a lowerbound for the norm of the differential of x → (exp p ( x ) , d exp p ( x )) in the ball orradius R in T p M .We compute: | β ( T ) | − | β (0) | = (cid:90) T ddt | β | = (cid:90) T b ( t ) ≤ T (1 − ε ) and if c < aε / , we get | β ( T ) | − | β (0) | ≤ T (1 − ca ) − T ε / ≤ | α (0) | − | α ( T ) | − T ε / . We are using the canonical parametrization for α , so T is the length of the curve exp ◦ α . By lemma 7.6, T is also | α (0) | − | α ( T ) | . The result follows with ε = ε / . (cid:3) With this lemma, we can perturb a CDC slightly to avoid some points: Definition 7.11. 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 amplitude c ,where c is the constant in the previous lemma for R = | α (0) | . Conjugate Locus Linking Curves. Let us assume that we have an ACDC α : [0 , t ] → T p M starting at a point x ∈ J , whose interior consists only of A points and ending up in an A point. We 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 is alocal diffeomorphism and we can lift any curve. However, we might be unable tocontinue the retort up to x if the returning curve hits the set of conjugate points.22inking curves, sutured manifolds and the Ambrose conjectureIf we hit an A point y = (cid:102) α ( t ) , we can take a ACDC β : [0 , t ] → V starting atthis point and ending in an A point. If β has a retort ˜ β : [0 , t ] → T p M that endsup in a non-conjugate point ˜ β ( t ) , we can continue with the retort (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 α , (cid:102) α , β , ˜ β and (cid:102) α is a linking curve (see figure 1).There are a few things that may go wrong with the above argument: the retort (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 problem can be avoided if theACDCs are built to dodge some small sets, as we will see later. Then, if we assumethat a retort never meets J \ A , we can iterate the above argument whenevera retort is interrupted upon reaching an A point. We will prove later that theargument only needs to be applied a finite number of times.This is the motivation for the following definitions: Definition 7.12. A finite conjugate linking curve (or FCLC, for short) is acontinuous linking curve α : [0 , t ] → T p M that is the concatenation α = α ∗ . . . ∗ α n of ACDCs and non-trivial retorts of those ACDCs, all of them of finite length . We will build the FCLCs in an iterative way, as hinted at the beginning of thissection, by concatenation of ACDCs and retorts of those ACDCs. Definition 7.13. An aspirant curve is an absolutely 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 the same order, we can reach a tuple with no retorts , by iterationof the following 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 . • The extremal points of the α i are called the vertices of α . The vertices of 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 7.4. − a splitter : a vertex that joins two ACDCs whose concatenation is alsoa ACDC. − a hit : a vertex that joins a retort that reaches A ( I ) transversally, andan ACDC starting at the intersection point. − a reprise : a vertex that joins a retort that completes its task of replyingto a ACDC α j , and the retort for a different ACDC α i (it follows fromthe first condition that i < j ).The tip of alpha is its endpoint α ( t ) .The loose ACDCs in α = α ∗ . . . ∗ α k are the ACDC curves α j for which thereis no retort in α .An aspirant curve is saturated if there are no loose ACDCs. The three new types of vertices: splitters , hits and reprises , always come in packs.We have already shown one example of how they could appear, but we formalizethat construction in the following definition. Definition 7.14. A standard T consists of three vertices: a splitter, a hit and areprise, such that the six curves α i contiguous to the three points map to a T -shapedcurve, with two curves mapping into each segment of the T (see figure 1). IIIIII IV V VI I II III IV VVI1 2 3 4 5 1 2 3 45 Figure 1. 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 and IVare ACDCs, III is the retort of II, V is the retort of IV, and VI isthe retort of I. Vertices 2 and 4 are A joins, vertex 1 is a splitter,vertex 3 is a hit and vertex 5 is a reprise. There can be more thantwo segments between a splitter and its matching hit, and betweena hit and its matching reprise. Proposition 7.15. Let α = α ∗ · · · ∗ α k be a saturated aspirant curve between x, y ∈ T p M . Then: • | x | > | y |• α is an FCLC.Proof. The first part follows trivially from lemma 7.6 and its generalization, lemma7.10. Each pair of a ACDC and its retort adds a negative amount to | α (0) | = | x | .For the second part, let x ∼ y and x ∼ y whenever x = α i ( s ) lies on an ACDC α i defined on [0 , l i ] and y = α j ( l i − s ) on its retort α j defined on the same interval,so that α i ( t ) = α j ( l i − t ) . We also identify the triples of vertices that belong to eachstandard T. Let T : [0 , l ] → Γ be the identification map associated to the relation ∼ . We must show that u = exp ◦ α is tree-formed with respect to T : let t , t suchthat T ( t ) = T ( t ) , and ϕ a continuous -form along u ( ϕ ( s ) ∈ T ∗ u ( s ) M ) that factorsthrough Γ . Then we claim that:(7.2) (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 image of itsmatching retort. The curves in each such pair have the same image, and the inte-grals cancel out, as the integral of a -form is independent of the parametrization,and only differs by sign.Suppose first that t is in the domain of an ACDC α i and t lies in the retort α j of α i . We recall it is possible to reach an empty tuple by canceling adjacentpairs of an ACDC and its retort. Thus, in order to cancel α i and α j , it must bepossible to cancel all the curves α k with i < k < j . These curves can be matchedin pairs { ( α n , α m ) } ( n,m ) ∈P of ACDC and retort, with i < n < m < j for each pair24inking curves, sutured manifolds and the Ambrose conjecture ( n, m ) ∈ P . Then we have: (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, respectively,and such that T ( t ∗ ) = T ( t ∗ ) . The result follows because the integral 7.2 dependscontinuously on t and t . (cid:3) Existence of FCLCs. The goal of this section is to prove the existence ofan FCLC starting at an arbitrary point x ∈ J . The set { y : | y | < | x | , exp p ( y ) =exp p ( x )) } = { y j } is finite. This follows because { y : | y | (cid:54) | x |} can be covered witha finite amount of neighborhoods of adapted coordinates, and in any of them thepreimage of any point is a finite set. At least one y j realizes the minimum distancefrom p to q = exp p ( x ) , and must be either A ( I ) or NC (in other words, y ∈ I ).We will show that there is an FCLC joining x and one y j ∈ I , though it may notbe the one with minimal radius. Definition 7.16. 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 preimages of exp p ( x ) with radius smaller than | x | are N C or A . Definition 7.17. A GACDC is an ACDC α such that • Im( α ) is contained in C ∩ V . • for any y ∈ B | α ( t ) | ∩ A such that exp p ( α ( t )) = exp p ( y ) , exp p ◦ α istransverse to exp p ( A ∩ B ε ( y )) at t , for some ε > . In words, all possible retorts of a GACDC avoid all singularities that are not A and only meet A transversally. Definition 7.18. The linking curve algorithm is a procedure that attempts to buildan FCLC starting at a given point x ∈ V (see figure 2).It starts with the trivial aspirant curve α = { x } and updates it at each segmentby addition of one or more segments, to get a new aspirant curve. It only stops ifthe aspirant curve is saturated, and its tip is in I .The aspirant curve α = α ∗ . . . ∗ α k is updated following the only rule in thefollowing list that can be applied: Descent: If the tip of α k is a point in J , let γ be a GACDC contained in V that starts at x and ends up in an A point. We know that γ inter-sects SA in a finite 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 itsextrema. The new curve α ∗ α k +1 ∗ · · · ∗ α k + r ends up in an A point. Thenext step is an A join. A join: 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 α k ( T ) . 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 tip of α . This is possible for the same reason as before and, again, the new tipof α ∗ α k +1 will be N C , A or A , and the latter can not happen unless α ∗ α k +1 is a linking curve. Success!: If α is saturated and its tip is in I , then α is an FCLC, so wereport success and stop the algorithm. For completeness, the algorithm alsoreports success if α = { x } , for x ∈ I . Tip is A Tip is A Tip is NC or A Is there anyremainingloose ACDC?A j oinReply to the lastACDC that wasadded, starting atthe A tip.The new tip may beA , A or NC.StartStart with thetrivial curve {x}Is x unequivocal? DescentDescend alonga GACDC up toan A point. Success!RepriseReply to the loose ACDCthat appears last.The new tip may beA , A or NC.YesNo NoYes Figure 2. Flow diagram for the linking curve algorithm Remark. The algorithm can also be presented in a recursive fashion. We startwith some definitions: • T ip ( α ) = α ( T ) , for any curve α defined in an interval [0 , T ] . • 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 following rules: • 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 )))) The reader may have noticed that γ to γ r are discarded, and only γ is kept(the ACDC up to the first point in SA ). This actually causes a small technicalproblem, so we will use only the iterative version of the algorithm. Theorem 7.19. Let M be a manifold with a Riemannian metric in H M . (1) For any R > there is L > such that any GACDC starting at x ∈ J ∩ B R has length at most L , and can be extended until it reaches an A point. The algorithm 7.18 always reports “success!” after a finite number of steps,for any starting point x ∈ J . Definition 7.20. A pair ( S, O ) of open subsets of T p M with ¯ S ⊂ O , is transient if and only if for any point x in S ∩ J , any aspirant curve that starts at x canbe extended to either an aspirant curve with endpoint outside of O or an FCLCcontained in O .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 ending at y .A transient pair is positive if it has positive gain.A transient pair is bounded if there is a uniform bound for the length of anyaspirant curve contained in O . Lemma 7.21. For any point x of type NC, A , A , A , D +4 or D − , there is apositive bounded transient pair ( S, O ) , with x ∈ S . Lemma 7.21 is all we need to complete the proof of Main Theorem B: Proof of theorem 7.19. We prove the second part first.Define: R = sup (cid:26) R : for all x ∈ B R , the algorithm starting at x reportssuccess! after a finite amount of iterations (cid:27) We will assume that R is finite and derive a contradiction, thus showing that thealgorithm always reports success after a finite amount of iterations. Using lemma7.21, we cover B R by a finite number of neighborhoods { S i } Ni =1 , where ( S i , O i ) are bounded positive 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 ( S i , O i ) .Take a point x ∈ B R + ε and assume x ∈ S . By hypothesis we can find anaspirant curve α with endpoint y outside of O .Thanks to the way we have chosen ε , we can assume | y | < R , and by hypothesisthere is a saturated aspirant curve β that joins y to some point z . Then α ∗ β is anaspirant curve starting at x and ending at z . If we want to complete this aspirantcurve to get a saturated one, it remains to reply to all the loose ACDCs in α . Eachof them, except possibly its endpoints, is contained in V \ SA . If, after replyingto one of them, we hit an A point y , then | y | < R , and thus we can appenda saturated aspirant curve that joins y to some z ∈ N C ∩ B | y | . Then we cancontinue to reply to the remaining loose ACDCs, and the process finishes in a finitenumber of steps. This is the desired contradiction that completes the proof of thesecond part.The first part follows trivially because the covering is by bounded pairs. (cid:3) Proof of theorem 7.2. The first part of theorem 7.19 guarantees that we can alwaysperform the “ descent ” step in the diagram. We have already shown why the othersteps can always be performed.The second part of that theorem shows that the algorithm always stops after afinite number of iterations.Thus, we can always produce an FCLC starting at any point in J . Theorem7.15 shows that an FCLC is a linking curve.This, together with lemma 7.3 completes the proof of theorem 7.2. (cid:3) It only remains to prove lemma 7.21. Before we can prove it, we need to look at A and D points more closely. 27ablo Angulo-Ardoy7.8. CDCs in adapted coordinates for A and D points. As we mentioned insection 6, the radial vector field, and the spheres of constant radius of T p M , whichhave very simple expressions in standard linear coordinates in T p M , are distortedin adapted coordinates. Thus, the distribution D and the CDCs do not alwayshave the same expression in adapted coordinates. In this section, we study themqualitatively. We will use the name R : T p M → R for the radius function, and r forthe radial vector field, and we assume that our conjugate point is a first conjugatepoint (it lies in ∂V ).7.8.1. A points. In a neighborhood O of an A point, T p M can be stratifiedas an isolated A point, inside a stratum of dimension of A points, inside asmooth surface consisting otherwise on A points. The conjugate points are givenby x + 2 x x + x = 0 , and the A points are given by the additional equation x + 2 x = 0 . The kernel is generated by the vector ∂∂x at any conjugate pointand we can assume that D is close to ∂∂x in O ∩ C .The radial vector field does not have a fixed expression in adapted coordinates,but the distribution D is a smooth line distribution and its integral curves aresmooth. Thus, the A point belongs to exactly one integral curve of D . A Branchof A (I) Branchof A (II) Figure 3. The distribution D and the CDCs at the conjugatepoints near an A point.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 can be showneasily that they must be of different types. Composing with the coordinate change( x , x , x ) → ( − x , x , x ) if necessary, we can assume that the CDCs travel inthe directions shown in figure 3.7.8.2. 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 kernel of de at the originis the plane x = 0 , which intersects this cone only at ( , . . . , ). Three generatricesof the cone consist of A points (they are given by the equations x = 0 , x − x = 0 and x + x = 0 , plus the equation of the cone), and the rest of the points are A .The radial vector field ( r , r , r ) at the origin must lie within the solid cone − r − r + r > , because the number of conjugate points (counting multiplicities)28inking curves, sutured manifolds and the Ambrose conjecturein a radial line through a point close to ( , , ), must be . In particular, | r | > .Composing with the coordinate change ( x , x , x ) → ( − x , − x , − x ) to the leftand ( x , x , x ) → ( x , x , − x ) to the right, if necessary, 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 a CDC startingat a first conjugate point moves away from the origin and may either hit an A point,or leave the neighborhood. Thus these points are not sinks of CDCs starting atpoints in V .We now claim that there are three CDCs that start at any D − point and flowout of O , and three CDCs that flow into any D − point, but the latter ones arecontained in the set of second conjugate points.Recall that the D − point is the origin. We write the radial vector as its value atthe 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 coordinatesin order to derive some qualitative behavior, even though these quantities do nothave 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 : thedeterminant is zero if and only if the angle is zero. The angle between k and r inthis coordinate system is bounded from below, and the norm of r is bounded closeto . Thus if we use unit vectors that span G and k , we get a number d ( x ) thatis comparable to the sine of the angle between G and the plane 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 generatrix of C at a point ( x , x , x ) ∈ C is the line through ( x , x , x ) and the origin. So d iscomputed 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 linesof A points correspond to x = − √ , x = √ , and the third line lies at ∞ . Weprove that p has three different roots, one in each interval: ( −∞ , − √ ), ( − √ , √ ),( √ , ∞ ). This follows immediately if we prove lim x →−∞ p ( x ) = −∞ , p ( − √ ) > ,29ablo Angulo-Ardoy A A A Figure 4. 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. p ( √ ) < and lim x →∞ p ( x ) = ∞ for all a and b such that a + b < . The firstand last one are obvious, so let us look at the second 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 andis attained only at the boundary a + b = 1 . The third inequality is analogous.Thus, there is exactly one direction where D is aligned with G en each sectorbetween two lines of A points. Take polar coordinates ( φ, r ) in C ∩ V . The rootsof d are transverse, and thus if φ corresponds to a root of d , then at a line indirection 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 line with angle φ , and sufficientlysmall r > , we move upwards in the direction of D (in the direction of increasingradius), we hit the line of A points, not the center. There are two CDCs startingat each side of every A point. A continuity argument shows that there must beone CDC in each sector that starts at the origin (see figure 4).Reversing the argument, we see that there are three CDCs that descend intothe elliptic umbilic point, one in each sector, all contained in the the set of secondconjugate points.7.8.3. D +4 points. The conjugate points in a neighborhood of adapted coordinateslie in the cone C given by x x − x = ( x + x ) − ( x − x ) − x . Thistime, the kernel of d exp p at the origin intersects this cone in two lines through theorigin, and the inside of the cone x x − x > is split into two parts. There is oneline of A points, the generatrix of the cone with parametric equations: 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 ) to the right, if necessary,we can assume that r > and r > . 30inking curves, sutured manifolds and the Ambrose conjectureWe write the radial vector as its value at the origin plus a first order perturbation: 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 starting ata first conjugate point might end up at the origin. Let F be the half cone of firstconjugate points (given by the equations x x = x and ( x + x ) < ). Let F + bethe points of F with radius greater than the origin. Its tangent cone at the originis F ∩ { x < } or F ∩ { x > } , depending on the sign of the third 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 thekernel k of e . This time, the kernel is spanned by ( − x , x , in the chart 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) Again, we look for the roots of the lower order ( -th order) approximation, whichis 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 = − in order 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 for apoint 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. But evenwhen p has two critical points, the local maximum may be negative, or the localminimum positive, with one real root.The vector r must satisfy r (cid:54) = 0 and x x − x > , or ab > . There are two chambers for r : r > and r < . We will say that a D +4 point such that 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 the kernel of de at the origin. The cubic polynomial p has limit ∓∞ at ±∞ , and p (0) > . The lineof 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 onepositive root, and two negative ones, one at each side of the line of A points. Thiscorresponds to the top right picture in figure 5, where the x axis is vertical, andthe CDCs descend, because r > .The positive root gives a direction that is tangent to a CDC that enters into the D +4 point, but moving to a nearby point we find CDCs that miss the origin, andapproach either of the two CDCs that depart from the origin, corresponding to thenegative 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. A CDCstarting at a point in F flows away from the stratum of A points and out of theneighborhood (see the bottom pictures at figure 5). It can be checked by examplethat both possibilities do occur.We want to remark that if there are three roots, the D +4 point is the endpointof the CDCs starting at any point in a set of positive H measure. Fortunately, allthese points are second conjugate points. This is the main reason why we build the31ablo Angulo-Ardoy Figure 5. An hyperbolic umbilic point.Explanation of figure 5. In the TopLeft corner, the cone C appears in blue,the line of A points in green, the radial vector at the origin in red, andthe CDCs in red.The other pictures show the CDCs in the parametrization of the half coneof first conjugate points, obtained by projecting onto the plane spannedby (1 , − , and (0 , , . The red dots indicate the directions where D isparallel to the generatrix of the cone. The A points lie in the half verticalline with x < . TopRight: a > , b > . BottomLeft: a < , b < , p has only one real root. BottomRight: a < , b < , p has three distinct real roots.synthesis as a quotient of V rather than all of T p M but more important: this isa hint of the kind of complications we might find in arbitrary dimension, or for anarbitrary metric, where we cannot list the normal forms and study each possiblesingularity 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, we foundalternative proofs and did not need to mention Sturm’s method in the proof. Theprecise boundary between the sets of a, b such that p has one or three real roots isfound by Sturm’s method. It is given by: p = − a b − a − b − ab + 243 = 0 Proof of lemma 7.21. Let x ∈ V be a point and O be a cubical neighborhoodof adapted coordinates around 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. Thegain 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 ina sufficiently small neighborhood S of x , there is a GACDC α that reaches y ∈ ∂O and has length at least ε/ .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 shortcurve will increase the radius at most | x | − | y | − δ for some δ > . Theinequality still holds with δ/ if instead of α we have a GACDC startingat 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/ .Any GACDC contained in O is the graph over any ACDC of a Lipschitzfunction with derivative bounded by c , so it has finite length. It followsthat the pair is bounded. A : We recall that the set of singular points C near an A point is an hyper-surface, and the stratum of A points is a smooth curve. An ACDC startingat any A point will flow either into the stratum of A points transversally(within C ), or into the boundary of O .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 it canbe replied in one step, and the algorithm stops. The algorithm alsostops 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.The length of any GACDC in O is bounded for the same reason as for A points, and this is enough to bound aspirant curves contained in 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 pointflows into the A point. H is a smooth curve, and splits U into two parts.One of them, U , contains only A points, while the other, U , contains allthe A points.Look at figure 6: a CDC starting at a point y ∈ U flows into theboundary of U without meeting any obstacle. A CDC α starting at a point x ∈ U , however, flows into the branch I of A . We can start a retort β at that point, but it will get interrupted when exp ◦ β reaches the stratumof the queue d’aronde that is the image of two strata of A points meetingtransversally. The retort cannot go any further because only the points“above exp( C ) ” (the side of ) have a preimage, and points in the main sheet33ablo Angulo-Ardoy U A Branch I of A Branch II of A xy exp p (x) exp p (y) U Figure 6. This picture shows a neighborhood of an A point in T p M , together with the linking curves that start at x and y (tothe left) and the image of the whole sketch by exp p (to the right).of exp( C ) have only one preimage, that is A . When he hit the stratum of A points, we follow a CDC to get a curve that leaves the neighborhood ina 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.8. Further questions We have proposed a new strategy for proving the Ambrose conjecture. If ouronly goal had been to prove that the Ambrose conjecture holds for a generic familyof metrics, we could have simplified the definitions of unequivocal point and linkedpoints. We have chosen the definitions so that the sutured property does not excludesome common manifolds.There is a weaker form of the sutured property that may be simpler to prove,allowing for a remainder set K consisting of points that are neither unequivocal norlinked to an unequivocal point, but such that the Hausdorff dimension of e ( K ) issmaller than n − . We have decided not to include it here, but the reader can finddetails in chapter 6.5 of [A].8.1. Bounding the length of the linking curves. It doesn’t seem likely that auniform bound can be found for the lengths of the FCLCs built with the linkingcurve algorithm. Let us show how a naive argument for bounding the length failsat giving a uniform bound.Let B R be the maximum length of a linking curve starting at a point x of radius R . The algorithm starting at x first adds a GACDC α of length l that leaves atransient neighborhood U of x . The next iterations of the algorithm add a linkingcurve β at the tip of α , and it only remains to reply to α . If this could be done in34inking curves, sutured manifolds and the Ambrose conjectureone step, we would have: B R < l + B R − ε but unfortunately, α = α ∗ . . . ∗ α k might cross SA k times. After adding β to thetip of α , we can always reply to α k , but then we may have to iterate the algorithmuntil we add a linking curve starting at the tip of α k − before we can reply to the α k − . This means we may have to add k linking curves, and our bound is only: B R < l + k · B R − ε This is of little use unless we can bound k .However, it may be enough to find a uniform bound of the composition of thelinking curve with e . Then a metric can be approximated by generic ones, obtain-ing sequences of linking curves for the approximate metrics, and then using [HL,Lemma 4.2], for instance. 35ablo Angulo-Ardoy References [A] W. Ambrose: Parallel translation of riemannian curvature . 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