On equifocal Finsler submanifolds and analytic maps
Marcos M. Alexandrino, Benigno Alves, Miguel Angel Javaloyes
OON EQUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTICMAPS
MARCOS M. ALEXANDRINO, BENIGNO ALVES, AND MIGUEL ANGEL JAVALOYES
Abstract.
A relevant property of equifocal submanifolds is that their parallelsets are still immersed submanifolds, which makes them a natural generaliza-tion of the so-called isoparametric submanifolds. In this paper, we prove thatthe regular fibers of an analytic map π : M m + k → B k are equifocal whenever M m + k is endowed with a complete Finsler metric and there is a restriction of π which is a Finsler submersion for a certain Finsler metric on the image. Inaddition, we prove that when the fibers provide a singular foliation on M m + k ,then this foliation is Finsler. Introduction
Roughly speaking, as the name might suggest, an equifocal submanifold L in aRiemannian manifold M is one whose “parallel sets” are always (immersed) sub-manifolds, even when they contain focal points of L (see the formal definitionbelow).From Somigliana’s article on Geometric Optics in 1919, and Segre and Cartan’swork in the 1930 s to the present day, this natural concept received different namesand was treated in different ways, but it has been playing an important role inthe theory of submanifolds and problems with symmetries; for surveys on the re-lation between the theory of isoparametric submanifolds and isometric actions seeThorbergsson’s surveys [37, 38, 39] and for books on these topics see e.g., [4, 11, 31].Isoparametric submanifolds and regular orbits of isometric actions are examplesof equifocal submanifolds. Both kinds of submanifolds are leaves of the so-calledsingular Riemannian foliations (SRF for short), i.e., singular foliations whose leavesare locally equidistant, see e.g., [5]. On the other hand, the regular leaves of anSRF are equifocal (see [6]). Therefore, so far, equifocality and SRF’s are closelyrelated.Also over the past decades, important examples have been presented as a pre-image of regular values of analytic maps, see [15] and [34]. And there are goodreasons for that. For example, as proved in [27], the leaves of SRF’s with closedleaves in Euclidean spaces are always pre-images of polynomial maps. Mathematics Subject Classification.
Primary 53C12, Secondary 58B20.
Key words and phrases.
Finsler foliations, Finsler submersion.The first and second authors were supported by Funda¸c˜ao de Amparo a Pesquisa do Estadode S˜ao Paulo-FAPESP (Tematicos: 2016/23746-6). The second author was supported by CNPq(PhD fellowship) and partially supported by PDSE-Capes (PhD sandwich program). This work isa result of the activity developed within the framework of the Programme in Support of ExcellenceGroups of the Regi´on de Murcia, Spain, by Fundaci´on S´eneca, Science and Technology Agencyof the Regi´on de Murcia. The third author was partially supported by MICINN/FEDER projectreference PGC2018-097046-B-I00 and Fundaci´on S´eneca (Regi´on de Murcia) project reference19901/GERM/15, Spain. a r X i v : . [ m a t h . DG ] F e b M.M. ALEXANDRINO, B. ALVES, AND M. A. JAVALOYES
Several problems related to symmetries and wavefronts presented in Riemanniangeometry are also natural problems in Finsler geometry. For example, the Finslerdistance (a particular case of transnormal functions) has been used to model forestwildfires and Huygens’ principle, see [29]. This is one of the several motivationsfor a recent development of Finsler’s isoparametric theory, and again isoparametricand transnormal analytic functions naturally appear; see [2, 13, 18, 19, 20, 21, 43].For other applications see [14] and [12].Very recently we have laid in [3] the foundations of the new theory of singu-lar Finsler foliations (SFF), i.e., a singular foliation where the leaves are locallyequidistant (but the distance from a plaque or leaf L a to L b does not need to beequal to the distance from a plaque or leaf L b to L a ), see Lemma 2.5. The partitionof a Finsler manifold into orbits of a Finsler action is a natural example of SFF. Wehave also presented non-homogeneous examples using analytic maps [3, Example2.13]. Among other results, we proved the equifocality of the regular leaves of anSFF on a Randers manifold, where the wind is an infinitesimal homothety. Never-theless, to prove the equifocality of the regular leaves of an SFF in the general casebecomes a challenging problem, presenting new aspects that did not appear in theanalogous problem in Riemannian geometry.For all these reasons, it is natural to ask if we can prove the equifocality of leavesof SFF’s described by analytic maps. In this article, we tackle this issue proving itunder natural conditions. During our investigation we generalize some Riemanniantechniques (like Wilking’s distribution) and obtain some results that, as far as weknow, do not appear in the literature even in the Riemannian case.Throughout this article, ( M, F ) will always be a complete analytical Finslermanifold.In order to formalize the concept of “parallel sets” and equifocality, let us startby recalling the definition of equifocal hypersurfaces.Let L be an (oriented) hypersurface with normal unit vector field ξ (in the non-reversible Finsler case, there could be two independent normal unit vector fields to L ). We define the endpoint map η ξ : L → M as η rξ ( p ) = γ ξ p ( r ), where γ ξ p is theunique geodesic with ˙ γ ξ p (0) = ξ p . In this case the parallel sets are L r = η rξ ( L ). The hypersurface L is said equifocal if dη rξ has constant rank for all r. From theconstant rank theorem, it is not difficult to check that if L is compact, then theparallel sets L r are immersed submanifolds (with possible intersections).If the codimension of L is greater than 1, we need to clarify which normal unitvector field ξ along L we consider as there are many different choices. We will followthe same approach as in the Riemannian case.When L is a regular leaf of an SFF denoted by F (i.e., L has maximal dimension),there exists a neighborhood U (in M ) of the point p ∈ L and a Finsler submersion ρ : U → S so that F U = F ∩ U are the pre-images of the map ρ . In this case, ξ canbe chosen as a projectable normal vector field along L ∩ U with respect to ρ , i.e.,an F - local basic vector field . Now, we say that L is an equifocal submanifold if, foreach local F -basic unit vector field ξ along L , the differential map dη rξ has constantrank for all r .Finally when π : M → B is an analytic map, c is a regular value and L = π − ( c ), ξ can be chosen as a basic vector field along L , i.e., π -basic . Once we want toconsider “parallel sets” of codimension bigger than one, it is natural to expect thatat least near L the fibers are equidistant. Therefore, in this case we will demand QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 3 that there exists a saturated neighborhood (cid:101) U of L where π is regular and the fibersof π restricted to (cid:101) U are the leaves of a regular Finsler foliation. In other words, therestriction π (cid:101) U : (cid:101) U → π ( (cid:101) U ) ⊂ B is a Finsler submersion for some Finsler metric on π ( (cid:101) U ) . With this assumption, we will say that L = π − ( c ) is an equifocal submanifoldif for each π -basic unit vector field ξ along L , the differential map dη rξ has constantrank for all r . Observe that in this definition, the partition F π = { π − ( c ) } c ∈ B of M by the fibers of π is not required to be a singular foliation. Theorem 1.1.
Let ( M, F ) be an analytic connected complete Finsler manifold and π : M m + k → B k an analytic map between analytic manifolds. Assume that thepre-image π − ( d ) is always a connected set and that there exists an open subset U where π is regular and such that F π | U is a Finsler foliation. We have that (a) if q ∈ π − ( c ) is a regular point (i.e., dπ q is surjetive), then all points of π − ( c ) are regular, i.e., c is a regular value, and the restriction of F π tothe set of regular points of π is a Finsler foliation. (b) Each regular level set π − ( c ) is an equifocal submanifold. In addition, theimage of the endpoint map η rξ : π − ( c ) → M (for r and ξ fixed) is containedin a level set. (c) If F π is a smooth singular foliation, then F π is a singular Finsler foliation.Remark . Item (a) shows how the hypothesis influences the singular fibers. Infact, consider f : R → R defined as f ( x , x ) = x x . Note that f − (0) is thestratified set { x = 0 } ∪ { x = 0 } . Also note that each point p ∈ f − (0) differentfrom (0 ,
0) is a regular point, i.e., ∇ f ( p ) (cid:54) = (0 , Remark . As far as we know, the theorem is new even in the Riemannian case.In fact, in [1] the first author dealt with transnormal analytic maps, assuming alsothat the normal bundle of the fibers was an integrable distribution (on the set ofregular points).There are a few interesting open questions. The first one is under which hypothe-ses can one ensure that the singular set is not just a stratification, but a submanifold.This problem was solved in [2], in the case where the map was an analytic function f : M → R . The second question is even when the singular sets are submanifolds,and the fibers are equidistant, under which conditions F π = { π − ( c ) } c ∈ B will be asmooth singular foliation, i.e., for each p ∈ π − ( d ) and v ∈ T p ( π − ( d )), there is asmooth vector field X tangent to the fibers with X p = v .This paper is organized as follows. In §
2, we briefly review a few facts on Finslergeometry and Finsler submersions. In §
3, we review the construction of Wilkings’distribution in the Finsler case. In §
4, we present the proof of Theorem 1.1. In § §
2) and some proofs of results (presented in §
3) on Jacobifields. In this way, we hope to make the paper accessible also to readers withoutprevious training in Finsler geometry, or who are more interested in understandingthe Riemannian case of the theorem.
Acknowledgements
We are very grateful to Prof. Daniel Tausk (IME-USP, S˜aoPaulo, Brasil) for explaining some aspects of the theory of Morse-Sturm systems tous.
M.M. ALEXANDRINO, B. ALVES, AND M. A. JAVALOYES Preliminaries
In this section we fix some notations and briefly review a few facts about FinslerGeometry that will be used in this paper. For more details see [9, 36, 35].2.1.
Finsler metrics.
Let M be a manifold. We say that a continuous function F : T M → [0 , + ∞ ) is a Finsler metric if(1) F is smooth on T M \ ,(2) F is positive homogeneous of degree 1, that is, F ( λv ) = λF ( v ) for every v ∈ T M and λ > p ∈ M and v ∈ T p M \ { } , the fundamental tensor of F definedas g v ( u, w ) = 12 ∂ ∂t∂s F ( v + tu + sw ) | t = s =0 for any u, w ∈ T p M is a nondegenerate positive-definite bilinear symmetricform.In particular, if M is a vector space V , and F : V → R is a function smooth on V \ { } and satisfying the properties (2) and (3) above, then ( V, F ) is called a
Minkowski space .Recall that the
Cartan tensor associated with the Finsler metric F is defined as C v ( w , w , w ) := 14 ∂ ∂s ∂s ∂s F ( v + (cid:88) i =1 s i w i ) | s = s = s =0 for every p ∈ M , v ∈ T p M \ , and w , w , w ∈ T p M. The Chern connection and the induced covariant derivative. In § ∇ V , with V a vector field without singularites on an open subset Ω(see [36, Eq. (7.20) and (7.21)] and also [22]):(1) ∇ V is torsion-free , namely, ∇ VX Y − ∇ VY X = [ X, Y ]for every vector field X and Y on Ω,(2) ∇ V is almost g-compatible , namely, X · g V ( Y, Z ) = g V ( ∇ VX Y, Z ) + g V ( Y, ∇ VX Z ) + 2 C V ( ∇ VX V, Y, Z ) , where X , Y , and Z are vector fields on open set Ω and g V and C V are thetensors on Ω such that ( g V ) p = g V p and ( C V ) p = C V p .It can be checked that the Christoffel symbols of ∇ V only depend on V p at every p ∈ M , and not on the particular extension of V . Therefore, the Chern connectionis an anisotropic connection. Moreover, it is positively homogeneous of degreezero, namely, ∇ λv = ∇ v for all v ∈ A and λ >
0. For an explicit expression ofthe Christoffel symbols of the Chern connection in terms of the coefficients of thefundamental and the Cartan tensors see [9, Eq. (2.4.9)]. The Chern connectionprovides an (anisotropic) curvature tensor R , which, for every p ∈ M and v ∈ T p M \ { } , determines a linear map R v : T p M × T p M × T p M → T p M . The Cherncurvature tensor can be defined using the anisotropic calculus as in Eq. (5.2), orin coordinates, with the help of the non-linear connection (see [10, Eq. (3.3.2)]). QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 5
Given a smooth curve γ : I ⊂ R → M and a smooth vector field W ∈ X ( γ ) along γ without singularites, where X ( γ ) denotes the smooth sections of the pullbackbundle γ ∗ ( T M ) over I , the Chern connection induces a covariant derivative D Wγ along γ , such that ( D Wγ X )( t ) = ∇ W ( t )˙ γ ( t ) X , with the identification X ( t ) = X γ ( t ) ,when X ∈ X ( M ). If the image of the curve is contained in a chart (Ω , ϕ ) (using theChristoffel symbols introduced in Eq. (5.1)) the covariant derivative is expressedas ( D Wγ X )( t ) = n (cid:88) k =1 ( ˙ X k ( t ) + n (cid:88) i,j =1 X i ( t ) ˙ γ j ( t )Γ kij ( W ( t ))) ∂ k , where X, W ∈ X ( γ ) and X l ( t ) and ˙ γ l ( t ) are the coordinates of X ( t ) and ˙ γ ( t ),respectively, for t ∈ I . When ˙ γ ( t ) (cid:54) = 0 for all t ∈ I , we can take as a referencevector W = ˙ γ . In such a case, we will use the notation X (cid:48) := D ˙ γγ X whenever thereis no possible confusion about γ .2.3. Geodesics and Jacobi fields.
We will say that a smooth curve γ : I ⊂ R → M is a geodesic of ( M, F ) if it is an auto-parallel curve of the covariant derivativeinduced by the Chern connection, namely, D ˙ γγ ˙ γ = 0. Given a vector v ∈ T M \ ,there is a unique geodesic γ v such that ˙ γ v (0) = v .When we consider a geodesic variation of γ , it turns out that the variationalvector field J is characterized by solving the differential equation(2.1) J (cid:48)(cid:48) ( t ) + R ˙ γ ( t ) ( J ( t )) = 0 , where, for every p ∈ M and v ∈ T p M \{ } , we define the operator R v : T p M → T p M as R v ( w ) = R v ( w, v ) v for every w ∈ T p M (see for example [24, Prop. 2.11]). Givena geodesic γ , the operator R ˙ γ , defined for vector fields along γ , and the solutions ofEq. (2.1) are known, respectively, as the Jacobi operator of γ , and the Jacobi fieldsof γ . When we consider a submanifold L , the minimizers from L are orthogonalgeodesics (see [8]), and variations of orthogonal geodesics will be given by L -Jacobifields, which will be introduced below. First let us define orthogonal vectors. Givena submanifold L ⊂ M , we say that a vector v ∈ T p M , with p ∈ L , is orthogonal to L if g v ( v, u ) = 0 for all u ∈ T p L . The subset of orthogonal vectors to L at p ∈ L ,denoted by ν p L , could be non-linear, but it is a smooth cone (see [25, Lemma 3.3]).We say that a geodesic γ : I → M is orthogonal to L at t ∈ I if ˙ γ ( t ) is orthogonalto L . Definition 2.1.
Let L be a submanifold of a Finsler manifold ( M, F ) and γ :[ a, b ) → M a geodesic orthogonal to L at p = γ ( a ). We say that a Jacobi field is L -Jacobi if • J ( a ) is tangent to L , • S ˙ γ ( a ) J ( a ) = tan ˙ γ ( a ) J (cid:48) ( a ),where S ˙ γ : T p L → T p L is the shape operator defined as S ˙ γ ( u ) = tan ˙ γ ( a ) ∇ ˙ γ ( a ) u ξ ,with ξ an orthogonal vector field along L such that ξ p = ˙ γ ( a ) and tan ˙ γ ( a ) , the g ˙ γ ( a ) -orthogonal projection into T p L . An instant t is called L -focal (and γ ( t ) afocal point ) if there exists an L -Jacobi field J such that J ( t ) = 0.One can check that the shape operator is symmetric since the connection issymmetric (recall [25, Eq. (16)]). Observe that to compute tan ˙ γ ( a ) ∇ ˙ γ ( a ) u ξ , one M.M. ALEXANDRINO, B. ALVES, AND M. A. JAVALOYES formally needs a vector field ξ along L which extends ξ p , but it turns out that thisquantity does not depend on the chosen extension, but only on ξ p (see [25, Prop.3.5]).2.4. Geodesic vector fields.
As we have stressed in § v ∈ T p M \ { } can be made using a vector field V without singularitesdefined on some neighborhood Ω of p ∈ M with the affine connection ∇ V . Thechoice of this V is arbitrary, so, for example, it is possible to choose V such that ∇ vu V = 0 for all u ∈ T p M (see [24, Prop. 2.13]). As, in this work, we will dealmainly with geodesics, it will be particularly convenient to choose a geodesic vectorfield V . In such a case, it is possible to relate some elements of the Chern connectionwith the Levi-Civita connection of the Riemannian metric g V . Proposition 2.2.
Let V be a geodesic field on an open subset U ⊂ M and ˆ g := g V denote the Riemannian metric on Ω induced by the fundamental tensor g , and let (cid:98) ∇ and (cid:98) R be the Levi-Civita connection and the Jacobi operator of ˆ g , respectively.Then, for any X ∈ X (Ω) ,(i) (cid:98) ∇ X V = ∇ VX V and (cid:98) ∇ V X = ∇ VV X ,(ii) (cid:98) R V X = R V X .As a consequence, the integral curves of V are also geodesics of (cid:98) g , and the FinslerianJacobi operator and Jacobi fields along the integral curves of V coincide with thoseof (cid:98) g . A proof of this result can be found in [35, Prop. 6.2.2] and also in [24, Prop.3.9].2.5.
Finsler submersions and foliations.
One of the first examples of Finslerfoliations is the partition of M into the fibers of a Finsler submersion. Recall that asubmersion π : ( M, F ) → ( B, ˜ F ) between Finsler manifolds is a Finsler submersion if dπ p ( B Fp (0 , B ˜ Fπ ( p ) (0 , , for every p ∈ M , where B Fp (0 ,
1) and B ˜ Fπ ( p ) (0 ,
1) arethe unit balls of the Minkowski spaces ( T p M, F p ) and ( T π ( p ) B, ˜ F π ( p ) ) centered at 0,respectively. Examples and a few properties of Finsler submersions can be foundin [3] and [7].Recall that we say that a geodesic γ : I ⊂ R → M is horizontal if ˙ γ ( t ) is anorthogonal vector to the fiber π − ( π ( γ ( t ))) for every t ∈ I . Here we just need torecall the lift property (see [7, Theorem 3.1]). Proposition 2.3.
Let π : ( M, F ) → ( B, ˜ F ) be a Finsler submersion. Then animmersed curve on B is a geodesic if and only if its horizontal lifts are geodesics on M . In particular, the geodesics of ( B, ˜ F ) are precisely the projections of horizontalgeodesics of ( M, F ) . This result allows us to get a geodesic field on (
M, F ) that projects on a geodesicfield on ( B, ˜ F ), which will simplify many computations. Proposition 2.4.
Let π : ( M, F ) → ( B, ˜ F ) be a Finsler submersion and V ∗ ageodesic field in some open subset ˜ U of B . Then the horizontal lift V of V ∗ is ageodesic vector field on U = π − ( ˜ U ) and the restriction π | U : ( U, g FV ) → ( ˜ U , g ˜ FV ∗ ) isa Riemannian submersion, where g F and g ˜ F are the fundamental tensors of F and ˜ F , respectively. QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 7
Proof.
That V is a geodesic field follows from Proposition 2.3, while the last state-ment is a consequence of [7, Prop. 2.2]. (cid:3) A Finsler foliation of M is a regular foliation F such that at every point p ∈ M ,there exists a neighborhood U in such a way that F| U can be obtained as the fibersof a Finsler submersion. This is equivalent to the property of transnormality : if a geodesic is orthogonal to one leaf of F , then it is orthogonal to all the (2.2) fibers it meets. Recall that a singular foliation of M is a partition of M by submanifolds, which arecalled leaves as in the case of regular foliations, with the following property: given p ∈ M , if v ∈ T p M is a vector tangent to a leaf, there exists a smooth vector field X in a neighborhood of p such that X p = v and X is always tangent to the leaves of thefoliation. This is equivalent to saying that for each p there exists a neighborhood U of p , a neighborhood B (0) of 0 in R k , and a submersion π : U → B (0) ⊂ R k sothat each fiber π − ( x ), with x ∈ B (0), is contained in a leaf, and P p = π − (0) is aprecompact open subset of the leaf L p that contains p (see [3, Lemma 3.8]). Thesubmanifold P p is called plaque . Note that k here is the codimension of L p and B (0) can be considered to be a transverse submanifold S p to P p . By definition,the leaves F on U must intersect S p . Finally, we say that the singular foliation is Finsler if the property (2.2) holds. If P q is a plaque, there exist future and pasttubular neighborhoods denoted, respectively, by O ( P q , ε ) and ˜ O ( P q , ε ) of a certainradius ε > C + r ( P q ) = { p ∈ M : d F ( P q , p ) = r } and C − r ( P q ) = { p ∈ M : d F ( p, P q ) = r } , where d F ( P q , p ) is computed as the infimumof the lengths of curves from P q to p and d F ( p, P q ) as the infimum of the lengths ofcurves from p to P q . Recall that given x ∈ O ( P q , ε ) (resp. ˜ O ( P q , ε )), the plaque P x is the connected component which contains x of the intersection of the leaf through x with O ( P q , ε ) (resp. ˜ O ( P q , ε )). Lemma 2.5.
A singular foliation F is Finsler if and only if its leaves are locallyequidistant, i.e., if its leaves satisfy the following property: if a point x belongs tothe future cylinder C + r ( P q ) (resp. the past cylinder C − r ( P q ) ), then the plaque P x ofthe future (resp. past) tubular neighborhood is contained in C + r ( P q ) (resp. C − r ( P q ) ). A proof of the above characterization can be found in [3, Lemma 3.7].
Remark . Given a (singular) Riemannian foliation F with closed leaves on acomplete Riemannian manifold ( M, h ) and an F -basic vector field W , then F isa (singular) Finsler foliation for the Randers metric with Zermelo data ( h , W ), c.f[3, Example 2.13]. The converse is also true. In fact, according with [3, Theorem1.1], every singular Finsler foliation with closed leaves on Randers spaces ( M, F ) isproduced in this way. 3.
Wilking’s construction
In [40], Wilking proved the smoothness of the Sharafutdinov’s retraction andstudied dual foliations of singular Riemannian foliations (SRF for short). For that,he used results on self-adjoint spaces, regular distributions along geodesics andMorse-Sturm systems along these distributions, see also [17]. These tools turn outto be quite useful in the study of the transversal geometry of SRF’s (see e.g. [28]).
M.M. ALEXANDRINO, B. ALVES, AND M. A. JAVALOYES
Figure 1.
Figure generated by the software geogebra.org, illustratinga few concepts of § C + r ( P q ) of a plaque P q ,other plaques contained in this future cylinder and the normal cone at q denoted by ν q P q , i.e, the set of vectors ξ ∈ T q M so that g ξ ( ξ, X ) = 0 forall X ∈ T q P q . Here the leaves are the fibers of the Finsler submersion π : ( M, F ) → ( B, ˜ F ), where M = R , B = R , π ( x ) = ( x , x ) and F and ˜ F are Randers metrics having the Euclidean metrics and the winds W = ( , , sin ( x )+14 ) and (cid:102) W = ( ,
0) as Zermelo data, respectively.
Roughly speaking, given an SRF F on M , for each horizontal geodesic γ (that maycross singular leaves) one can define a distribution t → H ( t ) along γ so that H ( t )is the normal distribution ν γ ( t ) ( L ) when γ ( t ) lies in a regular leaf; for the sake ofsimplicity we call this distribution Wilking’s distribution . In addition, when theleaves of F are closed and π : M → M/ F is the canonical projection, being thequotient M/ F a manifold with the induced Riemannian metric, one can identifythe Jacobi fields along π ◦ γ with the solutions of a Morse-Sturm system along t → H ( t ), the so-called transversal Jacobi field equation .In this section we review the Finsler version of these objects; see Definition 3.7.We also present a few results on Finsler Jacobi fields, whose proofs will be given in § Definition 3.1.
Let γ : I ⊆ R → M be a geodesic of a Finsler manifold ( M, F )and W a linear subspace of Jacobi fields so that γ is orthogonal to the elements of W , i.e., g ˙ γ ( ˙ γ, J ) = 0 for every J ∈ W . The vector space W is said to be self-adjoint if g ˙ γ ( J (cid:48) , J ) = g ˙ γ ( J , J (cid:48) ), for J , J ∈ W . Remark . One can prove that given two Jacobi fields J and J along γ then g ˙ γ ( J (cid:48) , J ) − g ˙ γ ( J , J (cid:48) ) is constant on the interval I . Therefore to show that a vectorspace W of Jacobi fields is self-adjoint, it suffices to check that g ˙ γ ( t ) ( J (cid:48) ( t ) , J ( t )) = g ˙ γ ( t ) ( J ( t ) , J (cid:48) ( t )) for some t ∈ I (see for example [25, Prop. 3.18]).In the next lemma, we present the distribution t → H ( t ) associated with a generalsubspace V of W as well as the (generalized) transversal Jacobi field equation. Lemma 3.3.
Let W be an ( n − -self-adjoint vector space of Jacobi fields orthogonalto a geodesic γ : I ⊆ R → M on a Finsler manifold ( M n , F ) . Let V be a linearsubspace of W and hence also a self-adjoint space. Define the subspace V ( t ) of T γ ( t ) M for every t ∈ I as V ( t ) := { J ( t ) | J ∈ V } ⊕ { J (cid:48) ( t ) | J ∈ V , J ( t ) = 0 } . (a) Then dim V ( t ) = dim V for every t ∈ I . Furthermore, the second summandis trivial for almost every t . QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 9 (b) Let t → H ( t ) be the orthogonal complement of V ( t ) with respect to g ˙ γ ( t ) ,i.e., w ∈ H ( t ) if and only if g ˙ γ ( t ) ( u, w ) = 0 for all u ∈ V ( t ) . Let ( · ) h and ( · ) v be the orthogonal projections (with respect to g ˙ γ ) into H ( t ) and V ( t ) ,respectively. Then if J ∈ W , J h fulfills the transversal Jacobi equation, i.e., (3.1) ( D h ) ( J h ) + ( R ˙ γ ( J h )) h − A γ ) ( J h ) = 0 , where D h is the induced connection on the horizontal bundle, which is de-fined as D h ( X ) = (( X h ) (cid:48) ) h , and A γ is the O’ Neill tensor along the geodesic γ , i.e., A γ ( X ) := (cid:0) ( X h ) (cid:48) (cid:1) v + (cid:0) ( X v ) (cid:48) (cid:1) h for every X ∈ X ( γ ) .(c) If J , . . . , J r is a basis of V with r = dim V such that at t ∈ I , J k ( t ) = . . . = J r ( t ) = 0 and J i ( t ) (cid:54) = 0 for i = 1 , . . . , k − , then J ( t ) , . . . , J k − ( t ) , t − t J k ( t ) , . . . , t − t J r ( t ) is a continuous basis of V ( t ) for every t in ( t − ε, t + ε ) ∩ I \ { t } for some ε > , with lim t → t t − t J i ( t ) = J (cid:48) i ( t ) for all i = k, . . . , r .Proof. First, observe that given t ∈ I , there exists a neighborhood U ⊂ M of γ ( t )which admits a geodesic vector field V such that γ ( t ) ⊂ U and V γ ( t ) = ˙ γ ( t ) for all t ∈ I ε = ( t − ε, t + ε ) ∩ I for some ε >
0. From Proposition 2.2, we conclude that W and V are also self-adjoint spaces of (Riemannian) Jacobi fields with respect to theRiemannian metric ˆ g := g V . Since the lemma is already true for self-adjoint spacesof Riemannian Jacobi fields (see [17, Chapter 1]), it is also valid in the Finsler casefor t ∈ I. Indeed, part (a) follows from [17, Lemma 1.7.1] (for the triviality of thesecond summand, observe that the interval I can be covered by a countable numberof intervals I ε i with geodesic vector fields V i , i ∈ N , as above). Part (b) followsfrom the equation above [17, Eq. (1.7.7)] observing that if A ( t ) : V ( t ) → H ( t )and its dual A ∗ ( t ) : H ( t ) → V ( t ) are the linear operators defined below [17, Eq.(1.7.4)] (which are also naturally defined for Finsler metrics), then A γ | H = − A ∗ and A γ | V = A . In fact, when t is regular (the second summand in part ( a ) is trivial),for every u ∈ V ( t ), there exists a Jacobi field J ∈ V such that J ( t ) = u . Then A ( t ) u = ( J (cid:48) ) h ( t ), but as J v = J , it follows that A γ | V ( t ) = A ( t ). Moreover, as g ˙ γ ( A γ | H ( t ) ( X ) , Z ) = − g ˙ γ ( X, A γ | V ( t ) ( Z )) = g ˙ γ ( X, − A ( t )( Z ))= g ˙ γ ( − A ∗ ( t )( X ) , Z ) , which allows us to conclude that A γ | H ( t ) = − A ∗ ( t ). This implies, taking intoaccount the equation above [17, Eq. (1.7.7)], that Eq. (3.1) holds for every regularinstant, and, by continuity, for all t ∈ I , as regular instants are dense by part( a ). We stress that, by Proposition 2.2, the O’Neill tensor of γ computed with g V coincides with that computed with F . Finally, for part (c), see the proof of [17,Lemma 1.7.1]. (cid:3) As we will see below, an example of ( n − W is the space of L -Jacobi fields for a submanifold L along a geodesic γ which is orthogonal to L . Inthe case where L is a fiber of a Finsler submersion π : M → B , V is the space ofholonomic Jacobi fields, i.e., those Jacobi fields whose π -projections are zero. Alsoin this case, the transversal Jacobi field equation along γ will be identified with aJacobi field equation along a geodesic in B (see Remark 3.9). Lemma 3.4.
Let L be a submanifold on a Finsler manifold ( M n , F ) and γ : I ⊂ R → M a geodesic orthogonal to L at t ∈ I. Consider W the vector space of L -Jacobifields orthogonal to γ . Then W is a self-adjoint space of dimension n − .Proof. See § (cid:3) Proposition 3.5.
Let ( M, F ) be a Finsler manifold and L a submanifold of M .Given a geodesic γ : I = [ a, b ] ⊂ R → M orthogonal to L at the instant t , we havethat a vector field J along γ is L -Jacobi if and only if it is the variation vector fieldof a variation whose longitudinal curves are L -orthogonal geodesics.Proof. See § (cid:3) When L is a fiber of a Finsler submersion and J is an L -Jacobi field tangent tothe fibers, then J turns out to be the variation vector field of a variation determinedby end-point maps η rξ , as we will see in the next lemma. Lemma 3.6.
Consider a Finsler submersion π : ( M, F ) → ( B, ˜ F ) , and set L = π − ( b ) for some b ∈ B . Let γ : I → M be a geodesic orthogonal to L at t ∈ I and t → J ( t ) be an L -Jacobi field along γ with J ( t ) tangent to T p L where p = γ ( t ) .Also assume that J is g ˙ γ -orthogonal to γ . Then the following items are equivalent:(a) J is a vertical (holonomy) Jacobi field, i.e., J is always tangent to the fibersof π . In other words J h = 0 .(b) There exists a curve β : ( − ε, ε ) ⊂ R → L with J ( t ) = ˙ β (0) , such that if ξ isthe normal basic vector field along L with ξ p = ˙ γ ( t ) and ψ : I × ( − ε, ε ) → M is the variation ( t, s ) → ψ ( t, s ) = γ ξ β ( s ) ( t − t ) = η t − t ξ ( β ( s )) , then J ( t ) = ∂∂s ψ ( t, .Proof. See § (cid:3) Definition 3.7.
Let F be a partition by stratified submanifolds of M such thatthere exists an open neighborhood U ⊂ M where the partition F| U is a regularFinsler foliation of U . Given a geodesic γ : I ⊆ R → M and t ∈ I such that γ ( t ) ∈ U and ˙ γ ( t ) ∈ ν ( L γ ( t ) ), denoting by L p the fiber of F| U which contains p ∈ U , let W be the self-adjoint space of L γ ( t ) -Jacobi fields orthogonal to γ definedin Lemma 3.4 and V the subspace of W such that J ∈ V iff J ( t ) ∈ T γ ( t ) L γ ( t ) for all t ∈ ( t − ε, t + ε ), where ε > γ ( t − ε, t + ε ) ⊂ U .Then the distributions t → V ( t ), t → H ( t ) and the differential equation (3.1)associated with the self-adjoint spaces W and V are called the Wilking’s distributionsand the transversal Jacobi field equation along γ starting at { γ ( t ) , L γ ( t ) } . Remark . Note that even when F is a singular foliation, the distributions t →V ( t ), t → H ( t ) and the differential equation (3.1) are well defined for all t includingthe case where γ ( t ) is a singular point. Also note that the transversal Jacobifield equation (3.1) is a Morse-Sturm system; see [16, 26, 28, 33]. Although thedistribution V ( t ) coincides with the tangent space of L γ ( t ) until γ crosses a singularleaf, it does not follow straightforwardly that V ( t ) should always be tangent to theregular leaves for all t (even at regular leaves after crossing the first singular leaf).This is the case when F is the foliation studied in § QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 11
Figure 2.
Figure generated by the software geogebra.org, illustrat-ing the Wilking’s distributions t → H ( t ) and t → V ( t ) associatedwith the singular Finsler foliation F π = { π − ( c ) } on the neighborhood B (0) ⊂ R where π : R → R is defined as π ( x ) = ( x + x , x ) . TheFinsler metric F restricted to B / (0) is the Randers metric having theEuclidean metric and the wind W = ( − x , x ,
0) as Zermelo data.
Remark . By applying Proposition 3.5 we can infer useful natural interpretationsof the transversal Jacobi field equation. Let π : M → B be a Finsler submersion and L a fiber, i.e., L = π − ( b ), for any b ∈ B . Let p ∈ L and γ p be a horizontal geodesicstarting at p , i.e., p = γ p (0). Then the transversal Jacobi equation along γ p startingat L γ p (0) can be identified with the Jacobi field equation along the geodesic π ◦ γ p in B . Indeed, the transversal Jacobi equation is the horizontal lift to M of the Jacobiequation on B . To check this, consider a geodesic vector field V whose integralcurves are horizontal geodesics of ( M, F ), being V ∗ = dπ ◦ V , and recall that, byProposition 2.4, π : ( M, g V ) → ( B, ˜ g V ∗ ) is a Riemannian submersion. Then usethe relation between the Levi-Civita connection of ( B, ˜ g V ∗ ) and the horizontal partof the Levi-Civita connection of ( M, g V ) when applied to basic vector fields (see[30, Lemma 1 (3)]) and the relation between the horizontal part of their curvaturetensors in [30, Theorem 2 { } ] taking into account Proposition 2.2. In addition, if x ∈ L and γ x is a horizontal geodesic starting at x so that π ◦ γ x = π ◦ γ p , then thetransversal Jacobi field equation along γ p starting at L p and the transversal Jacobifield equation along γ x starting at L x can be identified with each other. As we willsee from Lemmas 4.3 and 4.5, this interpretation will also hold for analytic singularFinsler submersions. 4. Proof of Theorem 1.1
In this section we prove Theorem 1.1 adapting an argument of [1] and usingWilking’s distributions, recall Definition 3.7. Let π : M n → B k be an analytic map. In the first subsection, assuming π is a Finsler submersion on an open subsetof M , we prove that π is a Finsler submersion on the regular part M and theregular fibers are equifocal submanifolds. In the second part, we will assume thatthe fibers constitute a singular foliation and will prove the transnormality.4.1. Equifocality.
Let q ∈ M be a regular point of π , c = π ( q ) and L = π − ( c ).We will consider a unit basic vector field ξ along a neighborhood of q in L , i.e., ξ is orthogonal to L , projectible and F ( ξ ) = 1. We will show that L is a regularfiber of π by extending this basic vector field along L (Lemma 4.1) and will checkpart ( b ) of Theorem 1.1. More precisely, in Lemma 4.3, we will prove that η rξ ( L ) iscontained in a fiber of π and in Lemmas 4.9 and 4.10, that dη rξ has constant rankfor all r . Lemma 4.1. If p ∈ M is a regular point of π and there exists a neighborhood U of p in (the regular part of ) L p such that F is constant on basic vector fields on U ,then L p is regular and F is constant on basic vector fields on L p .Proof. Let us denote by L Rp the connected component of p of this regular part(which is open) and ξ a basic vector field on L Rp , which is analytic as all the datais analytic. Then as, by hypothesis, the function F ◦ ξ : L Rp → R is constant ina neighborhood of p , it must be constant everywhere. Now consider a point q inthe boundary of L Rp . As F ◦ ξ is constant, if one considers a sequence of points { q n } ⊂ L Rp such that lim n →∞ q n = q , then the sequence of vectors ξ q n is precompactin T M and it converges to some v q ∈ T q M up to a subsequence. Let ˜ ξ = dπ ( ξ ),which is well-defined because ξ is basic. By continuity, the vector v q projects to ˜ ξ ,and since this can be done for every ˜ ξ ∈ T π ( p ) B (with the corresponding basic lift),it turns out that the points of the boundary are regular and F is constant on ξ . (cid:3) Remark . As a consequence of Lemma 4.1, the assumption that F π is a Finslerfoliation restricted to some open subset U ⊂ M implies that there exists a saturatedopen subset where π is a Finsler submersion. Namely, consider π ( U ), which is openbecause π is a submersion and then π | π − ( π ( U )) is a Finsler submersion for a certainFinsler metric on π ( U ). Lemma 4.3.
For all r ∈ R , there exists a value d ∈ B such that η rξ ( L ) ⊂ π − ( d ) .Proof. Assume that r > r < r = 0 is trivial).Consider x, y ∈ L and set ˜ I := { t ∈ [0 , r ] } such that π ( γ ξ x ( t )) = π ( γ ξ y ( t )) , for t ≤ t } . Note that the set ˜ I is not empty, because π restricted to a neighborhood of L isa Finsler submersion (recall Remark 4.2). The set ˜ I is closed due to continuity andit is an open set because π is an analytic map and geodesics of an analytic Finslermetric are also analytic. Therefore ˜ I = [0 , r ] and this concludes the proof. (cid:3) Fixed x ∈ L , and consider the geodesic γ ξ x . Since, by Remark 4.2, π is aFinsler submersion on a neighborhood of x , we can define along the geodesic γ ξ x the Wilking’s distribution pair t → ( H ( t ) , V ( t )) (recall Definition 3.7). Lemma 4.4. If x ∈ L and p = γ ξ x ( t ) is a regular point of π , then the Wilking’sdistribution V ( t ) coincides with the tangent space to L p . Moreover, the distribu-tions t → V ( t ) and t → H ( t ) are analytic and the singular points of γ ξ x (lying in asingular level set) are isolated. QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 13
Proof.
Recall by part (c) of Lemma 3.3 that there is a basis J , . . . , J r of V with r =dim V such that at t , J k ( t ) = . . . = J r ( t ) = 0 and J i ( t ) (cid:54) = 0 for i = 1 , . . . , k − J ( t ) , . . . , J k − ( t ) , t − t J k ( t ) , . . . , t − t J r ( t )is a continuous basis of V ( t ) for every t in ( t − ε, t + ε ) for some ε >
0, withlim t → t t − t J i ( t ) = J (cid:48) i ( t ) for all i = k, . . . , r . By Lemma 3.6, J i ( t ) = dη tξ ( v i ) forsome vector v i ∈ T x L for all i = 1 , . . . , r , and by Lemma 4.3, all J i ( t ) are tangentto L γ ξx ( t ) for all t . By continuity, J (cid:48) j ( t ) is also tangent to L γ ξx ( t ) for j = k, . . . , r .This implies that V ( t ) is contained in the tangent space to L p . As they have thesame dimension, they coincide. The analyticity of V and H follows from the basisof V ( t ) in (4.1), as the vector fields ˜ J j ( t ) = t − t J j ( t ) extend analytically to t making ˜ J j ( t ) = J (cid:48) j ( t ) as J j ( t ) = 0 and J j is analytic, for j = k, . . . , r . Up tothe singular points of γ ξ x , consider an analytic frame of the horizontal space H ,namely, analytic vector fields along γ ξ x , H , . . . , H n − r such that H ( t ) , . . . , H n − r ( t )is a basis of H ( t ) for every t (this can be obtained using the D h -parallel transport).Then when chosen an analytic frame along π ◦ γ ξ x , E , . . . , E n − r , the determinantof the transformation matrix with respect to dπ ◦ H , . . . , dπ ◦ H n − r can only haveisolated zeroes or being zero everywhere. As it is not zero close to t , it has onlyisolated zeroes. Even if one cannot ensure the existence of a global frame for π ◦ γ ξ x , it is enough to recover its domain with frames which intersect in opensubsets to conclude that the singular points along γ ξ x are isolated. Observe that,by Lemma 4.1, the singular points of π can only lie in singular levels. (cid:3) Lemma 4.5. If p = γ ξ x ( t ) is a regular point of π , then γ ξ x is orthogonal to L p .Proof. Observe that f i ( t ) = g ˙ γ ξx ( ˙ γ ξ x ( t ) , J i ( t )) and h i ( t ) = g ˙ γ ξx ( ˙ γ ξ x ( t ) , J (cid:48) i ( t )) areanalytic functions for each J i ∈ V in the basis of part (c) of Lemma 3.3, and if γ ξ x ( t ) is a regular point of π , then, by Lemma 4.4, T γ ξx ( t ) L γ ξx ( t ) = V ( t ). As π is a Finsler submersion in a neighborhood of L (recall Remark 4.2), f i and h i areidentically zero, and, in particular, by part (c) of Lemma 3.3, γ ξ x is orthogonal to L p . (cid:3) Lemma 4.6.
If there exists a neighborhood U of M such that F π | U is a Finslerfoliation, then F π is a Finsler foliation when restricted to the open subset of regularpoints of π .Proof. Fix q ∈ U . The first observation is that, by Remark 4.2, π is a Finslersubmersion when restricted to π − ( π ( U )) and L q is a closed submanifold. Given p ∈ M , a regular point of M , there exists a minimizing geodesic γ : [0 , b ] → M from L q to p , and it follows from the Finsler Morse index Theorem (see [32]) thatthere is no focal point on [0 , b ). As γ minimizes the distance from L q , it must beorthogonal to L q , and then, by Lemma 4.5, orthogonal to all the regular fibers of π . Now consider the basic vector field ξ along L q such that ξ γ (0) = ˙ γ (0). Thenthe map η tξ : L q → L γ ( t ) is a diffeomorphism on a neighborhood of γ (0) for every t ∈ [0 , b ). Fix an instant t ∈ (0 , b ) such that γ ( t ) is a regular point of π (recallthat, by Lemma 4.4, singular points are isolated along γ ), then by continuity andthe compactness of [0 , t ], it is possible to choose a neighborhood (cid:101) U ⊂ L q where η tξ : (cid:101) U → L γ ( t ) is a diffeomorphism onto the image for all t ∈ [0 , t ]. Given a vector v horizontal to (cid:101) U = η t ξ ( (cid:101) U ) at p = η t ξ ( q ), for some q ∈ ˜ U , consider the Wilking’sdistributions H q ( t ) and V q ( t ) along the horizontal curve γ ξ q , and let v h be theprojection of v to H q ( t ). Consider the D h -parallel vector field X along γ ξ q suchthat X ( t ) = v h and the vector field Y along γ ξ q orthogonal to η tξ ( (cid:101) U ) at each t ∈ [0 , t ] such that its Wilking’s projection to H q is X (recall [3, Lemma 2.9 (a)]).Let v be the orthogonal vector to L q at γ (0) such that dπ ( v ) = dπ ( Y (0)). Nowrepeat the process to obtain a vector field along H γ (0) (the horizontal Wilking’sdistribution along γ ) and ˜ Y horizontal to η tξ ( (cid:101) U ) such that its Wilking’s projectionto H γ (0) ( t ), ˜ X = ˜ Y h , is D h -parallel and ˜ X (0) = v h . Let us observe that if we provethat(4.2) dπ ( Y ( t )) = dπ ( ˜ Y ( t )) for all t ∈ [0 , t ] , then, for t close to 0, F ( Y ( t )) = F ( ˜ Y ( t )), because π is a Finsler submersion,but, by analyticity, this will be true for all t ∈ [0 , t ]. This implies that F ( v ) = F ( ˜ Y ( t )), and in particular, that F is constant on basic vector fields along (cid:101) U .By Lemma 4.1, F is constant on basic vector fields along the whole fiber L γ ( t ) .Finally, by continuity, this is also true for L p . So, let us prove Eq. (4.2), whichis equivalent to prove that dπ ( X ( t )) = dπ ( ˜ X ( t )) . This equivalence follows fromthe fact that dπ ( V q ( t )) = dπ ( V γ (0) ( t )) = { } , because T γ ξq ( t ) η tξ ( (cid:101) U ) = V q ( t ), T γ ( t ) η tξ ( (cid:101) U ) = V γ (0) ( t ), and η tξ ( (cid:101) U ) ⊂ L γ ξq ( t ) = L γ ( t ) (recall that we assume that η tξ | ˜ U is a diffeomorphism onto the image for t ∈ [0 , t ]). By analyticity, we only haveto prove that this holds for t close to 0. But this is true, because the D h -paralleltransport in a Riemannian submersion is the horizontal lift of the parallel transportin the base. In our case, the D h -parallel transport coincides with the lift of theparallel transport induced by ˜ g V ∗ , where V ∗ is a (local) geodesic vector field in thebase tangent to π ◦ γ , considering the Riemannian metric g V on M , where V is thehorizontal lift of V ∗ , being g and ˜ g , the fundamental tensors of the Finsler metricson M and B , respectively (recall Proposition 2.4). So, the proof is concluded. (cid:3) Remark . Observe that as a consequence of Lemmas 4.4, 4.5 and 4.6, given ageodesic which is horizontal in one regular point, then it is horizontal in all theregular points and, by continuity (using Lemma 4.4) the Wilking distribution doesnot depend on the regular instant chosen as initial point. Let us see that thesolutions to the transversal Jacobi equation of two horizontal geodesics with thesame projection can be identified.
Lemma 4.8.
Consider a geodesic γ of ( M, F ) which is horizontal at the regularpoints. It holds that(a) if X is a solution of the transversal Jacobi equation of γ in Eq. (3.1) , then alongthe regular instants of γ , dπ ◦ X is a Jacobi field of π ◦ γ , and the solutions ofthe transversal Jacobi equation are characterized by this property,(b) if α is another geodesic which is horizontal in the regular points with π ◦ γ = π ◦ α ,then there exists a vector field Y along α which fulfills the transversal Jacobiequation of α and such that dπ ◦ X = dπ ◦ Y and g ˙ γ ( X, X ) = g ˙ α ( Y, Y ) ,(c) γ and α provide the same conjugate points of the transversal Jacobi equation. QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 15
Proof.
For part ( a ), recall Remark 3.9, and use that the singular points are isolated(see Lemma 4.4). For part ( b ), observe that part ( a ) implies part ( b ) when werestrict γ and α to an interval with regular points. Analyticity implies that Y can be extended to the whole interval with the required properties. Part ( c ) is astraightforward consequence of part ( b ). (cid:3) Lemma 4.9. If d is a regular value, then η rξ : L → π − ( d ) is a diffeomorphism.Proof. By Lemma 4.5, ˙ γ ξ p ( r ) is orthogonal to π − ( d ). As π is a Finsler submersionon π − ( d ) (recall Lemma 4.6), we can extend the normal vector ˙ γ ξ p ( r ) to a unitbasic vector field ˜ ξ along the fiber π − ( d ). It is also possible to check that η − r ˜ ξ isthe inverse of η rξ . (cid:3) We have just proved in Lemma 4.9 that η rξ : π − ( c ) → π − ( d ) is a diffeomorphismwhen d is a regular value, which implies that dη rξ has constant rank. We have tocheck now that dη rξ has constant rank when d is a singular value. This will be donein Lemma 4.10. Lemma 4.10.
Consider η rξ : L → π − ( d ) with d ∈ B a singular value. Then dim rank dη rξ is constant along L .Proof. As L is connected, it is enough to prove that dim rank d ( η rξ ) x is locallyconstant. Consider a point p ∈ L . Since the transversal Jacobi field equation along γ ξ p starting at L is a Morse-Sturm system, there exists a δ > I δ = ( r − δ, r + δ ) are not conjugate to each other. In other words, if X is a solution of the transversal Jacobi field along γ ξ p , and s , s ∈ I δ so that X ( s ) = 0 = X ( s ), then X ( t ) = 0 for all t , see e.g. [33, Lemma 2.1]. Also notethat this δ > γ ξ x for x ∈ L as a consequence of part ( c ) ofLemma 4.8. Using again that the critical points of π on γ ξ x are isolated, we canalso suppose, reducing δ if necessary, that the point γ ξ x ( r ) is the only critical pointof π on γ ξ x | I δ for x ∈ L .Let ˜ s ∈ ( r − δ, r ) and ˜ p = γ ξ p (˜ s ). Define the unit vector field ˜ ξ along L ˜ p so that˜ ξ ˜ p = ˙ γ ξ p (˜ s ). Recall that η ˜ sξ : L p → L ˜ p is a diffeomorphism (see Lemma 4.9). Set˜ r := r − ˜ s . Also note that η rξ = η ˜ r ˜ ξ ◦ η ˜ sξ . Therefore to prove that x → dim rank d ( η rξ ) x is locally constant at p ∈ L , it suffices to prove that ˜ x → dim rank d ( η ˜ r ˜ ξ ) ˜ x is locallyconstant at ˜ p = η ˜ sξ ( p ) ∈ L ˜ p .We claim that L ˜ p -focal points along the horizontal segment of geodesic γ ˜ ξ ˜ x | (0 , ˜ r + δ ) are of tangential type. In other words, γ ˜ ξ ˜ x ( t ) is a focal point with multiplicity k (0 < t < ˜ r + δ ) if and only if ˜ x is a critical point of the endpoint map η t ˜ ξ anddim ker dη t ˜ ξ = k. In order to prove the claim, consider an L ˜ p -Jacobi field ˜ J along t → γ ˜ ξ ˜ x ( t ) = γ ξ x ( t + ˜ s ) and assume that ˜ J ( t ) = 0 for some 0 < t < ˜ r + δ . Let J be the Jacobifield along γ ξ x so that ˜ J ( t ) = J ( t + ˜ s ). Observe that J is the variation vector fieldof a variation by orthogonal geodesics to L ˜ p (see Prop. 3.5) and then its projectionis a Jacobi field of π ◦ γ ξ x . By part ( a ) of Lemma 4.8, J h is a solution of thetransversal Jacobi equation. As J h (˜ s ) = 0 = J h (˜ s + t ), from the choice of I δ andpart ( c ) of Lemma 4.8, we conclude that J h ( t ) = 0 for all t . Therefore ˜ J h ( t ) = 0 for all t . This fact and Lemma 3.6 imply that γ ˜ ξ ˜ x ( t ) is an L ˜ p -focal point if andonly if ˜ x is a critical point of the endpoint map η t ˜ ξ .From what we have discussed above, we have concluded that: m ( γ ˜ ξ ˜ x ) = dim ker dη ˜ r ˜ ξ (˜ x ) , (4.3)where m ( γ ˜ ξ ˜ x ) denotes the number of focal points on γ ˜ ξ ˜ x , each counted with itsmultiplicities.On the other hand, by continuity of the Morse index we have m ( γ ˜ ξ ˜ x ) ≥ m ( γ ˜ ξ ˜ p ) , (4.4)for all ˜ x in some neighbourhood of ˜ p in L ˜ p . Eq. (4.3) and (4.4) together with the elementary inequality dim ker dη ˜ r ˜ ξ (˜ x ) ≤ dim ker dη ˜ r ˜ ξ (˜ p ) (as dim Im dη ˜ r ˜ ξ (˜ x ) ≥ dim Im dη ˜ r ˜ ξ (˜ p )) imply that dim ker dη ˜ r ˜ ξ is con-stant in a neighborhood of ˜ p on L ˜ p . As η ˜ sξ : L p → L ˜ p is a diffeomorphism, it followsthat dim ker dη rξ is constant in a neighborhood of p on L p , which concludes. (cid:3) Finslerian character.
In this section, we will additionally assume that thefibers constitute a singular foliation and will prove its transnormality. This willfollow directly from Lemmas 4.11, 4.5 and 4.15.
Lemma 4.11. If F π is a singular foliation, γ is a geodesic with ˙ γ (0) ∈ ν ( L γ (0) ) and γ (0) is a regular point, then γ is horizontal.Proof. By Lemma 4.5, the geodesic γ is orthogonal to the regular leaves associatedwith regular values of π . Now suppose that γ ( t ) is a singular point and u ∈ T γ ( t ) L γ ( t ) . Then as ( M, F ) is a singular foliation, there exists a vector field X in a neighborhood of γ ( t ) such that X is tangent to the leaves and X γ ( t ) = u .By Lemma 4.4, the singular points along γ are isolated, so g ˙ γ ( t ) ( X γ ( t ) , ˙ γ ( t )) = 0for all t close enough to t . By continuity, we conclude that g ˙ γ ( t ) ( u, ˙ γ ( t )) = 0, asdesired. (cid:3) Note that the above lemma does not directly imply that the foliation F π isa singular Finsler foliation, because the lemma was only proved for a geodesic γ starting at a regular point p ∈ L . Lemma 4.12. If F π is a singular foliation and c is a regular value of π , then η rξ : π − ( c ) → π − ( d ) is surjective even when d is a singular value.Proof. The proof consists of two steps. In the first step we are going to check thatthe endpoint map η rξ : π − ( c ) → π − ( d ) is an open map. And in the second stepwe have to check that the set η rξ ( π − ( c )) is a closed subset of π − ( d ). This endsthe proof because the fibers are connected.For q = η rξ ( x ) ∈ π − ( d ), consider a plaque P q of π − ( d ) which contains q and a past tubular neighborhood U = ˜ O ( P q , ε ) of the plaque P q of radius ε > γ ξ x are isolated (seeLemma 4.4) and using Lemma 4.11, we can assume without loss of generality that x ∈ U and γ ξ x is the minimizing geodesic from x to P q . This implies that allthe geodesics γ ξ x with x ∈ ˜ U ⊂ U ∩ π − ( c ), being ˜ U a small enough open subset of π − ( c ) which contains x , are minimizing geodesics from x to P q , and then(4.5) η rξ = π ν | ˜ U , QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 17 where π ν is the (past) footpoint projection. Since F π is a singular foliation, themap π ν | ˜ U is open by [3, Lemma 3.10]. This fact and Eq. (4.5) imply that theendpoint map η ξ is an open map.Consider a sequence { y n } ⊂ η rξ ( π − ( c )) that converges to a point y ∈ π − ( d ).Let { x n } ⊂ π − ( c ) be a sequence so that η rξ ( x n ) = y n . Since, for n big enough, x n ∈ B − ( y, r ), denoting by B − ( y, r ) = { x ∈ M : d F ( x, y ) < r } , namely,the backward ball or radius 2 r centered at y , we have a convergent subsequence x n i → x ∈ B − ( y, r ) . Therefore η rξ ( x ) = lim η rξ ( x n i ) = lim y n i = y and hence y ∈ η rξ ( π − ( c )) . (cid:3) Lemma 4.13.
Let L p be a regular leaf, and L q be a singular leaf. Then theyare parallel, i.e., if y and y are points of L q , then d ( L p , y ) = d ( L p , y ) and d ( y , L p ) = d ( y , L p ) .Proof. For i = 0 ,
1, let γ i : [0 , r i ] → M be a minimizing segment of geodesic joining L p with y i , i.e., γ i (0) ∈ L p , γ i ( r i ) = y i and d ( L p , y i ) = (cid:82) r i F ( ˙ γ i ) dt =: (cid:96) ( γ i ).Since η r ξ is surjective by Lemma 4.12, we can transport γ and get a (maybenon-unique) new curve || γ joining L p to y such that (cid:96) ( || γ ) = (cid:96) ( γ ). Since γ is minimal, we conclude that (cid:96) ( γ ) ≤ (cid:96) ( γ ). An analogous argument allows us toconclude that (cid:96) ( γ ) ≤ (cid:96) ( γ ) and hence, (cid:96) ( γ ) = (cid:96) ( γ ) as required, which impliesthat d ( L p , y ) = d ( L p , y ). The other equality is analogous. (cid:3) Lemma 4.14.
Given a point ˜ q in a singular leaf L ˜ q , there exists a plaque P ˜ q of ˜ q admitting a future (resp. past) tubular neighborhood such that if P q is a (regular orsingular) plaque in this neighborhood, then P q ⊂ C + r ( P ˜ q ) (resp. P q ⊂ C − r ( P ˜ q ) ) foran appropriate r .Proof. We will consider only the future case, as the past one can be recovered byusing the reverse Finsler metric ˜ F ( v ) = F ( − v ). Let U be a totally convex neigh-borhood of ˜ q , namely, given two points p , p ∈ U there exists a unique geodesicjoining p to p and this geodesic is minimizing (see [41, 42] for their existence).Now consider a plaque ˜ P ˜ q of ˜ q , a future tubular neighborhood ˜ U ⊂ U , a smallerplaque P ˜ q of ˜ q and a future tubular neighborhood U ⊂ ˜ U of P ˜ q such that if D > U , then B + ( p, D ) ∪ B − ( p, D ) ⊂ ˜ U for every p ∈ U . Let q ∈ U , P q the plaque of q ∈ U , and y , y ∈ P q . Our goal is to check that d ( P ˜ q , y ) = d ( P ˜ q , y ).This will imply P q ⊂ C + r ( P ˜ q ).For i = 0 ,
1, let γ i : [0 , r i ] → M be a minimal segment of geodesic joining P ˜ q with y i , in particular d ( P ˜ q , y i ) = (cid:96) ( γ i ), for i = 0 ,
1. Choose an arbitrary regularpoint p ∈ M . Then there exists a minimizing geodesic from L p to ˜ q , whose singularpoints are isolated. As a consequence, we can consider a sequence of regular points p m of the future tubular neighborhood U converging to ˜ q . Moreover, consider asequence of minimal segments of geodesic β mi joining ˜ P p m to γ i (0) (even if ˜ P p m is notnecessarily closed, the minimizing geodesic β mi exists because p m ∈ ¯ B − ( γ i (0) , D ) ⊂ ˜ U by the choice of U ). In particular, d ( ˜ P p m , γ i (0)) = (cid:96) ( β mi ). Finally, considerminimal segments of geodesic γ mi : [0 , r m ] → M joining ˜ P p m to y i , which thereexist by the same reason as the β mi ’s. It follows that (cid:96) ( γ mi ) = d ( ˜ P p m , y i ) and (cid:96) ( γ mi ) ≤ (cid:96) ( γ i (cid:63) β im ), where (cid:63) stands for the concatenation of the two curves.Observe that the image of the curves γ mi is contained in ¯ B − ( y i , D ). As aconsequence, there exists a convergent subsequence γ n k i (0) converging to some P q P ˜ q P p m ˜ q p m y γ γ m β m P ˜ q D ˜ P ˜ q Figure 3.
The diagram to the left represents the different curvesused in the proof of Lemma 4.14. The diagram to the right depictsboth plaques P ˜ q and ˜ P ˜ P . The minimizing curves inside the tubularneighborhood U exist as short curves cannot reach the boundary,because of the hypothesis on the diameter.ˆ p ∈ ˜ P ˜ q ⊂ ˜ U (recall that the plaques ˜ P p m in U must necessarily converge to ˜ P ˜ q ).As all the sequence and the limit lie in the convex neighborhood U , we can guar-antee that γ n k i converges to a segment of geodesic (cid:98) γ i joining ˜ P ˜ q to y i . By Lemma4.13, (cid:96) ( β mi ) → m → ∞ , and hence we conclude that (cid:96) (ˆ γ i ) ≤ (cid:96) ( γ i ). Since d ( ˜ P ˜ q , y i ) = d ( P ˜ q , y i ) = (cid:96) ( γ i ), we infer that (cid:96) (ˆ γ i ) = (cid:96) ( γ i ).On the other hand, as the points p m are regular, one can apply the same tech-niques of parallel transport of the proof of Lemma 4.13 to prove that (cid:96) ( γ m ) = (cid:96) ( γ m )and hence (cid:96) (ˆ γ ) = (cid:96) (ˆ γ ). Therefore (cid:96) ( γ ) = (cid:96) ( γ ) and this concludes the proof ofthe lemma. (cid:3) Lemma 4.15.
Let x be a point of a singular leaf L ˜ q . Then the geodesic γ ξ x ishorizontal, i.e., orthogonal to the leaves of the singular foliation F = { π − ( c ) } .Proof. The proof follows directly from Lemmas 4.14 and 2.5. (cid:3) On Jacobi fields and curvature in Finsler Geometry
Anisotropic connection and curvature.
In this section we review a fewfacts about anisotropic connections, see [23, 24].As the tensors associated with Finsler metrics depend on the direction v ∈ T M \ , it is necessary a connection that take into account this fact, namely, itdepends on the direction. In the following, we will denote by X ( M ) the space ofsmooth vector fields on M and by F ( M ), the subset of smooth real functions on M . QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 19
Definition 5.1. An anisotropic connection is a map ∇ : A × X ( M ) × X ( M ) → T M, ( v, X, Y ) (cid:55)→ ∇ vX Y ∈ T p M \ , with v ∈ T p M ,where A = T M \ , such that for any X, Y ∈ X ( M ), the map A (cid:51) v → ∇ vX Y ∈ T M is smooth, and for all p ∈ M and v ∈ T p M \ { } ,(i) ∇ vX ( Y + Z ) = ∇ vX Y + ∇ vX Z , for any X, Y, Z ∈ X ( M ),(ii) ∇ vX ( f Y ) = X p ( f ) Y p + f ( p ) ∇ vX Y for any f ∈ F ( M ) and X, Y ∈ X ( M ),(iii) ∇ vfX + hY Z = f ( p ) ∇ vX Z + h ( p ) ∇ vY Z , for any f, h ∈ F ( M ) and X, Y, Z ∈ X ( M ).Observe that the properties ( ii ) and ( iii ) imply, respectively, that ∇ vX Y dependsonly on the value of Y on a neighborhood of p , and on the value of X p . So sometimeswe will put only X p and it will make sense to compute ∇ vX Y in a chart. If (Ω , ϕ =( x , . . . , x n )) is a chart of M , with its associated partial vector fields denoted by ∂ , . . . , ∂ n , the Christoffel symbols of ∇ are the functions Γ kij : T Ω \ → R , i, j, k = 1 , . . . , n , determined by(5.1) ∇ v∂ i ∂ j = n (cid:88) k =1 Γ kij ( v ) ∂ k , for i, j = 1 , . . . , n . Given a vector field V without singularities on an open setΩ ⊂ M , the anisotropic connection ∇ induces an affine connection ∇ V on Ω definedas ( ∇ VX Y ) p = ∇ V p ˜ X ˜ Y for any X, Y ∈ X (Ω), where ˜ X, ˜ Y ∈ X ( M ) are extensions of X and Y , respectively, namely, they coincide with X and Y in a neighborhood of p ∈ Ω. Moreover, if we define the vertical derivative of ∇ as the (1 , P given by P v ( X, Y, Z ) = ∂∂t (cid:16) ∇ v + tZ p X Y (cid:17) | t =0 , for p ∈ M , v ∈ T p M \ { } and X, Y, Z ∈ X ( M ), then it is possible to define thecurvature tensor of the anisotropic connection as(5.2) R v ( X, Y ) Z = ( R V ) p ( X, Y ) Z − ( P V ) p ( Y, Z, ∇ VX V ) + ( P V ) p ( X, Z, ∇ VY V ) , where V ∈ X (Ω) is an extension of v ∈ T p M \ , X, Y, Z ∈ X ( M ), P V is the tensorsuch that ( P V ) p = P V p and R V is the curvature tensor of the affine connection ∇ V .It is important to observe that the curvature tensor R v does not depend on thevector fields V, X, Y, Z chosen to make the computation, but only on v, X p , Y p and Z p , [24, Prop. 2.5]. With an anisotropic connection at hand, we can also computethe tensor derivative of any anisotropic tensor (see [23, § ∇ V (see [23, Remark 15]). In most of the references of FinslerGeometry, they use a connection along the vertical bundle of T M . For a relationshipbetween both types of connections see [23, § A few proofs about Finsler Jacobi fields.
Here, for the sake of complete-ness, we present the proof of a few results about Jacobi fields that we have used.5.2.1.
Proof of Lemma 3.4.
Consider J , J ∈ W . Using the shape operator S ˙ γ ( t ) (see Definition 2.1), it follows that g ˙ γ ( t ) ( J (cid:48) ( t ) , J ( t )) = g ˙ γ ( t ) ( S ˙ γ ( t ) ( J ( t )) , J ( t ))= g ˙ γ ( t ) ( J ( t ) , S ˙ γ ( t ) ( J ( t ))) = g ˙ γ ( t ) ( J ( t ) , J (cid:48) ( t )) , using that S ˙ γ ( t ) is self-adjoint (see [25, Prop. 3.5]). As we saw in Remark 3.2 thisimplies that W is self-adjoint for all t . In order to see that the dimension of W is equal to n −
1, observe first that thedimension of the L -Jacobi fields is n . Recall that for every t ∈ I , if we define˙ γ ( t ) ⊥ := { v ∈ T ˙ γ ( t ) M : g ˙ γ ( t ) ( ˙ γ ( t ) , v ) = 0 } , we have the splitting(5.3) T γ ( t ) M = span { ˙ γ ( t ) } + ˙ γ ( t ) ⊥ , and we will denote by tan γ : X ( γ ) → X ( γ ) and nor γ : X ( γ ) → X ( γ ) the firstand second projection of the splitting (5.3) at every t ∈ I . Observe that givenan arbitrary Jacobi field J , then J T = tan γ J and J ⊥ = nor γ J are again Jacobifields along γ (see for example [25, Lemma 3.17]). Moreover, as J is L -Jacobi, then J ( t ) is tangent to L and therefore g ˙ γ ( t ) -orthogonal to ˙ γ ( t ). As a consequence, J T ( t ) = 0. Taking into account that a Jacobi field is tangent to γ if and only if J ( t ) = ( a t + a ) ˙ γ ( t ) (see [25, Lemma 3.17 (i)]), we conclude that J T ( t ) = a t ˙ γ ( t ),which is an L -Jacobi field and then J ⊥ is also an L -Jacobi field, because J ⊥ = J − J T . Let J L ( γ ) be the space of L -Jacobi fields along γ and J TL ( γ ) = { J ∈J L ( γ ) : J ⊥ = 0 } , J ⊥ L ( γ ) = { J ∈ J L ( γ ) : J T = 0 } . Therefore, we have a map φ : J TL ( γ ) × J ⊥ L ( γ ) → J L ( γ ) , ( J , J ) → J + J , which is well-defined and one-to-one. As we have seen above that J TL ( γ ) has di-mension one, it follows that J ⊥ L ( γ ) has dimension n −
1, which concludes.5.2.2.
Proof of Proposition 3.5.
Observe that the dimension of the submanifold
T L ⊥ ⊂ T M of orthogonal vectors to L is n = dim M (see for example [25, Lemma3.3]). Now consider a curve ( − ε, ε ) (cid:51) s → N ( s ) ∈ T L ⊥ , with β ( s ) = ρ ( N ( s )) acurve in L such that N (0) = ˙ γ ( t ) (where ρ : T M → M is the natural projection).We can construct a variation Λ( t, s ) = γ N ( s ) ( t − t ), Λ : I × ( − ε, ε ) → M , whichis given by the L -orthogonal geodesics γ N ( s ) whose velocity at 0 is N ( s ). Observethat Λ( t , s ) = β ( s ), ∂ Λ ∂t ( t , s ) = N ( s ), ∂ Λ ∂s ( t , s ) = ˙ β ( s ). It is not difficult to showthat V ( t ) = ∂ Λ ∂s ( t,
0) is an L -Jacobi field along γ . Indeed, it is Jacobi, because itslongitudinal curves are geodesics (see [25, Prop. 3.13]). In order to show that it is L -Jacobi, observe that V ( t ) = ˙ β (0) and if (Ω , ϕ ) is a chart in a neighborhood of γ ( t ),with N i , β i , γ i , i = 1 , . . . , n , the coordinates of N , β and γ , and Γ ijk : T Ω \ → R the Christoffel symbols of the Chern connection in (Ω , ϕ ), then(5.4) V (cid:48) ( t ) = D ˙ γγ V ( t ) = D ˙ γβ N (0) = ( ˙ N i (0) + ˙ β j (0) ˙ γ k ( t )Γ ijk ( ˙ γ ( t ))) ∂ i , (using [25, Eq. (7)] in the second equality). Therefore V ( t ) is tangent to L and thesecond equality of (5.4) implies that tan L ˙ γ ( t ) V (cid:48) ( t ) = S ˙ γ ( t ) ( V ( t )), which concludesthat V is L -Jacobi. In order to see that all the L -Jacobi fields can be obtained withvariations of L -orthogonal geodesics, observe that the formula (5.4) allows us todefine a map ρ : T ˙ γ ( t ) T L ⊥ → T γ ( t ) M × T γ ( t ) M, defined as ρ ( ˙ N (0)) = ( V ( t ) , V (cid:48) ( t )). It is straightforward to check, using (5.4),that ρ is injective. Indeed, this follows from the expression of the coordinates of dNds (0) in a natural chart of T T M associated with (Ω , ϕ ), which are γ (0) , . . . , γ n (0) , ˙ γ ( t ) , . . . , ˙ γ n ( t ) , ˙ β (0) , . . . , ˙ β n (0) , ˙ N (0) , . . . , ˙ N n (0), taking into account that, bydefinition, N (0) = ˙ γ ( t ) and V ( t ) = ˙ β (0). Moreover, as we have seen above, theimage of ρ is contained in the initial values for the Cauchy problem of L -Jacobifields. As the dimension of T ˙ γ ( t ) T L ⊥ coincides with the dimension of the space of L -Jacobi fields (in both cases, the dimension is n ), this easily implies that every QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 21 L -Jacobi field can be realized as the variational vector field of a variation by L -orthogonal geodesics.5.2.3. Proof of Lemma 3.6.
The fact that (b) implies (a) follows from Proposition2.3, because all the geodesics in the variation ψ project into the same geodesic on B ,and as a consequence J ( t ) = ∂∂s ψ ( t,
0) is vertical for every t ∈ I . Let us check that(a) implies (b). First note that J is determined by its first initial condition J ( t ),because J is vertical. In fact assume by contradiction that there exists anothervertical L -Jacobi field ˜ J so that ˜ J ( t ) = J ( t ) and J (cid:54) = ˜ J . Set ˆ J := J − ˜ J . Henceˆ J is a vertical Jacobi field with ˆ J ( t ) = 0, and then ˆ J (cid:48) ( t ) = lim t → t t − t ˆ J ( t ),which implies that ˆ J (cid:48) ( t ) is also tangent to L . Being ˆ J L -Jacobi (recall Def. 2.1),it follows that the tangent part to L of ˆ J (cid:48) ( t ) is zero, which concludes that ˆ J = 0,as both initial conditions are zero. Now consider a curve β : ( − ε, ε ) → L such that˙ β (0) = J ( t ) and the Jacobi field ˜ J ( t ) = ∂∂s ψ ( t, ψ the variation definedin part (b). The fact that (a) implies (b) follows from the above discussion since J ( t ) = ˜ J ( t ) = ∂∂s ψ ( t, J = ˜ J . References [1] M. M. Alexandrino:
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Benigno O. AlvesInstituto de Matem´atica e Estat´ıstica, Universidade Federal da Bahia,Rua Bar˜ao de Jeremoabo ,40170-115 Salvador, Bahia, Brazil
Email address : [email protected];[email protected] QUIFOCAL FINSLER SUBMANIFOLDS AND ANALYTIC MAPS 23
Marcos M. AlexandrinoInstituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo,Rua do Mat˜ao 1010,05508 090 S˜ao Paulo, Brazil
Email address : [email protected], [email protected] Miguel ´Angel JavaloyesDepartamento de Matem´aticas, Universidad de Murcia,Campus de Espinardo,30100 Espinardo, Murcia, Spain
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