On equifocal Finsler submanifolds and analytic maps

A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $\pi:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $\pi$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, 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 a Riemannian manifold M is one whose "parallel sets" are always (immersed) submanifolds, even when they contain focal points of L (see the formal definition below).
From Somigliana's article on Geometric Optics in 1919, and Segre and Cartan's work in the 1930s to the present day, this natural concept received different names and was treated in different ways, but it has been playing an important role in the theory of submanifolds and problems with symmetries; for surveys on the relation between the theory of isoparametric submanifolds and isometric actions see Thorbergsson'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 examples of equifocal submanifolds.Both kinds of submanifolds are leaves of the so-called singular Riemannian foliations (SRF for short), i.e., singular foliations whose leaves are locally equidistant, see e.g., [5].On the other hand, the regular leaves of an SRF are equifocal (see [6]).Therefore, so far, equifocality and SRF's are closely related.
Also over the past decades, important examples have been presented as a preimage of regular values of analytic maps, see [15] and [34].And there are good reasons for that.For example, as proved in [27], the leaves of SRF's with closed leaves in Euclidean spaces are always pre-images of polynomial maps.
Several problems related to symmetries and wavefronts presented in Riemannian geometry are also natural problems in Finsler geometry.For example, the Finsler distance (a particular case of transnormal functions) has been used to model forest wildfires and Huygens' principle, see [29].This is one of the several motivations for a recent development of Finsler's isoparametric theory, and again isoparametric and 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 singular Finsler foliations (SFF), i.e., a singular foliation where the leaves are locally equidistant (but the distance from a plaque or leaf L a to L b does not need to be equal to the distance from a plaque or leaf L b to L a ), see Lemma 2.5.The partition of a Finsler manifold into orbits of a Finsler action is a natural example of SFF.We have also presented non-homogeneous examples using analytic maps [3,Example 2.13].Among other results, we proved the equifocality of the regular leaves of an SFF on a Randers manifold, where the wind is an infinitesimal homothety.Nevertheless, to prove the equifocality of the regular leaves of an SFF in the general case becomes a challenging problem, presenting new aspects that did not appear in the analogous problem in Riemannian geometry.
For all these reasons, it is natural to ask if we can prove the equifocality of leaves of SFF's described by analytic maps.In this article, we tackle this issue proving it under natural conditions.During our investigation we generalize some Riemannian techniques (like Wilking's distribution) and obtain some results that, as far as we know, do not appear in the literature even in the Riemannian case.
Throughout this article, (M, F ) will always be a complete analytical Finsler manifold.
In order to formalize the concept of "parallel sets" and equifocality, let us start by recalling the definition of equifocal hypersurfaces.
Let L be an (oriented) hypersurface with normal unit vector field ξ (in the nonreversible Finsler case, there could be two independent normal unit vector fields to L).We define the endpoint map η ξ : L → M as η r ξ (p) = γ ξp (r), where γ ξp is the unique geodesic with γξp (0) = ξ p .In this case the parallel sets are L r = η r ξ (L).The hypersurface L is said equifocal if dη r ξ has constant rank for all r.From the constant rank theorem, it is not difficult to check that if L is compact, then the parallel sets L r are immersed submanifolds (with possible intersections).
If the codimension of L is greater than 1, we need to clarify which normal unit vector field ξ along L we consider as there are many different choices.We will follow the 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 can be 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, for each local F-basic unit vector field ξ along L, the differential map dη r ξ has constant rank for all r.
Finally when π : M → B is an analytic map, c is a regular value and L = π −1 (c), ξ can be chosen as a basic vector field along L, i.e., π-basic.Once we want to consider "parallel sets" of codimension bigger than one, it is natural to expect that at least near L the fibers are equidistant.Therefore, in this case we will demand that there exists a saturated neighborhood U of L where π is regular and the fibers of π restricted to U are the leaves of a regular Finsler foliation.In other words, the restriction π U : U → π( U ) ⊂ B is a Finsler submersion for some Finsler metric on π( U ).With this assumption, we will say that L = π −1 (c) is an equifocal submanifold if for each π-basic unit vector field ξ along L, the differential map dη r ξ has constant rank for all r.Observe that in this definition, the partition F π = {π −1 (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 the pre-image π −1 (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 −1 (c) is a regular point (i.e., dπ q is surjetive), then all points of π −1 (c) are regular, i.e., c is a regular value, and the restriction of F π to the set of regular points of π is a Finsler foliation.(b) Each regular level set π −1 (c) is an equifocal submanifold.In addition, the image of the endpoint map η r ξ : π −1 (c) → M (for r and ξ fixed) is contained in a level set.(c) If F π is a smooth singular foliation, then F π is a singular Finsler foliation.Remark 1.2.Item (a) shows how the hypothesis influences the singular fibers.In fact, consider f : 0) is a regular point, i.e., ∇f (p) = (0, 0).In other words, in this very natural and simple example item (a) is not fulfilled.
Remark 1.3.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 also that the normal bundle of the fibers was an integrable distribution (on the set of regular points).
There are a few interesting open questions.The first one is under which hypotheses 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 −1 (c)} c∈B will be a smooth singular foliation, i.e., for each p ∈ π −1 (d) and v ∈ T p (π −1 (d)), there is a smooth 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 Finsler geometry 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 §5, we discuss the definition of the curvature tensor (related to the definition of Jacobi tensor presented in §2) and some proofs of results (presented in §3) on Jacobi fields.In this way, we hope to make the paper accessible also to readers without previous training in Finsler geometry, or who are more interested in understanding the Riemannian case of the theorem.

Preliminaries
In this section we fix some notations and briefly review a few facts about Finsler Geometry 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 is positive homogeneous of degree 1, that is, F (λv) = λF (v) for every v ∈ T M and λ > 0, (3) for every p ∈ M and v ∈ T p M \ {0}, the fundamental tensor of F defined as for any u, w ∈ T p M is a nondegenerate positive-definite bilinear symmetric form.In particular, if M is a vector space V , and F : V → R is a function smooth on V \ {0} 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 for every p ∈ M , v ∈ T p M \ 0, and w 1 , w 2 , w 3 ∈ T p M.

2.2.
The Chern connection and the induced covariant derivative.In §5.1, we will discuss the concept of anisotropic connection, which means that the Christoffel symbols are functions on the tangent bundle.Here we just briefly review the concept of Chern connection associated with a Finsler metric as a family of affine connections ∇ 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, , where X, Y , and Z are vector fields on open set Ω and g V and C V are the tensors on Ω such that (g V ) p = g Vp and (C V ) p = C Vp .
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 connection is an anisotropic connection.Moreover, it is positively homogeneous of degree zero, namely, ∇ λv = ∇ v for all v ∈ A and λ > 0. For an explicit expression of the Christoffel symbols of the Chern connection in terms of the coefficients of the fundamental and the Cartan tensors see [9,Eq. (2.4.9)].The Chern connection provides an (anisotropic) curvature tensor R, which, for every p ∈ M and v ∈ T p M \ {0}, determines a linear map R v : T p M × T p M × T p M → T p M .The Chern curvature tensor can be defined using the anisotropic calculus as in Eq. (5.2), or in coordinates, with the help of the non-linear connection (see [10,Eq. (3.3.2)]).
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 pullback bundle γ * (T M ) over I, the Chern connection induces a covariant derivative 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 the Christoffel symbols introduced in Eq. (5.1)) the covariant derivative is expressed as where X, W ∈ X(γ) and X l (t) and γl (t) are the coordinates of X(t) and γ(t), respectively, for t ∈ I.When γ(t) = 0 for all t ∈ I, we can take as a reference vector W = γ.In such a case, we will use the notation X := D γ γ X whenever there is 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 derivative induced by the Chern connection, namely, D γ γ γ = 0. Given a vector v ∈ T M \ 0, there is a unique geodesic γ v such that γv (0) = v.
When we consider a geodesic variation of γ, it turns out that the variational vector field J is characterized by solving the differential equation (2.1) where, for every p ∈ M and v ∈ T p M \{0}, 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]).Given a geodesic γ, the operator R γ , defined for vector fields along γ, and the solutions of Eq. (2.1) are known, respectively, as the Jacobi operator of γ, and the Jacobi fields of γ.
When we consider a submanifold L, the minimizers from L are orthogonal geodesics (see [8]), and variations of orthogonal geodesics will be given by L-Jacobi fields, which will be introduced below.First let us define orthogonal vectors.Given a 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 : 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 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 1 is called L-focal (and γ(t 1 ) a focal point) if there exists an L-Jacobi field J such that J(t 1 ) = 0.
One can check that the shape operator is symmetric since the connection is symmetric (recall [25, Eq. ( 16)]).Observe that to compute tan γ(a) ∇ γ(a) u ξ, one formally needs a vector field ξ along L which extends ξ p , but it turns out that this quantity 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 §2.1, all the computations at a direction v ∈ T p M \ {0} can be made using a vector field V without singularites defined on some neighborhood Ω of p ∈ M with the affine connection ∇ V .The choice of this V is arbitrary, so, for example, it is possible to choose V such that ∇ v u V = 0 for all u ∈ T p M (see [24,Prop. 2.13]).As, in this work, we will deal mainly with geodesics, it will be particularly convenient to choose a geodesic vector field V .In such a case, it is possible to relate some elements of the Chern connection with 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 ĝ := g V denote the Riemannian metric on Ω induced by the fundamental tensor g, and let ∇ and R be the Levi-Civita connection and the Jacobi operator of ĝ, respectively.Then, for any X ∈ X(Ω), As a consequence, the integral curves of V are also geodesics of g, and the Finslerian Jacobi operator and Jacobi fields along the integral curves of V coincide with those of g.

Finsler submersions and foliations. One of the first examples of Finsler
foliations is the partition of M into the fibers of a Finsler submersion.Recall that a submersion π : (M, F ) → (B, F ) between Finsler manifolds is a Finsler submersion if dπ p (B F p (0, 1)) = B F π(p) (0, 1), for every p ∈ M , where B F p (0, 1) and B F π(p) (0, 1) are the unit balls of the Minkowski spaces (T p M, F p ) and (T π(p) B, Fπ(p) ) centered at 0, respectively.Examples and a few properties of Finsler submersions can be found in [3] and [7].
Recall that we say that a geodesic γ : I ⊂ R → M is horizontal if γ(t) is an orthogonal vector to the fiber π −1 (π(γ(t))) for every t ∈ I.Here we just need to recall the lift property (see [7,Theorem 3.1]).
Proposition 2.3.Let π : (M, F ) → (B, F ) be a Finsler submersion.Then an immersed 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 horizontal geodesics of (M, F ).This result allows us to get a geodesic field on (M, F ) that projects on a geodesic field on (B, F ), which will simplify many computations.
) is a Riemannian submersion, where g F and g F are the fundamental tensors of F and F , respectively.
Proof.That V is a geodesic field follows from Proposition 2.3, while the last statement is a consequence of [7,Prop. 2.2].
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 fibers of 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) Recall that a singular foliation of M is a partition of M by submanifolds, which are called 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 the foliation.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 so that each fiber π −1 (x), with x ∈ B(0), is contained in a leaf, and P p = π −1 (0) is a precompact open subset of the leaf L p that contains p (see [3,Lemma 3.8]).The submanifold 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 past tubular neighborhoods denoted, respectively, by O(P q , ε) and Õ(P q , ε) of a certain radius ε > 0 (see [8]) and in these tubular neighborhoods a singular Finsler foliation can be characterized using the cylinders , where d F (P q , p) is computed as the infimum of the lengths of curves from P q to p and d F (p, P q ) as the infimum of the lengths of curves from p to P q .Recall that given x ∈ O(P q , ε) (resp.Õ(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.Õ(P q , ε)).
Lemma 2.5.A singular foliation F is Finsler if and only if its leaves are locally equidistant, i.e., if its leaves satisfy the following property: if a point x belongs to the future cylinder C + r1 (P q ) (resp. the past cylinder C − r2 (P q )), then the plaque P x of the future (resp.past) tubular neighborhood is contained in C + r1 (P q ) (resp.C − r2 (P q )).A proof of the above characterization can be found in [3,Lemma 3.7].
Remark 2.6.Given a (singular) Riemannian foliation F with closed leaves on a complete Riemannian manifold (M, h) and an F-basic vector field W , then F is a (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, Theorem 1.1], every singular Finsler foliation with closed leaves on Randers spaces (M, F ) is produced in this way.

Wilking's construction
In [40], Wilking proved the smoothness of the Sharafutdinov's retraction and studied dual foliations of singular Riemannian foliations (SRF for short).For that, he used results on self-adjoint spaces, regular distributions along geodesics and Morse-Sturm systems along these distributions, see also [17].These tools turn out to be quite useful in the study of the transversal geometry of SRF's (see e.g.[28]).) and W = ( 1 2 , 0) as Zermelo data, respectively.
Roughly speaking, given an SRF F on M , for each horizontal geodesic γ (that may cross 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 of simplicity we call this distribution Wilking's distribution.In addition, when the leaves of F are closed and π : M → M/F is the canonical projection, being the quotient M/F a manifold with the induced Riemannian metric, one can identify the 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 §5.2.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 Remark 3.2.One can prove that given two Jacobi fields J 1 and J 2 along γ then g γ (J 1 , J 2 ) − g γ (J 1 , J 2 ) is constant on the interval I. Therefore to show that a vector space W of Jacobi fields is self-adjoint, it suffices to check that g γ(t0) (J 1 (t 0 ), J 2 (t 0 )) = g γ(t0) (J 1 (t 0 ), J 2 (t 0 )) for some t 0 ∈ I (see for example [25,Prop. 3.18]).
In the next lemma, we present the distribution t → H(t) associated with a general subspace V of W as well as the (generalized) transversal Jacobi field equation.Lemma 3.3.Let W be an (n−1)-self-adjoint vector space of Jacobi fields orthogonal to a geodesic γ : I ⊆ R → M on a Finsler manifold (M n , F ). Let V be a linear subspace of W and hence also a self-adjoint space.Define the subspace V(t) of T γ(t) M for every t ∈ I as (a) Then dim V(t) = dim V for every t ∈ I. Furthermore, the second summand is trivial for almost every t.
(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., where D h is the induced connection on the horizontal bundle, which is defined as D h (X) = ((X h ) ) h , and A γ is the O' Neill tensor along the geodesic Proof.First, observe that given t 0 ∈ I, there exists a neighborhood U ⊂ M of γ(t 0 ) which admits a geodesic vector field V such that γ(t) ⊂ U and V γ(t) = γ(t) for all t ∈ I ε = (t 0 − ε, t 0 + 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 the Riemannian metric ĝ := g V .Since the lemma is already true for self-adjoint spaces of Riemannian Jacobi fields (see [17, In fact, when t 0 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 0 ) = u.Then A(t 0 )u = (J ) h (t 0 ), but as As we will see below, an example of (n − 1)-self-adjoint space W is the space of L-Jacobi fields for a submanifold L along a geodesic γ which is orthogonal to L. In the case where L is a fiber of a Finsler submersion π : M → B, V is the space of holonomic Jacobi fields, i.e., those Jacobi fields whose π-projections are zero.Also in this case, the transversal Jacobi field equation along γ will be identified with a Jacobi 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 0 ∈ I. Consider W the vector space of L-Jacobi fields orthogonal to γ.Then W is a self-adjoint space of dimension n − 1.
Proof.See §5.2.1.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 0 , we have that a vector field J along γ is L-Jacobi if and only if it is the variation vector field of a variation whose longitudinal curves are L-orthogonal geodesics.
When L is a fiber of a Finsler submersion and J is an L-Jacobi field tangent to the fibers, then J turns out to be the variation vector field of a variation determined by end-point maps η r ξ , as we will see in the next lemma.
Lemma 3.6.Consider a Finsler submersion π : (M, F ) → (B, F ), and set L = π −1 (b) for some b ∈ B. Let γ : I → M be a geodesic orthogonal to L at t 0 ∈ I and t → J(t) be an L-Jacobi field along γ with J(t 0 ) tangent to T p L where p = γ(t 0 ).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 fibers of π.In other words J h = 0. (b) There exists a curve β : (−ε, ε) ⊂ R → L with J(t 0 ) = β(0), such that if ξ is the normal basic vector field along L with ξ p = γ(t 0 ) and ψ : Definition 3.7.Let F be a partition by stratified submanifolds of M such that there exists an open neighborhood U ⊂ M where the partition F| U is a regular Finsler foliation of U .Given a geodesic γ : I ⊆ R → M and t 0 ∈ I such that γ(t 0 ) ∈ U and γ(t 0 ) ∈ ν(L γ(t0) ), denoting by L p the fiber of F| U which contains p ∈ U , let W be the self-adjoint space of L γ(t0) -Jacobi fields orthogonal to γ defined in 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 0 −ε, t 0 +ε), where ε > 0 is small enough to guarantee that γ(t 0 −ε, t 0 +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 distributions and the transversal Jacobi field equation along γ starting at {γ(t 0 ), L γ(t0) }.
Remark 3.8.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 including the case where γ(t) is a singular point.Also note that the transversal Jacobi field equation (3.1) is a Morse-Sturm system; see [16,26,28,33].Although the distribution V(t) coincides with the tangent space of L γ(t) until γ crosses a singular leaf, it does not follow straightforwardly that V(t) should always be tangent to the regular 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 §4 obtained as the fibers of an analytic Finsler submersion.x ∈ L and γ x is a horizontal geodesic starting at x so that π • γ x = π • γ p , then the transversal Jacobi field equation along γ p starting at L p and the transversal Jacobi field equation along γ x starting at L x can be identified with each other.As we will see from Lemmas 4.3 and 4.5, this interpretation will also hold for analytic singular Finsler submersions.

Proof of Theorem 1.1
In this section we prove Theorem 1.1 adapting an argument of [1] and using Wilking'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 subset of M , we prove that π is a Finsler submersion on the regular part M 0 and the regular fibers are equifocal submanifolds.In the second part, we will assume that the 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 −1 (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 regular fiber of π by extending this basic vector field along L (Lemma 4.1) and will check part (b) of Theorem 1.1.More precisely, in Lemma 4.3, we will prove that η r ξ (L) is contained in a fiber of π and in Lemmas 4.9 and 4.10, that dη r ξ has constant rank for 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 R p the connected component of p of this regular part (which is open) and ξ a basic vector field on L R p , which is analytic as all the data is analytic.Then as, by hypothesis, the function F • ξ : L R p → R is constant in a neighborhood of p, it must be constant everywhere.Now consider a point q in the boundary of L R p .As F • ξ is constant, if one considers a sequence of points {q n } ⊂ L R p such that lim n→∞ q n = q, then the sequence of vectors ξ qn is precompact in 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.Remark 4.2.As a consequence of Lemma 4.1, the assumption that F π is a Finsler foliation restricted to some open subset U ⊂ M implies that there exists a saturated open subset where π is a Finsler submersion.Namely, consider π(U ), which is open because π is a submersion and then π| π −1 (π(U )) is a Finsler submersion for a certain Finsler metric on π(U ).Lemma 4.3.For all r ∈ R, there exists a value d ∈ B such that η r ξ (L) ⊂ π −1 (d).Proof.Assume that r > 0 (the case r < 0 is analogous and r = 0 is trivial).Consider x, y ∈ L and set Ĩ := {t 0 ∈ [0, r]} such that π(γ ξx (t)) = π(γ ξy (t)), for t ≤ t 0 }.Note that the set Ĩ is not empty, because π restricted to a neighborhood of L is a Finsler submersion (recall Remark 4.2).The set Ĩ is closed due to continuity and it is an open set because π is an analytic map and geodesics of an analytic Finsler metric are also analytic.Therefore Ĩ = [0, r] and this concludes the proof.Fixed x ∈ L, and consider the geodesic γ ξx .Since, by Remark 4.2, π is a Finsler submersion on a neighborhood of x, we can define along the geodesic γ ξx the Wilking's distribution pair t → (H(t), V(t)) (recall Definition 3.7).Lemma 4.4.If x ∈ L and p = γ ξx (t 0 ) is a regular point of π, then the Wilking's distribution V(t 0 ) coincides with the tangent space to L p .Moreover, the distributions t → V(t) and t → H(t) are analytic and the singular points of γ ξx (lying in a singular level set) are isolated.
Proof.Recall by part (c) of Lemma 3.3 that there is a basis J 1 , . . ., J r of V with r = dim V such that at t 0 , J k (t 0 ) = . . .= J r (t 0 ) = 0 and J i (t 0 ) = 0 for i = 1, . . ., k − 1.Then (4.1) is a continuous basis of V(t) for every t in (t 0 − ε, t 0 + ε) for some ε > 0, with lim t→t0 1 t−t0 J i (t) = J i (t 0 ) for all i = k, . . ., r.By Lemma 3.6, J i (t) = dη t ξ (v i ) for some vector v i ∈ T x L for all i = 1, . . ., r, and by Lemma 4.3, all J i (t) are tangent to L γ ξx (t) for all t.By continuity, J j (t) is also tangent to L γ ξx (t) for j = k, . . ., r.This implies that V(t 0 ) is contained in the tangent space to L p .As they have the same dimension, they coincide.The analyticity of V and H follows from the basis of V(t) in (4.1), as the vector fields Jj (t) = 1 t−t0 J j (t) extend analytically to t 0 making Jj (t 0 ) = J j (t 0 ) as J j (t 0 ) = 0 and J j is analytic, for j = k, . . ., r.Up to the singular points of γ ξx , consider an analytic frame of the horizontal space H, namely, analytic vector fields along γ ξx , H 1 , . . ., H n−r such that H 1 (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 ξx , E 1 , . . ., E n−r , the determinant of the transformation matrix with respect to dπ • H 1 , . . ., dπ • H n−r can only have isolated zeroes or being zero everywhere.As it is not zero close to t 0 , it has only isolated zeroes.Even if one cannot ensure the existence of a global frame for π • γ ξx , it is enough to recover its domain with frames which intersect in open subsets to conclude that the singular points along γ ξx are isolated.Observe that, by Lemma 4.1, the singular points of π can only lie in singular levels.Lemma 4.5.If p = γ ξx (t) is a regular point of π, then γ ξx is orthogonal to L p .Proof.Observe that f i (t) = g γξx ( γξx (t), J i (t)) and h i (t) = g γξx ( γξx (t), J i (t)) are analytic functions for each J i ∈ V in the basis of part (c) of Lemma 3.3, and if γ ξx (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 are identically zero, and, in particular, by part (c) of Lemma 3.3, γ ξx is orthogonal to L p .Lemma 4.6.If there exists a neighborhood U of M such that F π | U is a Finsler foliation, then F π is a Finsler foliation when restricted to the open subset of regular points of π.
Proof.Fix q ∈ U .The first observation is that, by Remark 4.2, π is a Finsler submersion when restricted to π −1 (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]) that there is no focal point on [0, b).As γ minimizes the distance from L q , it must be orthogonal 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).Then the map η t ξ : L q → L γ(t) is a diffeomorphism on a neighborhood of γ(0) for every t ∈ [0, b).Fix an instant t 0 ∈ (0, b) such that γ(t 0 ) is a regular point of π (recall that, by Lemma 4.4, singular points are isolated along γ), then by continuity and the compactness of [0, t 0 ], it is possible to choose a neighborhood U ⊂ L q where η t ξ : U → L γ(t) is a diffeomorphism onto the image for all t ∈ [0, t 0 ].Given a vector v horizontal to U 0 = η t0 ξ ( U ) at p 0 = η t0 ξ (q 0 ), for some q 0 ∈ Ũ , consider the Wilking's distributions H q0 (t 0 ) and V q0 (t 0 ) along the horizontal curve γ ξq 0 , and let v h be the projection of v to H q0 (t 0 ).Consider the D h -parallel vector field X along γ ξq 0 such that X(t 0 ) = v h and the vector field Y along γ ξq 0 orthogonal to η t ξ ( U ) at each t ∈ [0, t 0 ] such that its Wilking's projection to H q0 is X (recall [3, Lemma 2.9 (a)]).Let v 1 be the orthogonal vector to L q at γ(0) such that dπ(v 1 ) = dπ(Y (0)).Now repeat the process to obtain a vector field along H γ(0) (the horizontal Wilking's distribution along γ) and Ỹ horizontal to η t ξ ( U ) such that its Wilking's projection to Let us observe that if we prove that then, for t close to 0, F (Y (t)) = F ( Ỹ (t)), because π is a Finsler submersion, but, by analyticity, this will be true for all t ∈ [0, t 0 ].This implies that F (v) = F ( Ỹ (t 0 )), and in particular, that F is constant on basic vector fields along U 0 .By Lemma 4.1, F is constant on basic vector fields along the whole fiber L γ(t0) .Finally, by continuity, this is also true for L p .So, let us prove Eq. (4.2), which is equivalent to prove that dπ(X(t)) = dπ( X(t)).This equivalence follows from the fact that dπ(V , and η t ξ ( U ) ⊂ L γ ξq 0 (t) = L γ(t) (recall that we assume that η t ξ | Ũ is a diffeomorphism onto the image for t ∈ [0, t 0 ]).By analyticity, we only have to prove that this holds for t close to 0. But this is true, because the D h -parallel transport in a Riemannian submersion is the horizontal lift of the parallel transport in the base.In our case, the D h -parallel transport coincides with the lift of the parallel transport induced by gV * , where V * is a (local) geodesic vector field in the base tangent to π • γ, considering the Riemannian metric g V on M , where V is the horizontal lift of V * , being g and g, the fundamental tensors of the Finsler metrics on M and B, respectively (recall Proposition 2.4).So, the proof is concluded.Remark 4.7.Observe that as a consequence of Lemmas 4.4, 4.5 and 4.6, given a geodesic which is horizontal in one regular point, then it is horizontal in all the regular points and, by continuity (using Lemma 4.4) the Wilking distribution does not depend on the regular instant chosen as initial point.Let us see that the solutions to the transversal Jacobi equation of two horizontal geodesics with the same projection can be identified.Lemma 4.8.Consider a geodesic γ of (M, F ) which is horizontal at the regular points.It holds that (a) if X is a solution of the transversal Jacobi equation of γ in Eq. (3.1), then along the regular instants of γ, dπ • X is a Jacobi field of π • γ, and the solutions of the 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 Jacobi equation 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.
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 we restrict 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 a straightforward consequence of part (b).Lemma 4.9.If d is a regular value, then η r ξ : Proof.By Lemma 4.5, γξp (r) is orthogonal to π −1 (d).As π is a Finsler submersion on π −1 (d) (recall Lemma 4.6), we can extend the normal vector γξp (r) to a unit basic vector field ξ along the fiber π −1 (d).It is also possible to check that η −r ξ is the inverse of η r ξ .
We have just proved in Lemma 4.9 that η r ξ : π −1 (c) → π −1 (d) is a diffeomorphism when d is a regular value, which implies that dη r ξ has constant rank.We have to check now that dη r ξ has constant rank when d is a singular value.This will be done in Lemma 4.10.
Proof.As L is connected, it is enough to prove that dim rank d(η r ξ ) x is locally constant.Consider a point p ∈ L. Since the transversal Jacobi field equation along γ ξp starting at L is a Morse-Sturm system, there exists a δ > 0 so that any two points in 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 ξp , and s 1 , s 2 ∈ I δ so that X(s 1 ) = 0 = X(s 2 ), then X(t) = 0 for all t, see e.g.[33,Lemma 2.1].Also note that this δ > 0 is the same for other γ ξx for x ∈ L as a consequence of part (c) of Lemma 4.8.Using again that the critical points of π on γ ξx are isolated, we can also suppose, reducing δ if necessary, that the point γ ξx (r) is the only critical point of π on γ ξx | I δ for x ∈ L.
Let s ∈ (r − δ, r) and p = γ ξp (s).Define the unit vector field ξ along L p so that ξp = γξp (s).Recall that η s ξ : L p → L p is a diffeomorphism (see Lemma 4.9).Set r := r − s.Also note that η r ξ = η r ξ •η s ξ .Therefore to prove that x → dim rank d(η r ξ ) x is locally constant at p ∈ L, it suffices to prove that x → dim rank d(η r ξ ) x is locally constant at p = η s ξ (p) ∈ L p.We claim that L p-focal points along the horizontal segment of geodesic γ ξx | (0,r+δ) are of tangential type.In other words, γ ξx (t 1 ) is a focal point with multiplicity k (0 < t 1 < r + δ) if and only if x is a critical point of the endpoint map η t1 ξ and dim ker dη t1 ξ = k.In order to prove the claim, consider an L p-Jacobi field J along t → γ ξx (t) = γ ξx (t + s) and assume that J(t 1 ) = 0 for some 0 < t 1 < r + δ.Let J be the Jacobi field along γ ξx so that J(t) = J(t + s).Observe that J is the variation vector field of a variation by orthogonal geodesics to L p (see Prop. 3.5) and then its projection is a Jacobi field of π • γ ξx .By part (a) of Lemma 4.8, J h is a solution of the transversal Jacobi equation.As J h (s) = 0 = J h (s + t 1 ), from the choice of I δ and part (c) of Lemma 4.8, we conclude that J h (t) = 0 for all t.Therefore Jh (t) = 0 for all t.This fact and Lemma 3.6 imply that γ ξx (t 1 ) is an L p-focal point if and only if x is a critical point of the endpoint map η t1 ξ .From what we have discussed above, we have concluded that: m(γ ξx ) = dim ker dη r ξ (x), (4.3)where m(γ ξx ) denotes the number of focal points on γ ξx , each counted with its multiplicities.
On the other hand, by continuity of the Morse index we have 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 constant in a neighborhood of p on L p.As η s ξ : L p → L p is a diffeomorphism, it follows that dim ker dη r ξ is constant in a neighborhood of p on L p , which concludes.

Finslerian character.
In this section, we will additionally assume that the fibers constitute a singular foliation and will prove its transnormality.This will follow directly from Lemmas 4.11, 4.5 and 4.15.
Proof.By Lemma 4.5, the geodesic γ is orthogonal to the regular leaves associated with regular values of π.Now suppose that γ(t 0 ) is a singular point and u ∈ T γ(t0) L γ(t0) .Then as (M, F) is a singular foliation, there exists a vector field X in a neighborhood of γ(t 0 ) such that X is tangent to the leaves and X γ(t0) = u.By Lemma 4.4, the singular points along γ are isolated, so g γ(t) (X γ(t) , γ(t)) = 0 for all t close enough to t 0 .By continuity, we conclude that g γ(t0) (u, γ(t 0 )) = 0, as desired.
Note that the above lemma does not directly imply that the foliation F π is a 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 r ξ : π −1 (c) → π −1 (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 that the endpoint map η r ξ : is an open map.And in the second step we have to check that the set η r ξ (π −1 (c)) is a closed subset of π −1 (d).This ends the proof because the fibers are connected.
For q = η r ξ (x 0 ) ∈ π −1 (d), consider a plaque P q of π −1 (d) which contains q and a past tubular neighborhood U = Õ(P q , ε) of the plaque P q of radius ε > 0. By Lemma 4.9 and the fact that the singular points along γ ξx 0 are isolated (see Lemma 4.4) and using Lemma 4.11, we can assume without loss of generality that x 0 ∈ U and γ ξx 0 is the minimizing geodesic from x 0 to P q .This implies that all the geodesics γ ξx with x ∈ Ũ ⊂ U ∩ π −1 (c), being Ũ a small enough open subset of π −1 (c) which contains x 0 , are minimizing geodesics from x to P q , and then where π ν is the (past) footpoint projection.Since F π is a singular foliation, the map π ν | Ũ is open by [3,Lemma 3.10].This fact and Eq.(4.5) imply that the endpoint map η ξ is an open map.Consider a sequence {y n } ⊂ η r ξ (π −1 (c)) that converges to a point y ∈ π −1 (d).Let {x n } ⊂ π −1 (c) be a sequence so that η r ξ (x n ) = y n .Since, for n big enough, x n ∈ B − (y, 2r), denoting by B − (y, 2r) = {x ∈ M : d F (x, y) < 2r}, namely, the backward ball or radius 2r centered at y, we have a convergent subsequence x ni → x ∈ B − (y, 2r).Therefore η r ξ (x) = lim η r ξ (x ni ) = lim y ni = y and hence y ∈ η r ξ (π −1 (c)).
Lemma 4.13.Let L p be a regular leaf, and L q be a singular leaf.Then they are parallel, i.e., if y 0 and y 1 are points of L q , then d(L p , y 0 ) = d(L p , y 1 ) and d(y 0 , L p ) = d(y 1 , L p ).
Proof.We will consider only the future case, as the past one can be recovered by using the reverse Finsler metric F (v) = F (−v).Let U be a totally convex neighborhood of q, namely, given two points p 1 , p 2 ∈ U there exists a unique geodesic joining p 1 to p 2 and this geodesic is minimizing (see [41,42] for their existence).Now consider a plaque Pq of q, a future tubular neighborhood Ũ ⊂ U, a smaller plaque P q of q and a future tubular neighborhood U ⊂ Ũ of P q such that if D > 0 is the diameter of U , then B + (p, 2D) ∪ B − (p, 2D) ⊂ Ũ for every p ∈ U .Let q ∈ U , P q the plaque of q ∈ U , and y 0 , y 1 ∈ P q .Our goal is to check that d(P q , y 0 ) = d(P q , y 1 ).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 ) = (γ i ), for i = 0, 1. Choose an arbitrary regular point p ∈ M .Then there exists a minimizing geodesic from L p to q, whose singular points 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 a sequence of minimal segments of geodesic β m i joining Ppm to γ i (0) (even if Ppm is not necessarily closed, the minimizing geodesic β m i exists because p m ∈ B− (γ i (0), D) ⊂ Ũ by the choice of U ).In particular, d( Ppm , γ i (0)) = (β m i ).Finally, consider minimal segments of geodesic γ m i : [0, r m ] → M joining Ppm to y i , which there exist by the same reason as the β m i 's.It follows that (γ m i ) = d( Ppm , y i ) and (γ m i ) ≤ (γ i β i m ), where stands for the concatenation of the two curves.Observe that the image of the curves γ m i is contained in B− (y i , D).As a consequence, there exists a convergent subsequence γ n k i (0) converging to some Pq Figure 3.The diagram to the left represents the different curves used in the proof of Lemma 4.14.The diagram to the right depicts both plaques P q and P P .The minimizing curves inside the tubular neighborhood U exist as short curves cannot reach the boundary, because of the hypothesis on the diameter.
p ∈ Pq ⊂ Ũ (recall that the plaques Ppm in U must necessarily converge to Pq ).As all the sequence and the limit lie in the convex neighborhood U, we can guarantee that γ n k i converges to a segment of geodesic γ i joining Pq to y i .By Lemma 4.13, (β m i ) → 0 when m → ∞, and hence we conclude that (γ i ) ≤ (γ i ).Since d( Pq , y i ) = d(P q , y i ) = (γ i ), we infer that (γ i ) = (γ i ).
On the other hand, as the points p m are regular, one can apply the same techniques of parallel transport of the proof of Lemma 4.13 to prove that (γ m 0 ) = (γ m 1 ) and hence (γ 1 ) = (γ 0 ).Therefore (γ 0 ) = (γ 1 ) and this concludes the proof of the lemma.Lemma 4.15.Let x be a point of a singular leaf L q .Then the geodesic γ ξx is horizontal, i.e., orthogonal to the leaves of the singular foliation F = {π −1 (c)}.
Proof.The proof follows directly from Lemmas 4.14 and 2.5.

5.
On Jacobi fields and curvature in Finsler Geometry 5.1.Anisotropic connection and curvature.In this section we review a few facts about anisotropic connections, see [23,24].
As the tensors associated with Finsler metrics depend on the direction v ∈ T M \ 0, it is necessary a connection that take into account this fact, namely, it depends on the direction.In the following, we will denote by X(M ) the space of smooth vector fields on M and by F(M ), the subset of smooth real functions on M .Definition 5.1.An anisotropic connection is a map Observe that the properties (ii) and (iii) imply, respectively, that ∇ v X Y depends only on the value of Y on a neighborhood of p, and on the value of X p .So sometimes we will put only X p and it will make sense to compute ∇ v X Y in a chart.If (Ω, ϕ = (x 1 , . . ., x n )) is a chart of M , with its associated partial vector fields denoted by ∂ 1 , . . ., ∂ n , the Christoffel symbols of ∇ are the functions Γ k for i, j = 1, . . ., n.Given a vector field V without singularities on an open set Ω ⊂ M , the anisotropic connection ∇ induces an affine connection , where X, Ỹ ∈ 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, 3)-anisotropic tensor P given by for p ∈ M , v ∈ T p M \ {0} and X, Y, Z ∈ X(M ), then it is possible to define the curvature tensor of the anisotropic connection as the tensor such that (P V ) p = P Vp 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 the vector 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 compute the tensor derivative of any anisotropic tensor (see [23, §3]), and this computation can be done using ∇ V (see [23,Remark 15]).In most of the references of Finsler Geometry, they use a connection along the vertical bundle of T M .For a relationship between both types of connections see [23, §4.4].

5.2.
A few proofs about Finsler Jacobi fields.Here, for the sake of completeness, 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 1 , J 2 ∈ W. Using the shape operator S γ(t0) (see Definition 2.1), it follows that [25,Prop. 3.5]).As we saw in Remark 3.2 this implies that W is self-adjoint for all t.
5.2.3.Proof of Lemma 3.6.The fact that (b) implies (a) follows from Proposition 2.3, 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 0 ), because J is vertical.In fact assume by contradiction that there exists another vertical L-Jacobi field J so that J(t 0 ) = J(t 0 ) and J = J.Set Ĵ := J − J. Hence Ĵ is a vertical Jacobi field with Ĵ(t 0 ) = 0, and then Ĵ (t 0 ) = lim t→t0 1 t−t0 Ĵ(t), which implies that Ĵ (t 0 ) is also tangent to L. Being Ĵ L-Jacobi (recall Def.2.1), it follows that the tangent part to L of Ĵ (t 0 ) is zero, which concludes that Ĵ = 0, as both initial conditions are zero.Now consider a curve β : (−ε, ε) → L such that β(0) = J(t 0 ) and the Jacobi field J(t) = ∂ ∂s ψ(t, 0), being ψ the variation defined in part (b).The fact that (a) implies (b) follows from the above discussion since J(t 0 ) = J(t 0 ) = ∂ ∂s ψ(t, 0), and then J = J.

Proposition 2 . 4 .
Let π : (M, F ) → (B, F ) be a Finsler submersion and V * a geodesic field in some open subset Ũ of B. Then the horizontal lift V of V * is a geodesic vector field on U = π −1 ( Ũ ) and the restriction π|

Figure 1 .
Figure 1.Figure generated by the software geogebra.org,illustrating Figure 1.Figure generated by the software geogebra.org,illustrating a few concepts of §2.5, such as a future cylinder C + r (Pq) of a plaque Pq, other plaques contained in this future cylinder and the normal cone at q denoted by νqPq, i.e, the set of vectors ξ ∈ TqM so that g ξ (ξ, X) = 0 for all X ∈ TqPq.Here the leaves are the fibers of the Finsler submersion π : (M, F ) → (B, F ), where M = R 3 , B = R 2 , π(x) = (x1, x2) and F and F are Randers metrics having the Euclidean metrics and the winds W = ( 1 2 , 0, sin 2 (x 1 )+1 4
[17, implies, taking into account the equation above[17, Eq.(1.7.7)], that Eq. (3.1) holds for every regular instant, 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].