On the Focal Locus of Submanifold of a Finsler Manifold
Aritra Bhowmick Sachchidanand Prasad


A. Bhowmick]Aritra Bhowmick
Department of Mathematics, Indian Institute of Science, Bangalore, India
aritrab@iisc.ac.in
S. Prasad]Sachchidanand Prasad
School of Mathematics, Jilin University, China
Faculty of Mathematics and Computer Science, Göttingen University, Germany
sachchidanand.prasad1729@gmail.com

Focal Locus of Finsler Submanifold]On the Focal Locus of Submanifold of a Finsler Manifold


[2020]Primary: 53C22, 53B40; Secondary: 53C60




[
    [
    September 9, 2024
=====================



§ ABSTRACT 
	In this article, we investigate the focal locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. The main goal is to show that the associated normal exponential map is regular in the sense of F.W. Warner (Am. J. of Math., 87, 1965). This leads to the proof of the fact that the normal exponential is non-injective near tangent focal points. As an application, following R.L. Bishop's work (Proc. Amer. Math. Soc., 65, 1977), we express the tangent cut locus as a closure of a certain set of points, called separating tangent cut points. This strengthens the results from the present authors' previous work (J. Geom. Anal., 34, 2024).




§ INTRODUCTION

One of the primary objects of study in Riemannian geometry are the geodesics, which are (locally) distance minimizing curves. A geodesic always arises as the solution to a first-order initial value problem, which gives rise to the exponential map. Given a complete Riemannian manifold, it is a smooth map from the tangent bundle to the base manifold. The singularities of the exponential map are of particular interest, as they are related to variations of geodesics. For a given submanifold, one can define the normal bundle to it, and the restriction of the exponential map is then called the normal exponential map. The focal locus of the submanifold consists of the critical values of the normal exponential map. A closely related concept is that of the cut locus, originally introduced by Henri Poincaré <cit.>. The cut locus of a submanifold consists of points, beyond which a globally distance-minimizing geodesic from the submanifold fails to be distance-minimizing. Both the cut locus and the focal locus have been studied extensively in the literature, see <cit.>.

Finsler manifolds are a natural generalization of the Riemannian ones, which were first studied by P. Finsler in his dissertation <cit.>. A Finsler metric on a manifold is a parametrized collection of Minkowski norms on each tangent space, which allows us to measure the length of a tangent vector. With this generality, we encounter certain challenges as well, primarily stemming from the fact that we cannot measure the angle between two tangent vectors, unlike the case of a Riemannian metric. Still, most of the results in Riemannian geometry can be translated to Finsler geometry, with suitable modifications. See <cit.> for a survey of results. In particular, the notions of cut and focal loci have their counterpart in a Finsler manifold.

The study of submanifolds in a Finsler manifold has been sporadic <cit.>. One of the first hurdles to cross is that in the absence of an inner product, the suitable generalization of a normal bundle of a submanifold is no longer a vector bundle, and it is only a topological manifold. In <cit.>, we have systematically studied the cut locus of a submanifold in a Finsler manifold, and showed that most of the well-known results in the Riemannian context still hold true. In the same spirit, in this article we study the focal locus of a submanifold in a Finsler manifold.

The primary goal of this article is to show that the normal exponential map associated to a submanifold in a Finsler manifold is regular in the sense of <cit.>, which is proved in <ref>. Along the way, we obtain <ref> and <ref>, which were stated without proof in <cit.> for submanifolds in Riemannian manifolds. For completeness, we give detailed proof in the Finsler setup. As a consequence, in <ref>, we obtain local normal forms for the normal exponential map near certain regular tangent focal points. This culminates in the following important result.

[<ref>]
    The normal exponential map of a submanifold in a (forward) complete Finsler manifold is not injective in any neighborhood of a tangent focal point.


Let us point out why the above result is quite striking. In general, a smooth map can have singularities, and yet be globally injective. As an example, consider the map x ↦ x^3 on ℝ, which has a singularity at the origin, and yet is (globally) injective. On the other hand, by the inverse function theorem, a non-singular map is locally injective. The above result can be thought of as a partial converse to this statement for the normal exponential map. This was originally proved for the exponential map exp_p : T_p M → M at a point p of an analytic Finsler manifold in <cit.> and later for C^∞ manifold in <cit.>. Later, Warner proved the same for any smooth Riemannian or Finsler manifold. In recent years, similar results have been proved for the sub-Riemannian exponential maps <cit.>.

Next, we look at the first tangent focal locus in particular, which, as the name suggests, consists of the first tangent focal locus encountered along a geodesic emanating from a submanifold. It is well-known that if a point is in the cut locus, then either it is the endpoint of at least two distinct distance minimizing geodesics, or it is a first focal locus. Furthermore, the points where two or more distance minimizing geodesic meet, called separating points in this article, are dense in the cut locus <cit.>. Both the cut points and separating points have their counterparts in the normal bundle, respectively called the tangent cut points, and the separating tangent cut points. In the same vein as <cit.>, we prove the following.

[<ref>]
    Given a compact submanifold of a forward complete Finsler manifold, the set of tangent cut points is the closure of the set of separating tangent cut points.


As a corollary, we reprove <cit.>, which states that the separating set is dense in the cut locus.



Conventions. Throughout this article, boldface symbols, e.g. 𝐮,𝐯,𝐱,𝐲, will always denote an element of a tangent space.

Organization of the article. Falana



§ DRAFT IDEAS

Some related articles : <cit.>.

    
  * Non-injectivity of the Normal Exponential Map following <cit.>. <ref>.
    
  * Decomposition of the Cut Locus following <cit.>, showing that tangent Cu(N) = Se(N). <ref>.
    
  * Conjugate locus of constant order <cit.>. Uses Morse theory. Proves a classification result.
    
  * Odd Dimensional Manifold with Regular Conjugate Locus <cit.>. Rules out a possibility in <cit.>.
    
  * Cartan-Ambrose-Hicks Theorem <cit.>. Uses Morse theory and <cit.>.
    
  * Regularity of the Focal Locus <cit.>. This is the one that “reports” (as stated in <cit.>) Warner's result can be easily proved for submanifolds of Riemannian manifolds. Then, it proves the disc bundle decomposition under extra hypothesis. Uses results from <cit.>.
    
  * Tangent cut locus of with exactly two separating vectors <cit.>. Uses <cit.> as a hypothesis.



§ PRELIMINARIES ON FINSLER GEOMETRY
 
In this section, we collect a few definitions and results on Finsler geometry. We refer to <cit.> as primary references.



 §.§ Finsler Metric


    Let M be a smooth manifold, and TM denotes its tangent bundle. A Finsler metric on M is a continuous function F: TM →ℝ satisfying the following properties.
    
        
  * F is smooth on TM;
        
  * for any p∈ TM, the restriction F_p   F|_T_pM is a Minkowski norm, i.e.,
        
            
  * (Positive 1-homogeneity) for any λ>0 and 𝐯∈ T_pM∖{0}, we have F_p(λ𝐯)=λ F_p(𝐯), and 
            
  * (Strong Convexity) for all 𝐯∈ T_pM∖{0}, the symmetric tensor g_𝐯 on T_pM, called the fundamental tensor, is positive definite, where 
            
    g_𝐯(𝐯_1,𝐯_2) .12∂^2∂ s_1 ∂ s_2|_s_1=s_2=0(F_p(𝐯 + s_1𝐯_1 + s_2𝐯_2))^2.

        
    
    F is reversible if F(-𝐯) = F(𝐯) holds for all 𝐯∈TM.


For 𝐯∈ T_p M ∖ 0, the associated Cartan tensor on T_p M is a symmetric 3-tensor defined as

    C_𝐯(𝐯_1,𝐯_2,𝐯_3) . 1/4∂^3/∂ s_1 ∂ s_2 ∂ s_3|_s_1=s_2=s_3=0( F_p(𝐯 + s_1 𝐯_1 + s_2 𝐯_2 + s_3 𝐯_3) )^2.

For each 𝐯∈ T_p M ∖ 0 and 𝐮, 𝐰∈ T_p M, we have the following relations

    C_𝐯(𝐯, 𝐮,𝐰) = C_𝐯(𝐮, 𝐯, 𝐰) = C_𝐯(𝐮, 𝐰, 𝐯) = 0.

We extend the definition of the fundamental tensor and the Cartan tensor to vector fields. For any V∈ΓTM and X,Y,Z ∈Γ TM defined near p ∈ M, we define

    g_V(X, Y)(p)   g_V_p(X_p, Y_p),      C_V(X, Y, Z)(p)  C_V_p( X_p, Y_p, Z_p ).




 §.§ Chern Connection
 Unlike the Levi-Civita connection in the Riemannian context, we do not get a canonical connection that is both torsion free and metric compatible in a suitable sense. In this article, we consider the Chern connection, which is a family of torsionless connections on a Finsler manifold.

<cit.>
    For each V ∈ΓTM, we have a unique affine connection 
    ∇^V: Γ TM ⊗Γ TM →Γ TM,
 called the Chern connection, satisfying the following conditions for any X, Y, Z ∈Γ TM.
    
        
  * (Torsion freeness) ∇^V_X Y - ∇^V_Y X = [X, Y].
        
  * (Almost metric compatibility) X (g_V(Y,Z)) = g_V(∇^V_X Y, Z) + g_V(Y, ∇^V_X Z) + 2 C_V(∇^V_X V, Y, Z).
    


The value of ∇^V_X Y|_p is independent of the values of V and X other than at p, and consequently ∇^𝐯_𝐱 Y|_p is well-defined for any vector 𝐯, 𝐱∈ T_p M with 𝐯 0, and for any vector field Y defined near p. Similarly, given a curve γ : [a, b] → M and V ∈Γγ^*TM, one can define ∇^V_X Y ∈Γγ^*TM for any X, Y ∈Γγ^*TM. This leads to a covariant derivative along a γ.
<cit.>
    Given a curve γ : [a,b] →  M and W ∈Γγ^*TM, the covariant derivative along γ is defined as 
    D^W_γ : Γγ^* TM →Γγ^* TM,

    which satisfies the following.
    
        
  * (Linearity) For any X, Y ∈Γγ^*TM and scalars α, β∈ℝ we have 
        
    D^W_γ(α X + β Y) = α D^W_γ X + β D^W_γ Y.


        
  * (Leibniz rule) For any X ∈Γγ^*TM and a smooth function f : [a,b] →ℝ, we have 
        
    D^W_γ(f X) = df/dt X + f D^W_γ X.

        
        
  * (Almost metric compatibility) For any X, Y ∈Γγ^* TM, we have 
    d/dtg_W(X,Y) = g_W ( D^W_γ X, Y ) + g_W ( X, D^W_γ Y ) + 2 C_W ( D^W_γ W, X, Y ).

    


If for some curve γ we have γ̇(t)  0 for all time, then we shall use the notations 

    Ẋ D^γ̇_γ X,   Ẍ D^ġ_γẊ = D^γ̇_γ D^γ̇_γ X,     for any X ∈Γγ^*TM,

provided the curve γ is understood from the context. A vector field X ∈Γγ^*TM is said to be parallel with respect to γ if Ẋ = 0.



 §.§ Geodesics and Jacobi Fields

Let us denote the space of piecewise smooth paths γ : [a,b] → M as 𝒫 = 𝒫([a,b]). We have two functionals defined on this path space, namely the length and the energy functionals.

    L : 𝒫   ⟶ℝ
    γ   ⟼∫_a^b F(γ̇(t)) dt
    ,         
        E : 𝒫   ⟶ℝ
    γ   ⟼1/2∫_a^b F(γ̇(t))^2 dt

A curve γ∈𝒫 is a geodesic if it is a critical point of the energy functional with respect to proper variations. Recall that given a piecewise smooth vector field W ∈Γγ^* TM, a W-variation of γ is a piecewise smooth map Λ : (-ϵ, ϵ) × [a,b] → M satisfying 
    Λ(0,t) = γ(t),   .∂/∂ s|_s=0Λ(s, t) = W(t),

for all (s,t) in the domain. The variation is proper if W(a) = 0 = W(b), or equivalently, if Λ(s, a) = γ(a) and Λ(s, b) = γ(b) for all s. Geodesics are locally distance minimizing, where we have the Finsler distance between p, q ∈ M defined as
    
    d(p, q) = inf{ L(γ)  | γ∈𝒫, γ(a) = p, γ(b) = q }.

In general, geodesics fail to be globally distance minimizing. We say a geodesic γ is a global distance minimizer (or simply a minimizer) if γ is unit-speed, and d(γ(a), γ(b)) = b - a = L(γ).

Geodesics are precisely the solutions to an initial value problem known as the geodesic equation (e.g., <cit.>), which can be written succinctly as γ̈= 0 with the notation as above. As such, geodesics are always smooth, and given a vector 𝐯∈ T_p M, we have a unique maximal geodesic γ_𝐯 : [0,ℓ] →  M satisfying 

    γ_𝐯(0) = p,     γ̇_𝐯(0) = 𝐯.

Note that, unlike Riemannian geometry, due to the asymmetry of the Finsler metric, the reversed curve γ̅:[0, ℓ] → M defined by γ̅(t) = γ(ℓ - t) need not be a geodesic. Nevertheless, γ̅ is a geodesic with initial velocity -γ̇(ℓ) for the reverse Finsler metric F̅ defined by F̅(𝐯) = F(-𝐯) for all 𝐯∈ TM <cit.>.

A Finsler manifold (M, F) is said to be forward complete if for all 𝐯∈ TM, the geodesic γ_𝐯 is defined for all time [0, ∞). We say (M, F) is backward complete if (M, F̅) is forward complete. By the Hopf-Rinow theorem (<cit.>), if a Finsler manifold (M, F) is forward or backward complete, then given any two points p, q ∈ M there exists a minimizer with respect to F, and another (possibly distinct) minimizer with respect to F̅ joining p to q. Throughout this article, we shall assume (M, F) to be a forward complete Finsler manifold, unless otherwise stated.


    The exponential map exp : TM →  M at a point p ∈ M is defined as exp_p(𝐯) = γ_𝐯(1) for any 𝐯∈ T_p M, where γ_𝐯 : [0,∞) → M is the unique geodesic starting at p with initial velocity 𝐯.


It follows from the theory of ordinary differential equation that the exponential map is smooth on TM, but only C^1 on the whole tangent bundle, see <cit.> for details.



  §.§.§ Jacobi Fields

Given a geodesic γ : [a,b] → M, consider a geodesic variation Λ : (-ϵ, ϵ) × [a,b] → M of γ, that is, for each -ϵ < s < ϵ we require the curve Λ_s : [a,b] → M given by Λ_s(t) Λ(s, t) to be a geodesic.


    Let γ : [a,b] → M be a geodesic. A vector field J ∈Γγ^* TM along γ is called a Jacobi field along γ if there exists a geodesic variation Λ : (-ϵ, ϵ) × [a,b] → M of γ satisfying 
    
    . ∂/∂ s|_s=0Λ(s,t) = J(t),    t∈ [a,b].



Given a geodesic γ : [a, b] → M, and for any U, W ∈Γγ^*TM, one can define a tensor field R^γ (γ̇, U) W along γ (see <cit.>). Then, a vector field J ∈Γγ^*TM is a Jacobi field if and only if it satisfies the Jacobi equation 
    
    D^γ̇_γ D^γ̇_γ J - R^γ(γ̇, J)γ̇= 0.

Since the Jacobi equation is a second order ODE, given the initial data, 𝐮, 𝐯∈ T_γ(a)M, there exists a unique Jacobi field J along γ satisfying, J(a) = 𝐮 and J̇(a) = D^γ̇_γ J(a) = 𝐯. In particular, the collection of all Jacobi fields along γ forms a vector space of dimension 2  M. Let us make an important observation regarding the zeros of a Jacobi field.


	Given a non-vanishing Jacobi field J along a geodesic γ : [0, ℓ] → M, if J(t_0) = 0 for some 0 ≤ t_0 ≤ℓ, then J̇(t_0)  0. Moreover, the zeros of J are isolated.


	Suppose J(t_0) = 0 for some 0 ≤ t_0 ≤ℓ. If possible, suppose J̇(t_0) = 0. If t_0 = 0, then we must J ≡ 0, by the uniqueness of Jacobi fields. Suppose t_0 > 0. 	We have geodesic variation Λ : (-ϵ, ϵ) × [0, ℓ] → M such that J(t) = ∂/∂ s |_s=0Λ(s, t). Define Λ̅ : (-ϵ, ϵ) × [0, t_0] → M by Λ̅(s, t) = Λ̅(s, t_0 - t). Then, Λ̅ is a geodesic variation with respect to the reverse Finsler metric F̅. In particular, J̅(t) ∂/∂ s |_s=0Λ̅(s, t) = -J(t) is a Jacobi field along the geodesic γ̅(t) = γ(t_0 - t) defined on [0, t_0], with respect to F̅. By our assumption, 
	
    0 = .D^γ/dt|_t=t_0 J(t) = .∂^2/∂ t∂ s|_(t_0,0)Λ(s, t) = - .∂^2/∂ t∂ s|_(0,0)Λ̅(s, t) = - .D^γ̅/dt|_t = 0J̅(t).

	But then J̅≡ 0 along γ̅, as the initial conditions are J̅(0) = 0 and D^γ̅/dt|_t=0J̅(t) = 0. Consequently, we have J ≡ 0 along γ, a contradiction. Hence, J(t_0) = 0 implies J̇(t_0)  0.

	Now, let us fix some frame of parallel vector fields { e_1(t), …, e_n(t) } along γ, near t_0, where n =  M. We can write, J = ∑_i = 1^n J^i(t) e_i (t) near t_0. Since ė_i = 0, we have 0 J̇(t_0) = ∑J̇^i(t_0) e_i(t_0). Hence, for some 1 ≤ i_0 ≤ n we get J^i_0(t_0) = 0 but J̇^i_0(t_0)  0. Consequently, J^i_0 is locally injective near t_0, and in particular, J^i_0 is non-zero in a deleted neighborhood of t_0. It follows that zeros of J are isolated.




 §.§ Submanifolds in Finsler Manifolds

Given a submanifold N of a Finsler manifold (M, F), we consider the normal cone bundle as the natural replacement for normal bundles.


    Given a submanifold N ⊂ M, the set 
    ν_p = ν_p(N) = {𝐯∈ T_p M ∖{0} |  g_𝐯(𝐯,𝐰) = 0  ∀𝐰∈ T_p N}∪{0}

    is called the normal cone of N at p ∈ N. The set ν = ν(N) = ∪_p ∈ Nν_p(N) is called the normal cone bundle of N. The unit normal cone bundle of N is denoted as S(ν) = ∪_p ∈ N S(ν_p), where S(ν_p) {𝐯∈ν_p  |  F_p(𝐯) = 1 }.


Given a 0 𝐧∈ν_p we have a direct sum decomposition of T_p N defined using the fundamental tensor g_𝐧 which is nondegenerate. In particular, for any 𝐯∈ T_p M we can uniquely write 

    𝐯 = 𝐯^⊤_𝐧 + 𝐯^⊥_𝐧∈ T_p N ⊕( T_p N )^⊥_𝐧.


It should be noted that ν(N) is not a vector bundle in general; in fact, it is a cone bundle. That is to say for any 0 𝐯∈ν we have λ𝐯∈ν for all λ≥ 0. We shall denote ν̂_p ν_p ∖{ 0 }, and ν̂∪_p∈ Nν̂_p is then called the slit cone bundle. It follows that ν̂ (resp. S(ν)) is a smooth submanifold of TM of dimension N +  N =  M (resp. M - 1), whereas ν is only a topological submanifold. In <ref>, we shall see a natural way to identify the tangent space to ν̂ at some 𝐯.

Recall the Legendre transformation ℒ : TM → T^*M <cit.>, which is a fiber-wise homeomorphism, restricting to a C^∞-diffeomorphism in the complement of the zero sections. For any 𝐯, 𝐰∈ TM we have

    ℒ(𝐯)(𝐰) = 
    
        g_𝐯( 𝐯, 𝐰),     𝐯 0, 
    
        0,     𝐯 = 0.

It is easily seen that ℒ maps ν bijectively onto the annihilator bundle of TN. Thus, the cone bundle ν is fiber-wise (non-linearly) homeomorphic to a vector bundle, and ν̂ is C^∞-diffeomorphic to the complement of the zero section in the annihilator bundle. As a consequence, for any 𝐯∈ν̂, we can define a smooth local extension V ∈Γν̂ of 𝐯 via ℒ.

Given 𝐧∈ν̂_p, we have the second fundamental form of N in the direction of 𝐧 defined as

    Π^𝐧 : T_p N ⊙ T_p N    ⟶( T_p N )^⊥_g_𝐧
    ( 𝐱, 𝐲)    ⟼  - ( ∇^𝐧̃_X Y |_p)^⊥_𝐧,

where X, Y ∈Γ TN and 𝐧̃∈Γν̂ are  arbitrary local extensions of 𝐱, 𝐲, 𝐧 respectively. The map is well-defined, as the assignment can be seen to be C^∞(N)-linear. Furthermore, as the Chern connection is torsion-free, we have 

    Π^𝐧( 𝐱, 𝐲) - Π^𝐧( 𝐱, 𝐲) = ( ∇^𝐧_X Y|_p - ∇^𝐧_Y X|_p )^⊥_𝐧 = ( [X, Y]_p )^⊥_𝐧 = 0.

In other words, Π^𝐧 is a symmetric linear map. Taking adjoint with respect to g_𝐧, we define the shape operator A_𝐧 : T_p N → T_p N of N along 𝐧 via the equation

    g_𝐧( A_𝐧𝐱, 𝐲) = g_𝐧( 𝐧, Π^𝐧( 𝐱, 𝐲) ),   𝐱, 𝐲∈ T_p N.


    For any 0 𝐧∈ν_p and 𝐱∈ T_p N, we have 
    
    A_𝐧(𝐱) = ( ∇^𝐧̃_X 𝐧̃|_p )^⊤_𝐧,

    where X ∈Γ TN, 𝐧̃∈Γν̂ are arbitrary local extensions of 𝐱, 𝐧 respectively.


    For any 𝐲∈ T_p N get an extension Y ∈Γ TN so that g_𝐧̃( 𝐧̃, T ) = 0. Then, 
    
    0 = X( g_𝐧̃( 𝐧̃, Y ) ) = g_𝐧̃( ∇^𝐧̃_X 𝐧̃, Y ) + g_𝐧̃( 𝐧̃, ∇^𝐧̃_X Y ) + 2C_𝐧̃( ∇^𝐧̃_X 𝐧̃, 𝐧̃, Y ).

    The last term vanishes by <ref>. Hence, evaluating at p we have
    
    g_𝐧( A_𝐧𝐱, 𝐲) 
               = g_𝐧( 𝐧, Π^𝐧( 𝐱, 𝐲) ) 
       = g_𝐧( 𝐧, - ( ∇^𝐧̃_X Y|_p )^⊥_𝐧) 
       = g_𝐧( 𝐧, -∇^𝐧̃_X Y|_p ),     as  g_𝐧( 𝐧, -( ∇^𝐧̃_X Y|_p )^⊤_𝐧) = 0 
       = g_𝐧( ∇^𝐧̃_X 𝐧̃|_p, 𝐲) 
       = g_𝐧( ( ∇^𝐧̃_X 𝐧̃|_p )^⊤_𝐧, 𝐲),     as  g_𝐧( ( ∇^𝐧̃_X 𝐧̃|_p )^⊥_𝐧, 𝐲) = 0.

    Since 𝐲∈ T_p N is arbitrary and g_𝐧|_T_p N is nondegenerate, we have the claim.



    Given N ⊂ M, the normal exponential map exp^ν: ν(N) →  M is defined as the restriction of the exponential map to the cone bundle ν(N).


As ν̂ is a smooth submanifold of TM, the restricted map exp^ν|_ν̂ is smooth. We shall denote the restriction as 

    ℰexp^ν|_ν̂.




 §.§ N-Geodesics and N-Jacobi Fields

Given a submanifold N ⊂ M, we have the subspace of piecewise smooth paths 
    𝒫_N = 𝒫_N([a,b]) = {γ∈𝒫([a,b])  | γ(a) ∈ N}
 starting at N. For any γ∈𝒫_N we say γ is a curve joining N to γ(b). Given q ∈ M, the distance from N to q is then defined as
    
    d(N,q) inf{ L(γ)  | γ∈𝒫_N,  γ(b) = q }.

We identify the tangent space of 𝒫_N at a given γ∈𝒫_N as the infinite dimensional vector space consisting of piecewise smooth vector fields W ∈Γγ^* TM with W(a) ∈ T_γ(a) N. Given W ∈ T_γ𝒫_N, a W-variation is then a piecewise smooth map Λ : (-ϵ,ϵ) × [a,b] →  M satisfying

    Λ(0, t) = γ(t),   . ∂/∂ s|_s=0Λ(s, t) = W(t),   Λ(s, a) ∈ N,

for all (s,t) in the domain of the definition. The variation is proper, if furthermore we have W(b) = 0, or equivalently if Λ(s, b) = γ(b) for all s.


    A piecewise C^1 curve γ : [a,b] → M in 𝒫(N) is called an N-geodesic if γ is a critical point of the restricted energy functional E|_𝒫(N) with respect to proper variations. An N-geodesic γ is called an N-segment joining N to γ(b) if γ is unit-speed, and d(N, γ(b)) = b - a = L(γ).


We recall the following useful lemma, which can be compared to <cit.>. 

 <cit.>
    Suppose, γ_i : [0, ℓ_i] → M are unit-speed minimizers joining p_i = γ_i(0) to q_i = γ_i(ℓ_i), where ℓ_i = d(p_i, q_i). Suppose ℓ_i →ℓ. If either
    

        (a) p_i → p and F is forward complete, .5cm or .5cm (b) q_i → q and F is backward complete,
    

    then a subsequence of γ_i converges uniformly to a minimizer γ, which satisfies L(γ) = ℓ.


As an immediate consequence, we get the following.

<cit.>
    Suppose (M, F) is a forward complete Finsler manifold, and N is a closed submanifold of M. Let γ_i : [0, ℓ_i] → M be N-segments joining N to q_i γ_i(ℓ_i). Suppose that either
    
    𝖧(a) N is compact, .5cm or .5cm (b) F is backward complete as well.

    If ℓ_i →ℓ and q_i → q, then there exists an N-segment γ : [0, ℓ]→ M joining N to q, with a subsequence γ_i →γ. In particular, for any q ∈ M there exists an N-segment γ joining N to q, with d(N, q) = L(γ).


We would like to point out that the hypothesis ([eq:hypothesisH]𝖧) is automatically satisfied for a closed (not necessarily compact) submanifold of a complete Riemannian manifold since the notion of forward and backward completeness coincides there. On the other hand, in a forward but not backward complete Finsler manifold, the distance from a closed submanifold to a point may not even be achieved on the submanifold, see <cit.> for an example.



  §.§.§ N-Jacobi Fields

Given a unit-speed N-geodesic γ : [a,b]→ M, a variation Λ : (-ϵ, ϵ) × [a,b] → M is said to be an N-geodesic variation if for each -ϵ < s < ϵ, the curve Λ_s : [a,b] → M is an N-geodesic.


    Let γ : [a,b] → M be a unit-speed N-geodesic. A vector field J ∈Γγ^*TM is called an N-Jacobi field of γ if there exists an N-geodesic variation Λ:(-ϵ,ϵ)× [a,b]→ M satisfying 
    
    .∂/∂ s|_s=0Λ(s,t) = J(t),    t∈[a,b].



We have the following characterization of an N-Jacobi field via the Jacobi equation.

<cit.>
    Given an N-geodesic γ:[a,b]→ M with initial velocity 𝐯 = γ̇(a), a vector field J ∈Γγ^*TM is an N-Jacobi field if and only if it satisfies the following initial value problem
    
    D^γ̇_γ D^γ̇_γ J - R^γ(γ̇, J) γ̇= 0,    J(a)∈ T_γ(a) N,   J̇(a) - A_𝐯( J(a) ) ∈( T_p N )^⊥_𝐧.




    Given an N-geodesic γ : [a,b]→ M, consider the vector field J(t) = t γ̇(t) along γ. Then, J̇ = γ̇+ t γ̈= γ̇⇒J̈ = 0. On the other hand, R^γ(γ̇, J)γ̇= t R^γ(γ̇, γ̇) γ̇= 0 by the anti-symmetry of the tensor. Thus, J satisfies the Jacobi equation. Also, J(a) = 0 ∈ T_γ(a)N and J̇(a) = γ̇(a) ∈ν_γ(a). Since, A_γ̇(a)( J(a) ) = 0, it follows that J is an N-Jacobi field along γ. Note that J(b)  0.


As an N-Jacobi field is itself a Jacobi field, by <ref>, zeros of N-Jacobi fields are isolated as well. We shall need the following result.

    Given an N-geodesic γ : [0, ℓ] → M and two N-Jacobi fields J, K along γ, we have g_γ̇(J, K̇) = g_γ̇(J̇, K).


    Since γ is a geodesic, we have D^γ̇_γγ̇= 0. Hence, for any two vector fields X, Y ∈Γγ^*TM we have 
    
    d/dt g_γ̇( X, Y ) = g_γ̇( Ẋ, Y ) + g_γ̇( X, Ẏ) + 2 C_γ̇( D^γ̇_γγ̇, X, Y  )_0.

    Now, for two Jacobi fields J, K along γ we compute
	
    d/dt[ g_γ̇( J, K̇) - g_γ̇( J̇, K )] 
    
    		=     g_γ̇( J, K̈) - g_γ̇( J̈, K ) 
    
    		=     g_γ̇( J, R^γ_γ̇(γ̇, K) γ̇) - g_γ̇( R^γ_γ̇(γ̇, J) γ̇, K ),   by <ref>
    
    		=     g_γ̇( J, R^γ_γ̇(γ̇, K) γ̇) - g_γ̇( R^γ_γ̇(γ̇, K) γ̇, J ),   by symmetry of the curvature tensor
    
    		=     0.

    Now, evaluating at t = 0, from <ref> and  <ref> we get
    
    g_𝐯( J(0), K̇(0) ) = g_𝐯( J(0), A_𝐯( K(0) ) ) = g_𝐯( 𝐯, Π^𝐯( J(0), K(0) )).

    Since the second fundamental form Π^𝐯 is symmetric, we get g_γ̇( J(0), K̇(0) ) - g_γ̇( J̇(0), K(0) ) = 0. Consequently, 
	g_γ̇( J, K̇) = g_γ̇( J̇, K ) holds for all time t.




 §.§ Cut Locus of a Submanifold

The cut locus of a point p ∈ M is the set consisting of all q ∈ M such that there exists a minimizer from p to q, any extension of which fails to be distance minimizing. We denote the cut locus of p by Cu(p). Generalizing this notion to an arbitrary submanifold, we have the following definition.


    Given a submanifold N ⊂ M, the cut locus of N consists of points q ∈ M such that there exists an N-segment joining N to q, whose extension fails to be an N-segment. Given 𝐯∈ S(ν), the cut time of 𝐯 is defined as 
    
    ρ(𝐯) sup{ t  |  d(N, γ_𝐯(t 𝐯)) = t },

    where γ_𝐯(t) = exp^ν(t 𝐯) is the unique N-geodesic with initial velocity 𝐯.


Note that we allow ρ to take the value ∞, although if M is compact then ρ(𝐯) < ∞ for any 𝐯∈ S(ν). The map ρ: S(ν ) →  [0, ∞] is continuous <cit.>. It follows from <cit.> that ρ is always strictly positive for any closed N, not necessarily compact. The tangent cut locus of N is defined as 

    Cu(N) {ρ(𝐯) 𝐯 | 𝐯∈ S(ν),  ρ(𝐯) ∞}⊂ν.

It follows from <ref> that Cu(N) = exp^ν(Cu(N)).
    
In order to characterize the points of Cu(N), we introduce the notion of separating sets, originally called the several geodesics set in <cit.>. In the terminology of <cit.>, this is also known as ordinary cut points, see <ref>.

    Given a submanifold N ⊂ M, a point p ∈ M is said to be a separating point of N if there exist two distinct N-segments joining N to p. The collection of all separating points of N is called the separating set of N, denoted Se(N).


Under hypotheses ([eq:hypothesisH]𝖧), it follows that Cu(N) = Se(N) <cit.>. In order to describe the points in Cu(N) ∖Se(N), we introduce the notion of focal points of N.



 §.§ Focal Locus of a Submanifold
 

    Given a submanifold N ⊂ M, a vector 𝐯∈ν̂ is said to be a tangent focal point of N if 
    
    d_𝐯(exp^ν|_ν̂) : T_𝐯ν̂→ T_exp^ν(𝐯)M

    is degenerate, i.e., if 𝐯 is a critical point of exp^ν|_ν̂. The set of tangent focal loci will be denoted by ℱ = ℱ(N) ⊂ν̂. The nullity of the map is known as the multiplicity of the tangent focal point 𝐯. The collection of all such vectors is called the tangent focal locus of N. The focal locus of N is defined as the image of the tangent focal locus under the exp^ν map.


In other words, the focal locus of N consists of the critical values of the (restricted) normal exponential map exp^ν|_ν̂. The (first) focal time for 𝐯∈ S(ν) is defined as 


    λ(𝐯) inf{ t  | d_t𝐯( exp^ν|_ν̂) is degenerate}.

The N-geodesic γ_𝐯 cannot be an N-segment beyond  λ(𝐯) <cit.>. Furthermore, under hypotheses ([eq:hypothesisH]𝖧), we have a (non-exclusive) dichotomy: a point in Cu(N) is either a point in Se(N) or it is a first focal locus along some N-segment <cit.>.

Focal locus can be naturally characterized via N-Jacobi fields with vanishing endpoint <cit.>. For 𝐯∈ν̂, denote the kernel 

    K_𝐯( d_𝐯( exp^ν|_ν̂) : T_𝐯ν̂→ T_exp^ν(𝐯) M = T_γ_𝐯(1) M ) =  d_𝐯ℰ.

On the other hand, consider the space of N-Jacobi fields

    𝒥_𝐯{ J  | J is an N-Jacobi field along γ_𝐯},

and its subspace 

    𝒦_𝐯{ J  | J is an N-Jacobi field along γ_𝐯, with J(1) = 0}.

For any 𝐱∈ T_𝐯ν̂, consider a curve α : (-ϵ, ϵ) →ν̂ such that α(0) = 𝐯 and α̇(0) = 𝐱. We then have a family of curves 

    Λ(s, t) exp^ν( t α(s) ),    0 ≤ t ≤ 1,   -ϵ < s < ϵ,

which is clearly an N-geodesic variation. In particular, 

    J_𝐱(t) = .∂/∂ s|_s=0Λ(s, t),    0 ≤ t ≤ 1,

is an N-Jacobi field along the N-geodesic γ_𝐯(t). Consider the map

    Φ : T_𝐯ν̂   →𝒥_𝐯
    𝐱   ↦ J_𝐱.



	Φ is a well-defined linear isomorphism T_𝐯ν̂≅𝒥_𝐯, which restricts to an isomorphism K_𝐯≅𝒦_𝐯.


	Let us denote c(s) = Λ(s, 0) = π∘α(s), which is a curve in N. Observe that ∂_t|_t = 0Λ(s, t) = α(s), as Λ(s, _) is an N-geodesic with initial velocity α(s). We compute the following.
	
    J(0)    = . ∂/∂ s|_s=0exp^ν (0 ·α(s)) = . ∂/∂ s|_s=0 c(s) = ċ(0) = dπ(α̇(0)) = dπ(𝐱).
    
    		D^γ̇_γ J (0)    = D^γ̇_γ |_t = 0∂_s Λ(s, t) = D^γ̇_c|_s = 0∂_t|_t = 0Λ(s, t) = D^γ̇_c α (0).

	Now, D^γ̇_c α (0) is determined by the first jet of α at 0, i.e., by 𝐱 = α̇(0) (see for example, <cit.>). Hence, J is uniquely determined by 𝐱, and in particular, Φ is a well-defined map. Note that for 𝐱∈ K_𝐯 =  d_𝐯ℰ, we have 
	
    J(1) = . ∂/∂ s|_s=0exp^ν(α (s)) = d_𝐯ℰ (α̇(0)) = d_𝐯ℰ(𝐱) = 0.

	Consequently, Φ restricts to a map K_𝐯→𝒦_𝐯.

	Let us show that Φ is linear. Clearly, 
	
    J_a𝐱 + b𝐲(0) = dπ(a𝐱 + b𝐲) = a   dπ(𝐱) + b   dπ(𝐲) = a J_𝐱(0) + b J_𝐲(0).

	Similarly, it follows from <cit.> that D^γ̇_γ J_a 𝐱 + b 𝐲(0) = a D^γ̇_γ J_𝐱(0) + b D^γ̇_γ J_𝐲(0). But then from the uniqueness of Jacobi fields, it follows that J_a 𝐱 + b 𝐲 = a J_𝐱 + b J_𝐲. Consequently, Φ is linear.

	If 𝐱 = 0, we have J_𝐱(0) = 0 and J̇_𝐱(0) = 0, which shows that Φ is injective. On the other hand, suppose J ∈𝒥_𝐯 is an N-Jacobi field. Then, J is given by an N-geodesic variation, say, Ξ : (-ϵ, ϵ) × [0, 1] → M. We have a curve β in ν̂ such that Ξ(s, t) = exp^ν(tβ(s)). The above computation then shows that J = J_𝐱, where 𝐱 = β̇(0). If J(1) = 0, we have 𝐱∈ K_𝐯, since d_𝐯ℰ(𝐱) = J_𝐱(1) = 0. Thus, Φ is a linear isomorphism, which restricts to an isomorphism K_𝐯→𝒦_𝐯.



    It is immediate that given a tangent focal point 𝐯∈ν̂, the focal multiplicity K_𝐯≤ T_𝐯ν̂ =  M. On the other hand, from <ref>, we have a Jacobi field J ∈𝒥_𝐯∖𝒦_𝐯. Thus, it follows from <ref> that K_𝐯 = 𝒦_𝐯⪇𝒥_𝐯 =  M.


Using <ref>, we get the following useful result, which is analogous to <cit.> in the Riemannian context.

    Suppose for 𝐯∈ S(ν), and let k =  d_ℓ𝐯ℰ for some ℓ > 0. Then, there exists a frame of N-Jacobi fields { J_1,… J_n } along γ_𝐯 such that
    
        
  * The subspaces Span⟨J̇_1(ℓ), … ,J̇_k(ℓ) ⟩ and Span⟨ J_k+1(ℓ), … , J_n(ℓ) ⟩ are g_ℓ𝐯 orthogonal.
        
  * T_γ_𝐯(ℓ) M = Span⟨J̇_1(ℓ), …J̇_k(ℓ), J_k+1(ℓ), … , J_n(ℓ) ⟩, and
        
  * { J_i(t) }_i=1^n is a basis of T_γ_𝐯(t) M for a deleted neighborhood 0 < |t - ℓ| < ϵ.
    
    In particular, focal points of N along γ_𝐯 are discrete.


    Fix a basis d_ℓ𝐯ℰ = Span⟨𝐱_1,… ,𝐱_k ⟩, and extend it to a basis T_ℓ𝐯ν̂ = Span⟨𝐱_1,… ,𝐱_n ⟩. Consider the N-Jacobi fields J_i  J_𝐱_i = Φ(𝐱_i) along γ_𝐯. By <ref>, { J_i } forms a basis of 𝒥_𝐯, and we have J_1(ℓ) = … = J_k(ℓ) = 0. Suppose, if possible, ∑_i=1^k a^i J̇_i(ℓ) = 0 for some scalars a^i. Consider the N-Jacobi field J = ∑_i=1^k a^i J_i. Then, J̇(ℓ) = ∑_i=1^k a^i J̇_i(ℓ) = 0. By <ref>, we have J = 0, which forces a^1 = … = a^k = 0. Thus, {J̇_1(ℓ), … , J̇_k(ℓ) } are linearly independent. Next, suppose ∑_i = k+1^n b^i J_i (ℓ) = 0 for some scalars b^i. Consider N-Jacobi field K = ∑_i > k b^i J_i. As Φ in <ref> is a linear isomorphism, we have K = Φ(𝐲), where 𝐲 = ∑_i > k b^i 𝐱_i. Now, d_ℓ𝐯ℰ(𝐲) = K(ℓ) = ∑_i > k b^i J_i(ℓ) = 0 implies 𝐲∈ d_ℓ𝐯ℰ. By our choice of 𝐱_i, we must have b^k+1 = … = b^n = 0. Thus, { J^k+1(ℓ), … , J^n(ℓ) } are linearly independent as well. Lastly, for some J = ∑_i=1^k a^i J_i and some K = ∑_i > k b^i K_i, we have from <ref>
    
    g_ℓ𝐯( J̇(ℓ), K(ℓ) ) = g_ℓ𝐯( J(ℓ), K̇(ℓ) ) = 0.

    Consequently, Span⟨J̇_1(ℓ),…J̇_k(ℓ) ⟩ and Span⟨ J_k+1(ℓ), … J_n(ℓ) ⟩ are g_ℓ𝐯 orthogonal. A simple dimension counting then proves that T_γ_𝐯(ℓ) = Span⟨J̇_1(ℓ), …J̇_k(ℓ), J_k+1(ℓ), … , J̇_n (ℓ) ⟩.

    Now, for 1 ≤ i ≤ k, as J_i(ℓ) = 0 and J̇_i(0)  0, i.e., J_i(t) has a 0 of order 1 at t = ℓ. We then have an N-Jacobi field 
    
    Y_i(t) = 
        J_i(t)/t - ℓ,    t ℓ
    J̇_i(ℓ),    t = ℓ.

    Set Y_i = J_i for k < i ≤ n. Clearly, Span⟨ Y_i(t) ⟩ = Span⟨ J_i(t) ⟩ for all t ℓ. Now, { Y_i(ℓ) } are linearly independent, and hence, in some neighborhood |t - ℓ| < ϵ, we have { Y_i(t) } are linearly independent as well. But then for 0 < |t - ℓ| < ϵ, we get { J_i(t) } are linearly independent.
    
    Lastly, suppose t_0 𝐯 is a focal point of N along γ_𝐯, for some 0 < |t_0 - ℓ| < ϵ. But then there exists some non-vanishing N-Jacobi field J along γ_𝐯 with J(t_0) = 0. Write J = ∑ c^i J_i for some scalars c^i. But then ∑ c^i J_i(0) = 0, which implies c^i = 0, a contradiction. Thus, there are no focal point of N along γ_𝐯 for 0 < |t - ℓ| < ϵ, i.e., focal points along γ_𝐯 are discrete.





 §.§ Index Form and the Morse Index Theorem

From the second variation formula of the restricted energy functional E|_𝒫_N, we get the index form <cit.>.


    Let γ : [a,b] → M be a unit-speed N-geodesic. Then, the index form ℐ_γ is defined for X, Y ∈ T_γ𝒫_N as 
    
    ℐ_γ(X,Y) ∫_a^b [ g_γ̇(D^γ̇_γ X, D^γ̇_γ Y) - g_γ̇( R^γ(γ̇, X) Y, γ̇) ] - g_γ̇(a)( π^⊥∇^γ̇(a)_X Y, γ̇(a) ),

    where π^⊥ is the second component projection of the canonical splitting T_γ(a)M = T_γ(a)N ⊕( T_γ(a)N )^⊥_g_γ̇(a).


It follows that ℐ_γ is a symmetric 2-form, and the kernel of the index form ℐ_γ consists of precisely the N-Jacobi fields J along γ, with J(b) = 0. In particular, γ is a nondegenerate critical point (i.e., an N-geodesic) of the energy functional, precisely when there are no N-Jacobi fields along γ vanishing at the endpoint. In other words, γ is nondegenerate if and only if γ(b) is not a focal point of N along γ. 


    The index of an N-geodesic γ is defined as 
    
    Ind(γ) max{ K  | ℐ _γ is negative definite restricted to some K ⊂ T_γ𝒫_N}.
    


By the Morse index lemma for Finsler submanifolds <cit.>, it follows that for a unit-speed N-geodesic γ : [0, T] → M given as γ(t) = exp^ν(t 𝐯), such that γ(T) is not a focal point of N along γ, one has 

    Ind(γ) = ∑_0 < t < T( d_t𝐯( exp^ν|_ν̂) ) < ∞.

In other words, Ind(γ) equals the number of focal points of N along γ, counted with multiplicities, excluding the endpoints. Note that by <ref>, only finitely many non-zero terms can appear in the sum. For a fixed unit vector 𝐯∈ S(ν), based on the index of γ_𝐯, we can now define the following.



    Given N ⊂ M and 𝐯∈ S(ν), the k^th focal time in the direction of 𝐯 is defined as 
    
    λ_k(𝐯) sup{ t  | Ind(γ_𝐯|_[0, t]) ≤ k - 1 },

    where γ_𝐯 : [0, ∞) → M is the unit speed N-geodesic given by γ_𝐯(t) = exp^ν(t 𝐯). The k^th focal locus of N is defined as the set 
    
    {exp^ν( λ_k(𝐯) 𝐯) |𝐯∈ S(ν), λ_k(𝐯) ∞}.



Clearly, λ_1(𝐯) = λ(𝐯) as defined in <ref>. In general, we have 0 < λ_1(𝐯) ≤λ_2(𝐯) ≤…. If γ_𝐯(T) is not a focal point of N along γ_𝐯, we can immediately see 

    Ind(γ_𝐯|_[0, T]) = ∑ K_λ_k(𝐯)𝐯,

where the sum runs over the finite set {λ_k(𝐯) < T }, and K_t 𝐯 is as in <ref>. We now prove a useful result about the index being locally constant near a non-focal point, analogous to <cit.>.


    Suppose, for 𝐯_0 ∈ S(ν), we have γ_𝐯(T) is not a focal point of N along γ_𝐯. Then, there exists a neighborhood 𝐯_0 ∈ U ⊂ S(ν) such that, 
    
    Ind(γ_𝐯|_[0, T]) = Ind(γ_𝐯_0|_[0, T]),     for all 𝐯∈ U.



    Let us first show that for Ind(γ_𝐯|_[0, T]) ≥Ind(γ_𝐯_0|_[0, T]) for 𝐯 sufficiently near 𝐯_0. As S(ν) is a smooth manifold of dimension n - 1, consider a chart 𝐯_0 ∈ U ⊂ S(ν), where U ≅𝔻^n-1. Then, we have a (n-1)-dimensional N-geodesic variation 
    
    Λ : U × [0, T]    ⟶ M 
    
            (𝐯, t)    ⟼exp^ν(t 𝐯).

    Clearly, γ_𝐯 = Λ(𝐯, _). Suppose for some X ∈Γγ_𝐯_0^*TM, we have I_γ_𝐯_0(X, X) < 0. Since U is contractible, we can get an extension, say, X̃∈ΓΛ^* TM. Denoting X̃_𝐯(t) = X̃(𝐯, t), we see that X̃_𝐯∈Γγ_𝐯^* TM. Since the index form <ref> is continuous, shrinking U if necessary, we may assume that I_γ_𝐯(X̃_𝐯, X̃_𝐯) < 0 for all 𝐯∈ U. Now, suppose Ind(γ_𝐯_0|_[0, T]) = k. Choose a frame X^1, … , X^k ∈Γγ_𝐯_0^*TM spanning a maximal subspace on which I_γ_𝐯_0 is negative definite, and satisfying
    
    I_γ_𝐯_0(X^i, X^i) < 0,      1 ≤ i ≤ k.

    As described above, shrinking U as necessary for finitely many times, we have X̃^i_𝐯∈Γγ_𝐯^* TM such that 
    
    I_γ_𝐯( X̃^i_𝐯, X̃^i_𝐯) < 0,      1 ≤ i ≤ k,  𝐯∈ U.

    Since { X^i } are linearly independent, possibly shrinking U further, we have {X̃^i_𝐯} are linearly independent for all 𝐯∈ U. But then, Ind(γ_𝐯|_[0, T]) ≥ k = Ind(γ_𝐯_0|_[0, T]) for all 𝐯∈ U. Note that this inequality is true irrespective of whether γ_𝐯_0(T) is focal of N along γ_𝐯_0 or not.

    Since d_T𝐯_0ℰ is nonsingular, we can assume that d_T 𝐯ℰ is nonsingular for all 𝐯∈ U as well. Thus, γ_𝐯(T) is not a focal point of N along γ_𝐯. We show that Ind(γ_𝐯) is constant on some neighborhood of 𝐯_0 in U. If not, choose some sequence 𝐯_j ∈ U such that Ind(γ_𝐯_j|_[0, T]) Ind(γ_𝐯_0|_[0, T]). Passing to a subsequence, we may assume that 𝐯_j →𝐯_0, and
    
    *Ind(γ_𝐯_j|_[0, T]) > Ind(γ_𝐯_0|_[0, T])   ∀ j.

    Let us denote K(j, k)  K_λ_k(𝐯_j) 𝐯_j. Thus, K(j, k) is precisely the multiplicity of the k^th focal time of 𝐯_j. Clearly, 1 ≤ K(j, k) < n =  M (see <ref>). Thus, inducting on k, by passing onto subsequence as necessary, we may assume that for each k fixed, K(j, k) = η_k is constant for all j ≥ 1. It then follows that there are precisely, say, r many distinct focal times in (0, T) along γ_𝐯_j for all j ≥ 1. Furthermore, for each k we have exactly η_k many values of λ_i(𝐯_j) coincide. Let us rename these unique focal times as 0 < μ_1(𝐯_j) < … < μ_r(𝐯_j) < T. Note that K(j, k) = ( d_μ_k(𝐯_j)𝐯_jℰ) for 1 ≤ k ≤ r, and we have 
    
    Ind(γ_𝐯_j|_[0, T]) = ∑_i=1^r  K(j, k),    j ≥ 1.

    
    Let us now consider K(j, k) as points in the Grassmann bundle Gr_η_k(Tν̂). Since the base-points 𝐯_j →𝐯_0, passing to a subsequence, we may assume that K(j, k) converges to an η_k-dimensional subspace Z_k ⊂ T_t_k 𝐯_0ν̂ for some time t_k > 0. Clearly μ_k(𝐯_j) 𝐯_j → t_k 𝐯_0 ⇒μ_k(𝐯_j) → t_k, as F(𝐯_j) = 1 = F(𝐯_0). Since we only consider μ_k(𝐯_j) ≤ T, it follows that t_k ≤ T as well. Now, d_μ_k(𝐯_j)ℰ vanishes on K(j, k), and hence by continuity, d_t_k 𝐯_0ℰ vanishes on the limit subspace Z_k. In particular, t_k 𝐯_0 is a focal point of N along γ_𝐯_0. Since, by assumption, γ_𝐯(T) is not a focal point, we must have 0 < t_k < T.

    For distinct values of t_k, then each Z_k are subspaces of the tangent spaces at distinct points of ν̂ along the ray R_𝐯_0 = { t 𝐯_0 }. If they are not distinct, let us consider the situation t_k_1 = … = t_k_s = t_0 for some s ≥ 2, where no other t_k equals t_0. We show that Z_k_1 + … + Z_k_s is a direct sum in the tangent space T_t_0 𝐯_0ν̂. Clearly, ∑ Z_k_i⊂ K_t_0 𝐯_0. Fix some αβ∈{ k_1, … , k_r }. For some 𝐚∈ K_α and 𝐛∈ K_β, get sequences 𝐚_j ∈ K(j, α) and 𝐛_j ∈ K(j, β) such that 𝐚_j →𝐚, 𝐛_j →𝐛. Consider the N-Jacobi fields J_𝐚_j = Φ(𝐚_j),    J_𝐛_j = Φ(𝐛_j) along γ_𝐯_j. By <ref>, we have g_γ̇_𝐯_j( J̇_𝐚_j, J_𝐛_j) = g_γ̇_𝐯_j( J_𝐚_j, J̇_𝐛_j) for all time t. As J_𝐚_j( t ) = 0 for t = μ_α(𝐯_j), we have 
    
    g_γ̇_𝐯_j(μ_α(𝐯_j))( J̇_𝐚_j( μ_α(𝐯_j) ), J_𝐛_j( μ_α(𝐯_j) ) ) = 0,      j ≥ 1.

    On the other hand, it follows from <ref>, that there is an N-Jacobi field K_𝐛_j along γ_𝐯_j satisfying 
    
    K_𝐛_j(t) = 
        J_𝐛_j(t)/t - μ_β(𝐯_j) ,    t μ_β(𝐯_j) 
    J̇_𝐛_j( μ_β(𝐯_j) ),    t = μ_β(𝐯_j).

    Then we have, 
    
    g_γ̇_𝐯_j(μ_α(𝐯_j))( J̇_𝐚_j( μ_α(𝐯_j) ), K_𝐛_j( μ_α(𝐯_j) ) ) = 0,      j ≥ 1.

    Since μ_α(𝐯_j) → t_α = t_0 and μ_β(𝐯_j) → t_β = t_0 as j →∞, taking limit we get 
    
    g_γ̇_𝐯_0(t_0)( J̇_𝐚 (t_0), J̇_𝐛 (t_0) ) = 0.

    Now, it follows from <ref> and <ref> that the linear map
    
    Ψ : K_t_0 𝐯_0   ⟶ T_γ_𝐯_0(t_0) M 
    𝐱   ⟼J̇_𝐱(t_0)

    is injective. The above then discussion shows that for any αβ∈{ k_1,… k_s }, the subspaces Ψ(Z_α), Ψ(Z_β) are g_γ̇_𝐯_0(t_0)-orthogonal. Consequently, ∑_i=1^s Ψ( Z_k_i) is a direct sum. Since Ψ is injective, it follows that _i=1^s Z_k_i is a direct sum as well. Hence, we have obtained that Ind(γ_𝐯_0|_[0, T]) ≥∑_k=1^r  Z_k.

    Now, for each j ≥ 1 we get
    
    Ind(γ_𝐯_j|_[0, T]) = ∑_k=1^r  K(j, k) = ∑_k=1^r  Z_k ≤Ind((γ_𝐯_0|_[0, T])),

    which is a contradiction to (<ref>). Hence, we have some neighborhood of 𝐯_0 in S(ν) on which Ind(γ_𝐯|_[0, T]) is constant.



    For each k≥ 1, the functions λ_k : S(ν) → (0, ∞] is continuous.


    Fix some 𝐯_0 ∈ S(ν). Suppose λ_k(𝐯_0) = T < ∞. Then, for ϵ > 0 small, we have from <ref> that γ_𝐯_0((T+ϵ)) and γ_𝐯_0((T-ϵ)) are not tangent focal points of N along γ_𝐯_0. It follows from <ref> that there exists some neighborhood 𝐯_0 ∈ U ⊂ S(ν) such that for all 𝐯∈ U we have
    
    Ind(γ_𝐯|_[0, T- ϵ]) = Ind(γ_𝐯_0|_[0, T-ϵ]) ≤ k - 1,   and  Ind(γ_𝐯|_[0, T+ϵ]) = Ind(γ_𝐯_0|_[0, T+ϵ])≥ k.

    Consequently, for all 𝐯∈ U we have T - ϵ≤λ_k(𝐯) ≤ T + ϵ. Since ϵ > 0 is arbitrary, we see that λ_k is continuous at 𝐯_0 if λ_k(𝐯_0) < ∞.

    Lastly, suppose λ_k(𝐯_0) = ∞. Then, for all T large we have Ind(γ_𝐯_0|_[0, T]) ≤ k - 1, and we may also assume that γ_𝐯_0(T) is not a focal point of N along γ_𝐯_0. Then, for some 𝐯_0 ∈ U ⊂ S(ν) we have, Ind(γ_𝐯|_[0, T]) = Ind(γ_𝐯_0|_[0, T]) ≤ k - 1, and hence λ_k(𝐯) ≥ T. Since T is arbitrary, we have λ_k is continuous at 𝐯_0 with λ_k(𝐯_0) = ∞ as well. This concludes the proof.




§ WARNER'S REGULARITY OF THE NORMAL EXPONENTIAL MAP
 
Suppose N is a closed submanifold of a forward complete Finsler manifold (M, F). In this section, we prove that ℰ is regular in the sense of Warner <cit.> while modifying the author's definition suitably. For any 𝐯∈ν̂, define the ray 

    R_𝐯 = { t 𝐯 |  t > 0 }⊂ν̂.

We also need the following definition.

	A subset C ⊂ν̂ is said to be radially convex if C is connected, and if for each 𝐯∈ C the subset { t  | exp^ν(t𝐯) ∈ C }⊂ (0, ∞) is connected.


In this section, we prove the following.

	Given a closed submanifold N of a forward complete Finsler manifold (M, F), the normal exponential map exp^ν: ν(N) → M satisfies the following three regularity properties.
	
		
  (R1)  For each 𝐯∈ν, the derivative d_𝐯ℰ is nonvanishing on T_𝐯R_𝐯.
		
  (R2)  The map 
		
    K_𝐯   ⟶ T_ℰ(𝐯)M/Imd_𝐯ℰ
    𝐱   ⟼J̇_𝐱(1)

		is a linear isomorphism.
		
  (R3)  For each 𝐯∈ν̂, there exists a radially convex open neighborhood 𝐯∈ U ⊂ν̂, such that for each 𝐮∈ U, the number of critical points (counted with multiplicity) of ℰ on R_𝐮∩ U is a constant independent of 𝐮, and equals to the dimension of K_𝐯.
	


	To prove ([R1]R1), consider the curve η : (-ϵ, ϵ) → R_𝐯 given by η(t) = t𝐯, so that η(0) = 𝐯 and T_𝐯R_𝐯 = Span⟨η̇(0) ⟩γ̇(0). Now, 
	
    d_𝐯ℰ(η̇(0)) = . d/dt|_t=0 (ℰ∘γ) = . d/dt|_t=0exp^ν(t 𝐯) = 𝐯 0.

	This proves ([R1]R1).

	For ([R2]R2), pick a basis K_𝐯 = Span⟨𝐱_1, … , 𝐱_k ⟩ and extend it to a basis T_𝐯ν̂ = Span⟨𝐱_1,…𝐱_n ⟩. It follows from <ref> that d_𝐯ℰ(x_i) = J_𝐱_i(1). But then ([R2]R2) follows immediately from <ref> and <ref>.

	Lastly, to show ([R3]R3), set 𝐯_0 = 𝐯/F(𝐯)∈ S(ν). Then, 𝐯 = T 𝐯, where T = F(𝐯). If 𝐯 is not a focal locus of N along γ_𝐯_0, then one can easily choose a radially convex open neighborhood 𝐯∈ U ⊂ν̂ so that d_𝐯_0ℰ is nonsingular for each 𝐮∈ U. The claim is then immediate. Now, suppose T𝐯_0 is a focal point. Since focal points along γ_𝐯_0 are discrete (<ref>), we have some ϵ > 0 so that T 𝐯_0 is the only focal point of N along γ_𝐯_0 for |T - t| ≤ϵ. Applying <ref> twice for both γ_𝐯_0(T - ϵ) and γ_𝐯_0(T + ϵ), we have a neighborhood 𝐯_0 ∈ V ⊂ S(ν) so that for all 𝐯∈ V we have
	
		
  * γ_𝐯(T ±ϵ) are not focal points of N along γ_𝐯,
		
  * Ind(γ_𝐯|_[0, T - ϵ]) = Ind(γ_𝐯_0|_[0, T - ϵ]), and
		
  * Ind(γ_𝐯|_[0, T + ϵ]) = Ind(γ_𝐯_0|_[0, T + ϵ])
	
	
	
	Consider the radially convex open set 
	
    U = { t𝐯 |  T - ϵ < t < T + ϵ,  𝐯∈ V}⊂ν̂.

	For any 𝐮∈ U, we have R_𝐮∩ U = { t 𝐮_0  |  T - ϵ < t < T + ϵ}, where 𝐮_0 = 𝐮/F(𝐮)∈ V. By the Morse index lemma, the number of focal points counted with multiplicity on R_𝐮∩ U then equals 
	
    Ind(γ_𝐮_0|_[0, T + ϵ]) - Ind(γ_𝐮_0|_[0, T - ϵ]) = Ind(γ_𝐯_0|_[T + ϵ]) - Ind(γ_𝐯_0|_[0, T - ϵ]) =  d_T𝐯_0ℰ =  d_𝐯ℰ.

	This concludes the proof of ([R3]R3).


In view of ([R3]R3), one defines a certain regularity for tangent focal points.


	A tangent focal point 𝐯∈ν̂ is said to be regular if there exists a radially convex open neighborhood 𝐯∈ U ⊂ν̂ such that for each 𝐮∈ U, the map ℰ has at most one critical point on R_𝐮∩ U, where the multiplicity necessarily equals that of 𝐯. The set of regular tangent focal loci will be denoted as ℱ^reg⊂ℱ. A tangent focal point that is not regular is called a singular focal point, and the set of all singular tangent focal points will be denoted as ℱ^sing⊂ℱ.



	Suppose 𝐯∈ν̂ is a tangent focal point of multiplicity k. Then, there exist coordinate charts (U, x^1,… ,x^n) and (V, y^1,… ,y^n) around 𝐯 and γ_𝐯(1) = ℰ(𝐯) respectively, satisfying 
	
    d_t𝐯ℰ( . ∂/∂ x^i|_t𝐯) = f_i(t) . ∂/∂ y^i|_ℰ(t𝐯),    for t with t𝐯∈ U,

	where f_i(t) are smooth functions satisfying f_i(t)  0 for 1 ≤ i ≤ k, and the only zeros of f_i(t) for k + 1 ≤ i ≤ n are at t = 1, and thence ḟ_i(1) > 0.


	Pick a basis {𝐱_1, …𝐱_k } of K_𝐯 =  d_𝐯ℰ, and extend it to a basis {𝐱_1,…𝐱_n } of T_𝐯ν̂. Let us write 𝐯 = ℓ𝐯_0 for 𝐯_0 ∈ S(ν) and ℓ = F(𝐯). As in the proof of <ref>, we get N-Jacobi fields J_1,… J_n along the N-geodesic γ_𝐯_0. In particular, J_1(ℓ) = … = J_k(ℓ) = 0. Next, as in <ref>, we define the N-Jacobi fields Y_i for 1 ≤ i ≤ k, and set Y_i = J_i for i > k. Then, { Y_i(t) } is basis for T_γ_𝐯_0(t) M in some small neighborhood of ℓ.
	
	Consider some α_i : (-ϵ, ϵ) →ν̂ such that α_i(0) = 𝐯 and α̇_i(0) = 𝐱_i. Then, we have a vector field A_i(t) = . ∂/∂ s|_s = 0( t α_i(s) ) along the ray R_𝐯_0 = { t 𝐯_0 }. It is immediate that 
	
    d_t 𝐯_0ℰ( A_i(t) ) = . ∂/∂ s|_s=0ℰ( t α_i(s)) = J_i(t).

	By construction, { A_i(ℓ) } = {𝐱_i } is a basis at T_𝐯ν̂ = T_ℓ𝐯_0ν̂. Also, in a deleted neighborhood of ℓ, we have { d_t 𝐯_0ℰ( A_i(t) ) } = { J_i(t) } is a basis for T_ℰ(t𝐯_0) M, and hence, { A_i(t) } is a basis of T_t 𝐯_0ν̂ for a neighborhood of ℓ. Then, by a standard argument (see <cit.> for a detailed proof), we have a coordinate system (U, x^1, …, x^n) around 𝐯 so that . ∂/∂ x^i|_t 𝐯 = A_i(t ℓ). Similarly, we get coordinate chart (V, y^1, …  , y^n) around γ_𝐯(1) so that . ∂/∂ y^i|_γ_𝐯(t) = Y_i(tℓ). If we set f̃_i(t) = t - ℓ for 1 ≤ i ≤ k and f̃_i(t) = 1 for i > k, then J_i = f̃_i Y_i. We get,
	
    d_t𝐯ℰ( . ∂/∂ x^i|_t𝐯) = d_tℓ𝐯_0ℰ( A_i(t ℓ) ) = J_i(t ℓ) = f̃_i(t ℓ) Y_i(t ℓ) = f_i(t ℓ) . ∂/∂ y^i|_γ_𝐯(t).

	Setting f_i(t) = f̃(t ℓ) concludes the proof.


We now state and give a sketch of proof for the following result, which is a direct generalization of <cit.>.

	ℱ^reg is open and dense in ℱ. Furthermore, ℱ^reg is an embedded codimension 1 submanifold of ν̂, with T_𝐯ν̂ = T_𝐯ℱ^reg⊕ T_𝐯 R_𝐯 for all 𝐯∈ℱ^reg.


	Given any open set U ⊂ν̂, let us denote ℱ^reg_U = ℱ^reg∩ U and ℱ^sing_U = ℱ^sing∩ U. By definition of the regular focal loci, given any 𝐯∈ℱ^reg we have some (radially convex) open set p ∈ U ⊂ν̂ such that every focal point in U is regular. In other words, ℱ^reg_U = ℱ∩ U, proving that ℱ^reg is open in ℱ. Next, let 𝐯_0 ∈ = ℱ^sing = ℱ∖ℱ^reg be a singular focal point. By [R3]R3, we get an arbitrarily small radially convex neighborhood 𝐯_0 ∈ U ⊂ν̂ such that for each 𝐮∈ U, the number of focal points on the sub-ray R_𝐮∩ U equals the multiplicity of 𝐯. Since 𝐯 is singular, there is some focal point 𝐯_1 ∈ U with at least two distinct focal points on R_𝐯_1∩ U. But then the focal multiplicity of 𝐯_1 is strictly less than that of 𝐯_0. If 𝐯_1 is regular, we are done. Otherwise, we repeat this argument, choosing the successively nested neighborhoods. Eventually, after some finitely many steps, we either get a regular focal point, or we get a focal point of order 1. But every order 1 focal point is automatically regular. Thus, U ∩ℱ^reg∅, proving that ℱ^reg is dense in ℱ.

	In order to show that ℱ^reg is a codimension 1 submanifold of ν̂, we locally realize ℱ^reg as the zero set of some submersion. Fix 𝐯∈ℱ^reg with focal multiplicity, say, k. By <ref> get coordinate charts (U, 𝐱^i) and (V, 𝐲^j), respectively around 𝐯 and ℰ(𝐯) = γ_𝐯(1). With these coordinates, we have the eigenvalues of dℰ, say, f_1, …, f_n as smooth functions on U. Denote Δ : U →ℝ as the (n-k+1)^th-elementary symmetric polynomial in n many variables, evaluated on the eigenvalues. Thus, 
	
    Δ(𝐮) = ∑_1 ≤ i_1 < … < i_k ≤ n f_1(𝐮) …f_i_1(𝐮)…f_i_k-1(𝐮)… f_n(𝐮),   𝐮∈ U.

	It follows from <ref> that along the ray R_𝐯, we have 
	
    f_1(𝐯) = … = f_k(𝐯) = 0,    f_k+1(𝐯)  0, … , f_n(𝐯)  0.

	Furthermore, denoting the curve σ(t) = t𝐯, we see the radial derivatives, 
	
    σ̇(1)( f_i ) = d f_i ( σ̇(1) ) > 0,    1 ≤ i ≤ k.

	Consequently, a simple computation gives us Δ(𝐯) = 0 and σ̇(1)(Δ) > 0. Shrinking U as necessary, while keeping it radially convex, we may assume that the radial derivative of  Δ along each of the rays intersecting U is non-vanishing on U. In particular, Δ is a submersion on U. We show that D^-1(0) = ℱ^reg_U. Indeed, for any 𝐮∈ℱ^reg we have the multiplicity equals k, and thus d_𝐮ℰ has rank n - k. Consequently, at least one of the eigenvalues vanishes at 𝐮 in each of the product terms appearing in the sum of products Δ. Thus, ℱ^reg⊂ D^-1(0). For the converse, suppose Δ(𝐮) = 0 for some 𝐮∈ U. Pick the (unique) focal point 𝐮_0 ∈ R_𝐮∩ U. We have Δ(𝐮_0) = 0 = Δ(𝐮). Also, it follows from the radial convexity of U, that the radial line joining 𝐮 and 𝐮_0 is contained in U, and moreover the radial derivative of Δ is nonvanishing on this line. Then, by the mean value theorem, we must have 𝐮 = 𝐮_0 ∈ℱ^reg_U. Thus, ℱ^reg_U = Δ^-1(0). The proof then follows.


Fix a connected component, say, 𝒞⊂ℱ^reg. Clearly 𝒞 is open in ℱ^reg, and hence a codimension 1 submanifold of ν̂. Furthermore, the focal multiplicities of points in 𝒞 are constant, say, k. At any 𝐯∈𝒞, we have two subspaces of T_𝐯ν̂, namely, T_𝐯𝒞 = T_𝐯ℱ^reg and K_𝐯 =  d_𝐯ℰ. Let us denote, 

    𝒩_𝐯 K_𝐯∩ T_𝐯ℱ^reg,   𝐯∈𝒞.

We have 𝒩_𝐯 is either k-1 or k, in which case K_𝐯⊂ T_𝐯ℱ^reg. Based on 𝒩_𝐯, we have a decomposition 

    𝒞 = 𝒞(k) ⊔𝒞(k-1).

Since dℰ attains its maximum value n - (k-1) on 𝒞(k-1), we have 𝒞(k-1) is an open submanifold of 𝒞. Also, since the union ∪_𝐯∈𝒞(k-1)𝒩_𝐯 is precisely the kernel of d( ℰ|_𝒞(k-1)), it is a rank (k-1) involutive distribution on 𝒞(k-1). By a similar argument, the union ∪_𝐯∈𝒞(k)𝒩_𝐯 = ∪_𝐯∈𝒞(k) K_𝐯 is also an involutive distribution of rank k on the open submanifold 𝒞(k) ⊂𝒞. We have the following result, generalizing <cit.>.


	For any 𝐯∈ℱ^reg with focal multiplicity ≥ 2, we have K_𝐯⊂ T_𝐯ℱ^reg, and thus 𝒞(k-1) = ∅.



Note that for a component 𝒞⊂ℱ^reg consisting of tangent focal points of multiplicity 1, both 𝒞(k) and 𝒞(k-1) can be non-empty. We are now in position to state the local form of the normal exponential map ℰ near a regular focal point.


	Suppose 𝐯∈ν̂ is a regular tangent focal locus of N with multiplicity k. Let 𝒞 be the connected component of ℱ^reg containing 𝐯, and consider the decomposition 𝒞 = 𝒞(k) ⊔𝒞(k-1) (<ref>). Then, there exist coordinates { x^1,… x^n } and { y^1,… y^n } near 𝐯∈ν̂ and ℰ(𝐯) ∈ M, which can be arranged so that ℰ has special forms in the following cases.
	
		
  *  If 𝐯 has multiplicity k ≥ 2, then one can arrange so that 
		
    y^i ∘ℰ = 
    		
    			x^n · x^i,    i = 1,… k, 
    
    			x^i,    i = k + 1, … , n.

		
		
  *  If 𝐯 has multiplicity k = 1 and furthermore, 𝐯∈𝒞(k) then one can arrange so that 
		
    y^i ∘ℰ = 
    		
    			x^n · x^1,    i=1,
    
    			x^i,    i = 2, … , n.

		
		
  *  If 𝐯 has multiplicity k = 1 and furthermore, K_𝐯⊄T_𝐯ℱ^reg (i.e., if 𝐯∈𝒞(k-1) = 𝒞(0)), then one can arrange so that
		
    y^i ∘ℰ = 
    		
    			x^1 · x^1,    i = 1,
    
    			x^i,    i = 2, … , n.

	


The proofs of the above two theorems follows in the same vein as <cit.>, which uses the notion of higher order tangent vectors. To keep the article self-contained, we have provided sketches of proof in the appendix (<ref>).


	The only regular tangent focal points 𝐯 not considered in <ref>, are of the form 𝒞∖( 𝒞(0) ⊔𝒞(1) ) = 𝒞(1) ∖𝒞(1), where 𝒞 is a connected component of ℱ^reg consisting of multiplicity 1 tangent focal points. Clearly, then form a nowhere dense subset of ℱ^reg.


As an immediate corollary, we have the following.

	Let N be a submanifold of a forward complete Finsler manifold (M, F). Then the normal exponential map ℰ = exp^ν|_ν̂ is not injective on any neighborhood of a tangent focal point 𝐯∈ν̂.


	Since ℱ^reg is dense in ℱ, we only proof the statement for 𝐯∈ℱ^reg. Assume 𝒞⊂ℱ^reg is the connected component containing 𝐯, and denote the decomposition 𝒞 = 𝒞(k) ⊔𝒞(k-1). If 𝐯 has multiplicity k ≥ 2, then by <ref>, we have a k-dimensional foliation in 𝒞, such that ℰ maps each leaf to a single point. The claim is then immediate. Suppose k = 1. If 𝐯∈𝒞(k), we again have a one dimensional foliation in 𝒞, so that each leaf is mapped to a single point by ℰ. If 𝐯∈𝒞 (k-1), then by <ref> ([thm:normalExponentialLocalForm:3]3), we have a coordinate system for which ℰ looks like <ref>. Consequently, ℰ is not locally injective in a neighborhood of 𝐯. Lastly, suppose 𝐯∈𝒞∖( 𝒞 (k-1) ⊔𝒞 (k) ), which is nowhere dense by <ref>. Then in any neighborhood of 𝐯, there exists a 𝐯 ' ∈𝒞 (k-1) ⊔𝒞 (k) and hence in that neighborhood, the map ℰ fails to be injective. This concludes the proof. 



	In <cit.>, it was proved that under hypothesis ([eq:hypothesisH]𝖧), we have Cu(N) = Se(N). In the course of the proof, we reached a dichotomy, whose case (a) leads to a contradiction. We would like to point out that in view of <ref>, case (a) is immediately ruled out. In the next section, we shall prove an even stronger result following the ideas of <cit.>.



§ DECOMPOSITION OF THE TANGENT CUT LOCUS
 
As mentioned earlier, the cut locus Cu(N) consists of points that are either a first focal locus along an N-geodesic, or points admitting at least two N-segments, i.e., points of Se(N). In the normal bundle ν, we have the set of tangent cut points Cu(N), which maps to Cu(N) under the normal exponential map.


    A vector 𝐯∈Cu(N) is called an ordinary (or separating) tangent cut point if there exists some 𝐰∈Cu(N) with 𝐰𝐯 such that exp^ν(𝐯) = exp^ν(𝐰). Otherwise, 𝐯 is called a singular tangent cut point. We denote the set of all ordinary tangent cut points of N by Se(N).


It is clear that Se(N) is mapped to Se(N) under the normal exponential map. Let us denote by ℱ_1 ⊂ℱ the set of all first tangent focal locus. Then, it follows that 

    Cu(N) ∖Se(N) ⊂ℱ_1(N).

Specializing <ref> to the first tangent focal locus, we obtain the following. Set ℱ^reg_1 ℱ^reg∩ℱ_1, the set of regular first tangent focal loci.


	ℱ^reg_1 is open in both ℱ_1 and ℱ ^reg, and dense in ℱ _1. Furthermore, ℱ^reg_1 is an embedded codimension 1 submanifold of ν̂.


	Suppose 𝐯∈ℱ^reg_1 has multiplicity k. As 𝐯 is regular, there exists arbitrarily small neighborhood 𝐯∈ U ⊂ν̂ so that for each 𝐮∈ U, there is a unique tangent focal point on R_𝐮∩ U, which has multiplicity k. Choose a neighborhood basis, say, U_n so that ∩_n ≥ 1 U_n = {𝐯}. Suppose, if possible, for each n there is some 𝐯_n ∈ U_n such that the first tangent focal locus along 𝐯_n is 𝐰_n 𝐯_n. Since 𝐯_n →𝐯, by a similar argument as in <ref>, passing to a subsequence as necessary, we get 𝐰_n →𝐰, where 𝐰 is a tangent focal point on R_𝐯. As 𝐰_n is a first tangent focal point, we have λ_1(𝐰̂_n) = F(𝐰_n), where 𝐱̂ = 𝐱/F(𝐱)∈ S(ν). By <ref>, we have λ_1(𝐰̂_n) →λ_1(𝐰̂), and hence, λ_1(𝐰̂) = limλ_1(𝐰̂_n) = lim F(𝐰_n) = F(𝐰). But on R_𝐯, we have λ_1(𝐰̂) = λ_1(𝐯̂) = F(𝐯), which shows that 𝐰 = 𝐯, i.e., 𝐰_n →𝐯. Consequently, in every neighborhood U of 𝐯, for n large, we have two tangent focal points 𝐰_n 𝐮_n on R_𝐮_n∩ U, which contradicts the regularity of 𝐯. Hence, for some n_0 we must have that the unique tangent focal point on R_𝐮∩ U_n_0 is a first tangent focal point, for each 𝐮∈ U_n_0. Thus, ℱ∩ U_n_0 = ℱ^reg_1 ∩ U_n_0 = ℱ^reg∩ U_n_0, proving that ℱ^reg_1 is open in both ℱ^reg and ℱ_1.

	To show that ℱ^reg_1 is dense in ℱ_1, consider some 𝐯∈ℱ_1 ∖ℱ^reg_1. Thus, 𝐯 is a singular first tangent focal point. In particular, in every neighborhood 𝐯∈ U ⊂ν̂, there exists some 𝐮∈ U so that R_𝐮∩ U has at least two tangent focal points. Choose a sequence 𝐯_n ∈ U of such tangent focal points, so that 𝐯_n →𝐯, and denote the first tangent focal point on R_𝐯_n by 𝐰_n. As argued in the previous paragraph, we can assume that 𝐰_n →𝐯, and in particular, 𝐰_n ∈ U for some n large. Also, 𝐰_n has multiplicity strictly less than that of 𝐯. Repeating finitely many times, we can find some 𝐰∈ U which is a first tangent focal point and has multiplicity. But then 𝐰 is regular. In other words, 𝐰∈ U ∩ℱ^reg_1, proving that ℱ^reg_1 is dense in ℱ_1.

	Since ℱ^reg_1 is open in ℱ^reg, it follows from <ref> that ℱ^reg_1 is a codimension 1 submanifold of ν̂.



Following <cit.>, we now prove the following.

    Let N be a closed submanifold of a forward complete Finsler manifold (M, F), and suppose hypothesis ([eq:hypothesisH]H) holds. Then, Cu(N) = Se(N).


    Since every non-focal tangent cut point is necessarily ordinary (<ref>), we consider some 𝐯∈Cu(N) ∩ℱ = Cu(N) ∩ℱ_1. We have the following cases to consider 
    
        
  * Suppose 𝐯 is a regular focal locus, and of type ([thm:normalExponentialLocalForm:1]1) or ([thm:normalExponentialLocalForm:2]2) in <ref>. Then, we have an involutive distribution on ℱ^reg, defined near 𝐯, such that each leaf is mapped to a single point by exp^ν. In particular, the leaf passing through 𝐯, say, ℒ is mapped to q exp^ν(𝐯). Consider a sequence 𝐯_i ∈ℒ∖{𝐯} with 𝐯_i →𝐮. We have the tangent cut points 𝐰_i ρ(𝐯_i)𝐯_i, and the tangent focal points λ_1(𝐯_i) 𝐯_i = 𝐯_i. From <cit.>, we have ρ≤λ_1. If ρ(𝐯_i) = λ_1(𝐯_i) for some n, we get 𝐯_i ∈Cu(N). But then 𝐯∈Se(N), as exp^ν(𝐯) = exp^ν(𝐯_i) and 𝐯𝐯_i.
        
        Otherwise, assume that ρ(𝐯_i) < λ_1(𝐯_i) holds for all n. As the cut time map ρ is continuous, we get 𝐰_i →𝐯. As 𝐯_i is a first tangent focal locus, we must have 𝐰_i ∈Se(N). But then in any neighborhood of 𝐯 we have some element of Se(N), proving 𝐯∈Se(N).
        
        
  * Next, suppose 𝐯 is a regular focal locus of type ([thm:normalExponentialLocalForm:3]3) as in <ref>. That is, 𝐯 has multiplicity 1, and K_𝐯∩ T_𝐯ℱ^reg = 0. From the normal form <ref>, we get coordinate systems x^1,…, x^n and y^1,… ,y^n around 𝐯∈ U ⊂ν̂ and exp^ν(𝐯) ∈ V ⊂ M respectively, so that exp^ν folds the neighborhood U around the { x^1 = 0 } hyperplane. In other words, the image of U under exp^ν is contained in the region { y^1 ≥ 0 } of V. Furthermore, the singularities of exp^ν in U can only appear along { x^1 = 0 }, and thus the focal points in V appears on { y^1 = 0 }. Set ℓ d(N, q), and denote Û{𝐮 | 𝐮∈ U }⊂ S(ν). Without loss of generality, we can assume that U is radially convex, so that Û is an open neighborhood of 𝐯 in S(ν), and furthermore, U = { t𝐮 | 𝐮∈Û,   |t - ℓ| < ϵ} for some ϵ > 0.
        
        Now, pick a sequence q_i ∈ V with y^1(q_i) < 0, such that q_i → q = exp^ν(𝐯).  Say, γ_𝐰 : [0, ℓ_i]→ M is a unit-speed N-segment joining N to q_i, where ℓ_i = d(N, q_i) and 𝐰∈ S(ν). As q_i → q, we have ℓ_i →ℓ. Then, for i large, we must have 𝐰∉Û, since other ℓ_i 𝐰∈ U ⇒ y^1(q_i) ≥ 0, a contradiction. As S(ν) is compact and the set of all such 𝐰 as above avoids an open set Û, passing to a subsequence, we get a convergent sequence 𝐰_i →𝐰 in S(ν) ∖ U such that exp^ν(𝐰_i) = q_i. By <ref>, we see that γ_𝐰(t) = exp^ν(t 𝐰) is an N-segment of length ℓ, joining N to q. Clearly, 𝐯𝐯. Hence, we get 𝐯∈Se(N) in this case.

        
  * Lastly, assume that 𝐯 is a focal point not covered in <ref>. Since 𝐯∈ℱ_1, by <ref>, we have 𝐯 is the limit point of 𝐯_i ∈ℱ^reg_1. Also, in view of <ref>, we may assume that each 𝐯_i is of type ([thm:normalExponentialLocalForm:1]1), ([thm:normalExponentialLocalForm:2]2) or ([thm:normalExponentialLocalForm:3]3) in as in <ref>. Denote 𝐰_i ρ(𝐯_i) 𝐯_i, and note that 𝐰_i →𝐯 as in case (1). If ρ(𝐯_i) = λ_1(𝐯_i) holds for some i, we have 𝐯_i ∈Cu(N), and hence by the previous two cases, 𝐰_i = 𝐯_i is a limit point of Se(N). If ρ(𝐯_i) < λ_1(𝐯_i), then 𝐰_i ∈Cu(N) ∖ℱ_1 = Se(N). By standard Cantor's diagonalization argument, we then get a sequence in Se(N) converging to 𝐯, again showing that 𝐯∈Se(N)
    
    Thus, we have obtained Cu(N) ⊂Se(N).
    
    To show the equality, let us consider some 𝐯∈Se(N)∖Se(N). Then, we have a sequence 𝐯_i ∈Se(N) such that 𝐯_i →𝐯. Furthermore, we have 𝐰_i 𝐯_i such that q_i exp^ν(𝐰_i) = exp^ν(𝐯_i). Clearly, q_i → q exp^ν(𝐯). Also, passing to a subsequence, we have 𝐰_i →𝐰. Since q = exp^ν(𝐯) = lim_i exp^ν(𝐯_i) = lim_i exp^ν(𝐰_i) = exp^ν(𝐰), we must have 𝐰 = 𝐯, as 𝐯∉Se(N). But then in any neighborhood of 𝐯, the map exp^ν fails to be injective. By the inverse function theorem, we must have d_𝐯exp^ν|_ν̂ is singular, i.e., 𝐯∈ℱ. Also, by <ref>, we see that γ_𝐯 being a limit of N-segments γ_𝐯_i, is an N-segment itself. But then q = γ_𝐯(1) is a first focal locus of N along γ_𝐯 <cit.>. In particular, q is then a cut point, and consequently 𝐯∈Cu(N). Hence, we have Cu(N) = Se(N), concluding the proof.


Since exp^ν is continuous, we immediately get
    
    Cu(N) = exp^ν( Cu(N) ) = exp^ν( Se(N)) ⊂exp^ν( Se(N) ) = Se(N).

Consequently, Se(N) is dense in Cu(N). In fact, Cu(N) = Se(N) holds as well <cit.>.


§ HIGHER ORDER TANGENT VECTOR

In <cit.>, the author used the notion second order tangent vectors to state the regularity condition ([R2]R2). In this appendix, we show that our condition ([R2]R2) is directly comparable to that of <cit.>. See also <cit.> for a similar discussion.

Let M be a manifold with M = n. For p ∈ M, denote by ℐ_p the collection of germs at p of functions M →ℝ. A germ 𝐟∈ℐ_p is assumed to be represented by a function f : M→ℝ locally defined near p, and in particular, the evaluation 𝐟(p) = f(p) is well-defined. It is a local ring, with the maximal ideal 𝔪_p {𝐟 |  f(p) = 0 }. The k^th order tangent space is denoted as 

    T^k_p M ( 𝔪_p / 𝔪_p^k+1)^* = ( 𝔪_p / 𝔪_p^k+1, ℝ).

Given a map f : M → N, one has the k^th derivative map

    d^k_p f : T^k_p M    → T^k_f(p) N 
    θ   ↦( 𝐠 + 𝔪_f(p)^k+1↦θ( f_*(𝐠) + 𝔪^k+1_p ) ),

where the push-forward f_*(𝐠) is defined as the germ of g ∘ f at the point p. There is a canonical isomorphism T^k+1_p M / T^k_p M = ⊙^k+1 T_p M, which gives rise to the (non-canonical) isomorphism T^k_p M = ⊕_i = 1^k ⊙^i T_p M. Here ⊙^i denotes the i^th symmetric tensor product.

We are particularly interested in the identification T_p M ⊙ T_p M = T^2_p M / T_p M,
which is explicitly given as 

    𝐱⊙𝐲⟼( 𝐟↦ X( Y(f) ) |_p )  T_p M

for arbitrary extensions X, Y of 𝐱, 𝐲∈ T_p M, and for 𝐟∈𝔪_p/𝔪^3_p represented by some f. The Leibniz rule shows that the map is well-defined, whereas the symmetry is a consequence of the identity [X, Y](f) = XY(f) - YX(f) where 𝐟↦ [X, Y](f)|_p is a first order vector. The second order derivative of some f : M → N induces the following natural map,

    d^2_p f : T_p M ⊙ T_p M    ⟶ T^2_f(p) M / Im d_p f 
    𝐱⊙𝐲   ⟼( 𝐠↦ X ( Y ( g ∘ f ) ) |_p ) Im d_p f,

which we still denote by d^2 f.

Let us now consider the restricted normal exponential map ℰ = ( exp^ν|_ν̂) : ν̂→ M. For any 𝐯∈ν̂, we have the geodesic γ_𝐯(t) = ℰ(t𝐯). Using the identification above, we have the maps

    d_𝐯ℰ : T_𝐯ν̂→ T_γ_𝐯(1) M,    d^2_𝐯ℰ : T_𝐯ν̂⊙ T_𝐯ν̂→ T^2_γ_𝐯(1)M / Im d_𝐯ℰ.

Denote the ray σ(t) = t𝐯 and the vector 𝐫 = σ̇(1) ∈ T_𝐯ν̂. Now, the condition (R2) of <cit.> translates to the following

    
  (R2') The map
    
    K_𝐯 =  d_𝐯ℰ∋𝐱⟼ d^2_𝐯ℰ(𝐫⊙𝐱) ∈ T_γ_𝐯(𝐯)M / Im d_𝐯ℰ⊂ T^2_γ_𝐯(1) M /Imd_𝐯ℰ

    is a linear isomorphism onto the image.



    For any 𝐱∈ T_𝐯ν̂, consider the N-Jacobi field J_𝐱 along γ_𝐯 (<ref>). Then, we have the following.
    
        
  *  d_𝐯ℰ(𝐱) = J_𝐱(1), and in particular d_𝐯ℰ( 𝐫 ) = γ̇_𝐯(1).
        
  *  d^2_𝐯ℰ( 𝐫⊙𝐱) = J̇_𝐱(1), and thus J̇_𝐱(1) = RX(ℰ) for any extensions
    
    Consequently, ([R2]R2) is equivalent to ([warnerR2]R2').


    Let α : (-ϵ, ϵ) →ν̂ be a curve such that α(0) = 𝐯 and α̇(0) = 𝐱. Then, we immediately have 
    
    d_𝐯ℰ(𝐱) = . ∂/∂ s|_s=0ℰ( α(s) ) = J_𝐱(1),

    which proves ([prop:secondOrderTangent:firstOrder]1). Now, consider a germ 𝐟∈𝔪_p^2/𝔪_p, represented by some f defined near ℰ(𝐯) = γ_𝐯(1). Considering the map Ξ(s, t) = t α(s), we have an extension X(t)= . ∂/∂ s|_s=0Ξ(s, t) of 𝐱 along σ. Extend this X arbitrarily in a neighborhood of 𝐯, and similarly extend 𝐫 to some R. Then,
    
    d^2_𝐯ℰ( 𝐫⊙𝐱)(𝐟)    = R X (f ∘ℰ)|_𝐯 = . ∂/∂ t|_t = 1( X(f ∘ℰ) ) (t 𝐯) = . ∂/∂ t|_t = 1 d(f ∘ℰ) ( X_t 𝐯) 
       = . ∂/∂ t|_t = 1. ∂/∂ s|_s = 0 d(f ∘ℰ) ( t α(s) ) = d_ℰ(𝐯)f ( J̇_𝐱(1) ) = J̇_𝐱(1) ( 𝐟).

    This proves ([prop:secondOrderTangent:secondOrder]2). We conclude the proof from <ref> and <ref>.




	Suppose 𝐯∈ℱ^reg be a regular focal point, such that K_𝐯⊄T_𝐯ℱ^reg. Let 𝐲∈ K_𝐯∩ T_𝐯ℱ^reg. We shall show that J̇_𝐲(1) = 0, whence we get 𝐲 = 0 by [R2]R2.
	
	Consider the vector 𝐫∈ T_𝐯ν̂ given by 𝐫 = σ̇(1), where σ(t) = t𝐯 is the ray. Since 𝐫 is transverse to both T_𝐯ℱ^reg and K_𝐯 (by <ref> and [R1]R1, respectively), and since T_𝐯ν̂ = K_𝐯 + T_𝐯ℱ^reg by hypothesis, we have 𝐫 = 𝐱 + 𝐳 for some 𝐱∈ K_𝐯∖ T_𝐯ℱ^reg and 𝐳∈ T_𝐯ℱ^reg∖ K_𝐯. Then, it follows from <ref> ([prop:secondOrderTangent:secondOrder]2) that 
	
    J̇_𝐲(1) = d^2_𝐯ℰ(𝐫⊙𝐲) = d^2_𝐯ℰ (𝐱⊙𝐲) + d^2_𝐯ℰ(𝐳⊙𝐲),

	where d^2_𝐯ℰ denotes the second order derivative of ℰ (see <ref>).
	
	As 𝐳∈ T_𝐯ℱ^reg, we can choose an extension Z of 𝐳 around 𝐯, such that Z is tangential to ℱ^reg on points of ℱ^reg. Similarly, we can extend 𝐲 to some Y near 𝐯 so that Y is in K_𝐯 on points of ℱ^reg, which is possible since K_𝐯 has constant rank on ℱ^reg. Then, for any function f defined near ℰ(𝐯) we have
	
    Z ( Y ( f ∘ℰ) )(𝐯) = Z ( df   dℰ (Y))(𝐯) = 0,

	as dℰ(Y) = 0 on ℱ^reg. But then d^2_𝐯ℰ(𝐳⊙𝐲) = 0 follows from the definition (<ref>). Since 𝐲∈ T_𝐯ℱ^reg as well, we get from the symmetry, d^2_𝐯ℰ(𝐱⊙𝐲) = d^2_𝐯ℰ(𝐲⊙𝐱) = 0. Thus, J̇_𝐲(1) = 0, which shows that 𝐲 = 0 by ([R2]R2). In other words, K_𝐯∩ T_𝐯ℱ^reg = 0, showing that K_𝐯 = 1. Thus, for K_𝐯≥ 2 we must have K_𝐯⊂ T_𝐯ℱ^reg.



	Suppose we are in either case ([thm:normalExponentialLocalForm:1]1) or case ([thm:normalExponentialLocalForm:2]2). In both the cases, we have a k-dimensional foliation on 𝒞 near 𝐯, induced by the distribution d( ℰ|_𝒞), which is of constant rank k and involutive near 𝐯. We can get a coordinate system { u^1, … , u^n } on ν̂ near 𝐯 such that the codimension 1 submanifold 𝒞 is locally given as { u^n = 0 }. Furthermore, we can arrange so that the foliation on 𝒞 is given by 
	
    { u^k+1 = … = u^n-1 = constant,    u^n = 0.}.

	Recall that each leaf of this foliation is mapped to a constant by ℰ, and d_𝐯ℰ has full rank in the transverse directions. Hence, we can fix a coordinate { v^1,… ,v^n } near ℰ(𝐯) so that image of the slice { u^1 = … = u^k = 0,    u^n = 0 } under ℰ is the slice { v^1 = … = v^k = 0,    v^n = 0 }, and furthermore, d_𝐯ℰ( . ∂/∂ u^i|_𝐯) = . ∂/∂ v^i|_ℰ(𝐯) holds for i = k+1,… , n. In particular, for i = 1,…, k the function v^i ∘ℰ has a zero of order 1 at 𝐯. Define the functions 
	
    w^i = 
    	
    		u^i,    i = 1,… ,k 
    
    		v^i ∘ℰ,    i = k + 1, … , n,

	near 𝐯. Then, { w^1,… , w^n } is a coordinate near 𝐯. Next, consider the projection π : ℝ^n →ℝ^n onto the last n-k-coordinates, preceded by k many zeros. Near ℰ(𝐯) ∈ M, we then have functions, 
	
    y^i = 
    	
    		v^i - v^i ∘ℰ∘( ϕ^-1∘π∘ψ),    i=1,… k
    
    		v^i,    i=k+1, … , n,

	where ϕ, ψ are respectively the coordinate charts { w^1,… w^n } and { v^1,… v^n } on ν̂ and M. By construction, { y^1,… ,y^n } is a coordinate system near ℰ(𝐯), since 𝐯 is an order 1 zero of v^i ∘ℰ for i=1,… ,k. Also, for i = 1,… k, we have y^i ∘ℰ = 0 on { w^n = 0 }. Hence, we have functions x^i near 𝐯 such that y^i ∘ℰ = w^n · x^i for i=1,… k. Set x^i = w^i for i = k+1,… n. Thus, we have
	
    y^i ∘ℰ = 
    	
    		x^n · x^i,    i=1,… ,k
    
    		x^i,    i=k+1,… n.

	We check that { x^1,… , x^n } is a coordinate near 𝐯. As x^i = w^i for i = k+1,… ,n, it follows that the n × n matrix ( . ∂ x^i/∂ w^j|_𝐯) has full rank if the k × k block given by 1 ≤ i, j ≤ n has full rank. We have 
	
    . ∂^2 ( y^i ∘ℰ)/∂ w^n ∂ w^j|_𝐯 = . ∂ x^i/∂ w^j|_𝐯 ,      1 ≤ i, j ≤ k,

	since y^i ∘ℰ has order 1 zero at 𝐯, and y^i ∘ℰ = w^n · x^i. Consider the vectors 𝐰^i = . ∂/∂ w^i|_𝐯. By construction, K_𝐯 =  d_𝐯ℰ = Span⟨𝐰^1,… ,𝐰^k ⟩. Furthermore, 𝐰^n is transverse to T_𝐯𝒞. Consider the radial vector 𝐫∈ T_𝐯ν̂ given by 𝐫 = σ̇(1), where σ(t) = t𝐯 is the ray. Then, we have 𝐫 = a 𝐰^n + 𝐳 for some scalar a  0 and some 𝐳∈ T_𝐯ℱ^reg. As argued in <ref>, Im d_𝐯ℰ we have 
	
    J̇_𝐰^i(1) = d^2_𝐯ℰ( 𝐫⊙𝐰^i ) = a d^2_𝐯ℰ( 𝐰^n ⊙𝐰^j ) + d^2_𝐯ℰ( 𝐳⊙𝐰^j )_0 = a d^2_𝐯ℰ( 𝐰^n ⊙𝐰^j ) .

	By [R2]R2, we have { d^2_𝐯ℰ(𝐰^n ⊙𝐰^j) Imd_𝐯ℰ,    1 ≤ i ≤ k } are linearly independent. Now, by construction 
	
    Im d_𝐯ℰ = Span⟨. ∂/∂ v^i|_ℰ(𝐯) ,   i = k+1,… ,n ⟩ = Span⟨. ∂/∂ y^i|_ℰ(𝐯) ,   i = k+1,… ,n ⟩.

	Evaluating on y^i for i = 1,…,k we then get that the k × k matrix 
	
    [ . ∂^2(y^i ∘ℰ)/∂ w^n ∂ w^j|_𝐯 ] = 
    	[ d^2_𝐯ℰ(𝐰^n ⊙𝐰^j)(y^i) ]

	has full rank. This shows that { x^1,… ,x^n } is a coordinate system near 𝐯, concluding the proof of ([thm:normalExponentialLocalForm:1]1) and ([thm:normalExponentialLocalForm:2]2).

	Now, suppose the hypothesis of ([thm:normalExponentialLocalForm:3]3) holds. Thus, 𝐯∈ℱ^reg is such that in a neighborhood 𝐯∈ U ⊂𝒞 we have K_𝐮∩ T_𝐯𝒞 = 0. We show that near 𝐯, the normal exponential map ℰ is a submersion with folds (see <cit.> for terminology). This amounts to checking the following two properties.
	
		
  * As in the proof of <ref>, near 𝐯, the codimension 1 submanifold 𝒞 is given as the zero of the function Δ (<ref>). For k = 1, this Δ is precisely the product of eigenfunctions of dℰ, and thus Δ =  d ℰ. It follows from [R2]R2 that along the radial direction, the derivative of Δ is non-vanishing. This shows that the first jet j^1 ℰ is transverse to the codimension 1 submanifold S_1 ⊂ J^1(ν̂, M) consisting of jets of rank n - 1.
		
  * The singularities of ℰ are precisely the neighborhood U, along which we have T_𝐮ν̂ = K_𝐮 + T_𝐮𝒞 =  d_𝐮ℰ + T_𝐮( Δ^-1(0) ). This shows that every singularity of ℰ is a fold point.
	
	Then, the normal form <ref> for ℰ follows immediately from <cit.>, finishing the proof of ([thm:normalExponentialLocalForm:3]3).






§ ACKNOWLEDGEMENT
 The authors would like to thank to J. Itoh for many fruitful discussions. They would like to thank ICTS for the wonderful hosting and a research-friendly environment during the workshop Zero Mean Curvature, where the bulk of this work was done. The first author was supported by the NBHM grant no. 0204/1(5)/2022/R &D-II/5649 and the second author was supported by the Jilin University. 

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