# On the Cut Locus of Submanifolds of a Finsler Manifold

Aritra Bhowmick  Department of Mathematics and Statistics, IISER Kolkata
West Bengal, India<EMAIL_ADDRESS>and  Sachchidanand Prasad  ICTS-TIFR
Bangalore, Karnataka, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the cut locus of submanifolds in a Finsler
manifolds, a natural generalization of Riemannian manifolds. We explore the
deformation and characterization of the cut locus, extending the results of
[BP21]. We also obtain a generalization of Klingenberg’s lemma for closed
geodesic ([Kli59]) in the Finsler setting.

###### Key words and phrases:

cut locus, conjugate locus, closed geodesics, Finsler geometry, Finsler
submanifolds, cone bundles

###### 2020 Mathematics Subject Classification:

Primary: 53C22, 53B40; Secondary: 53C60

##  1\. Introduction

In the field of Riemannian geometry, the study of geodesics plays a
fundamental role in comprehending the geometric properties of the underlying
space. _Geodesics_ are (locally) the shortest paths between two points on a
Riemannian manifold, akin to straight lines in Euclidean spaces. One important
concept related to geodesics is the _cut locus_. The cut locus of a point (or
a submanifold) of a Riemannian manifold is the set of points beyond which a
distance minimizing geodesic from the point (or the submanifold) ceases to be
distance minimal. Cut locus of a point, a notion introduced by Henri Poincaré
[Poi05], has been extensively studied (see [Kob67] for a survey as well as
[Buc77], [Mye35], [Wol79], and [Sak96]). Prior to Poincaré, its implicit
appearance maybe traced back to a paper by Mangoldt [vM81]. Other articles
such as [Whi35] and [Mye35, Mye36] describe topological properties of the cut
locus. The exploration of the cut locus of a Riemannian manifold plays a
crucial role in understanding the manifold’s underlying geometry, leading to
many interesting results in Riemannian geometry. For instance, Rauch’s proof
of the “Sphere theorem” [Rau59] heavily relies on estimates of distances to
the cut locus. Additionally, it was realized that much of a manifold’s
topological properties lie in its cut locus. A very good account of these
results and methods are contained in articles [Kli59, Kob67, Wei68].

Finsler manifolds provide a generalization of Riemannian manifolds by
considering a broader class of metrics that capture both the length and the
direction of tangent vectors. The study of Finsler geometry has yielded
significant insights and applications in various fields, including physics,
optimization theory, and robotics. Essentially, a _Finsler manifold_ is a
manifold $M$ where each tangent space is equipped with a Minkowski norm, which
is a norm not necessarily induced by an inner product. In sharp contrast to
the Riemannian case, this norm induces canonical inner products that are not
parameterized by points of $M$, but by directions in $TM$. Thus, one can think
of a Finsler manifold as a space where the inner product does not only depend
on where you are but also in which direction you are looking. While a
Riemannian manifold is itself trivially a Finsler manifold, where the inner
product at a point is the same in all the tangent directions, Finsler geometry
represents quite a large step away from the Riemannian realm. Nonetheless,
Finsler geometry contains analogs for many of the natural objects in
Riemannian geometry. For example, the length of a curve, geodesics, curvature,
connections, and covariant derivatives all generalize; however, normal
coordinates do not [Run59]. In this article, we focus on investigating the cut
locus of a submanifold within a Finsler manifold. For a Finsler manifold, the
cut locus of a point has been extensively studied, see [War65, Wei68, Has73,
ST16].

The second author with Basu in [BP21] showed that the complement of the cut
locus of a submanifold in a Riemannian manifold topologically deforms to $N$.
In this article, we prove the same for Finsler manifolds as well. Let $(M,F)$
be a Finsler manifold $M$ with a Finsler metric $F$.

######  Theorem A (4.10).

Let $N$ be a closed submanifold of $(M,F)$ and $\mathrm{Cu}(N)$ denotes the
cut locus of $N$. Then,

  1. (1)

$M\setminus\mathrm{Cu}(N)$ strongly deformation retracts onto $N$, and

  2. (2)

$M\setminus N$ strongly deformation retracts onto $\mathrm{Cu}(N)$.

Along with the above result, we give two characterizations of the cut locus.
The first one is in terms of the focal locus (4.6), and the second one is in
terms of separating set (4.8). Recall that the _focal locus_ is the collection
of points where the normal exponential map fails to be of full rank (3.22),
whereas the _separating set_ of a submanifold is the collection of all points
which can be joined by more than one minimizing geodesics from the submanifold
(4.1). The second characterization is particularly important in calculations,
as shown in [Pra23].

######  Theorem B (4.6).

Let $N$ be a closed submanifold of a Finsler manifold $(M,F)$. Then,
$q\in\mathrm{Cu}(N)$ if and only if $\gamma:[0,\ell]\to M$ is an $N$-segment
with $q=\gamma(\ell)$ and at least one of the following holds true.

  1. (1)

There exists two distinct $N$-segments joining $N$ to $q$, i.e,
$q\in\mathrm{Se}(N)$, or

  2. (2)

$q$ is the first focal point of $N$ along $\gamma$.

######  Theorem C (4.8).

Let $N$ be a submanifold of a Finsler manifold $(M,F)$. Then, the separating
set of $N$ is dense in the cut locus of $N$. Furthermore, if $N$ is closed,
then the cut locus is a closed set, and we have
$\mathrm{Cu}(N)=\overline{\mathrm{Se}(N)}$.

In Finsler manifolds, geometric objects possess a two-sided nature, exhibiting
both forward and backward characteristics. This duality arises due to the
inherent asymmetry of the distance function in Finsler geometry. Consequently,
Klingenberg’s celebrated result on closed geodesics loops in Riemannian
geometry ([Kli59]), does not generalize directly to the Finsler setup, and one
needs to look at _reversible_ Finsler metrics, which are symmetric in nature.
For submanifolds, geodesics emanating from them and also ending in them,
possibly at a point different from the origin, has been studied by Omori in
[Omo68]. Generalizing results from [ISS12] and [Xu15], we have the following,
which is new even for the Riemannian setup.

######  Theorem D (4.17).

Let $N$ be a submanifold of a reversible Finsler manifold $(M,F)$. Suppose
$x_{0}\in\mathrm{Cu}(N)$ is a (local) minima of the distance function from $N$
on $\mathrm{Cu}(N)$. Then, either $x_{0}$ is a focal point of $N$, or we have
a geodesic starting and ending at $N$ crossing the cut locus exactly at the
midpoint.

The above result follows from 4.15, which provides a criterion for the
existence of exactly two distance minimizing geodesics from a manifold to its
cut locus and might be of independent interest.

### Organization of the paper

In section 2, we begin by recalling basic definitions from Finsler geometry,
including the Finsler metric (2.1) and the Chern connection (2.3). These
fundamental concepts lay the groundwork for our subsequent discussions. In the
next section (section 3), we introduce the cut locus (3.16) and the focal
locus (3.22) of a submanifold, and derive some basic properties of them. In
section 4, we present our main results including the characterization of the
cut locus, and the deformation of its complement. The last part (subsection
4.5) of this section, we address the generalization of Klingenberg’s lemma to
the Finsler setup.

##  2\. Preliminaries on Finsler Geometry

This section recalls some basic notions about Finsler geometry and cut locus.
Primary references for this section are [She01, Pet06, JS15, Oht21].
Throughout this article, we shall use the notation $\widehat{TM}=TM\setminus
0$ to denote the slit tangent bundle of a manifold $M$. Given a bundle $E$
over a manifold, $\Gamma E$ will be considered as the sheaf of sections of
$E$, in particular, $X\in\Gamma E$ denotes a _local_ section of $E$ with some
unspecified open set in $M$ as the domain of definition.

###  2.1. Finsler Metric

Finsler metric was first studied by P. Finsler in his dissertation [Fin51]. In
his thesis, he studied the variational problems of Finsler metrics.

######  Definition 2.1.

Let $M$ be a smooth manifold, and $TM$ denotes its tangent bundle. A Finsler
metric on $M$ is a continuous function $F:TM\to\mathbb{R}$ satisfying the
following properties.

  1. (1)

$F$ is smooth on $\widehat{TM}$;

  2. (2)

for any $p\in TM$, the restriction $F_{p}\vcentcolon=F\big{|}_{T_{p}M}$ is a
Minkowski norm, i.e.,

    1. (a)

(Positive $1$-homogeneity) for any $\lambda>0$ and $\mathbf{v}\in
T_{p}M\setminus\\{0\\}$, we have $F_{p}(\lambda\mathbf{v})=\lambda
F_{p}(\mathbf{v})$, and

    2. (b)

(Strong Convexity) for all $\mathbf{v}\in T_{p}M\setminus\\{0\\}$, the
symmetric tensor $g_{\mathbf{v}}$ on $T_{p}M$, called the _fundamental tensor_
, is positive definite, where

$g_{\mathbf{v}}(\mathbf{v}_{1},\mathbf{v}_{2})\vcentcolon=\left.\dfrac{1}{2}\dfrac{\partial^{2}}{\partial
s_{1}\partial
s_{2}}\right|_{s_{1}=s_{2}=(0,0)}\left(F_{p}(\mathbf{v}+s_{1}\mathbf{v}_{1}+s_{2}\mathbf{v}_{2})\right)^{2}.$

$F$ is called a _pseudo-Finsler_ (or, _semi-Finsler_) metric if the
fundamental tensor is only nondegenerate. $F$ is _reversible_ if
$F(-\mathbf{v})=F(\mathbf{v})$ holds for all $\mathbf{v}\in\widehat{TM}$.

######  Remark 2.2.

Although we only state them for Finsler manifolds, all the results presented
in this article remain true for pseudo-Finsler manifolds, without any
significant changes to the proof, unless mentioned otherwise.

As a consequence of the homogeneity, the fundamental tensor satisfies the
useful identity

$g_{\mathbf{v}}(\mathbf{v},\mathbf{v})=F_{p}(\mathbf{v})^{2},\quad\mathbf{v}\in
T_{p}M\setminus 0.$

For $\mathbf{v}\in T_{p}M\setminus 0$, the associated _Cartan tensor_ on
$T_{p}M$ is a symmetric $3$-tensor defined as

$C_{\mathbf{v}}(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3})\vcentcolon=\left.\frac{1}{4}\frac{\partial^{3}}{\partial
s_{1}\partial s_{2}\partial
s_{3}}\right|_{s_{1}=s_{2}=s_{3}=0}\left(F_{p}(\mathbf{v}+s_{1}\mathbf{v}_{1}+s_{2}\mathbf{v}_{2}+s_{3}\mathbf{v}_{3})\right)^{2}.$

For each $\mathbf{v}\in T_{p}M\setminus 0$ and $\mathbf{u},\mathbf{w}\in
T_{p}M$, we have

$C_{\mathbf{v}}(\mathbf{v},\mathbf{u},\mathbf{w})=C_{\mathbf{v}}(\mathbf{u},\mathbf{v},\mathbf{w})=C_{\mathbf{v}}(\mathbf{u},\mathbf{w},\mathbf{v})=0.$

We extend the definition of the fundamental tensor to vector fields as
follows:

$g_{V}(X,Y)(p)\vcentcolon=g_{V_{p}}(X_{p},Y_{p}),$

for any $V\in\Gamma\widehat{TM}$ and $X,Y\in\Gamma TM$ defined near $p\in M$.
Similarly, we extend the definition of the Cartan tensor as well.

####  2.1.1. Chern Connection

Unlike the Levi-Civita connection in the Riemannian context, we have a family
of torsionless connections on a Finsler manifold.

######  Definition 2.3.

[Rad04, Jav14] For each $V\in\Gamma\widehat{TM}$, we have a unique affine
connection

$\nabla^{V}:\Gamma TM\otimes\Gamma TM\rightarrow\Gamma TM,$

called the _Chern connection_ , satisfying the following conditions for any
$X,Y,Z\in\Gamma TM$.

  * •

(Torsion freeness) $\nabla^{V}_{X}Y-\nabla^{V}_{Y}X=[X,Y]$

  * •

(Almost metric compatibility)
$X(g_{V}(Y,Z))=g_{V}(\nabla^{V}_{X}Y,Z)+g_{V}(Y,\nabla^{V}_{X}Z)+\frac{1}{2}C_{V}(\nabla^{V}_{X}V,Y,Z)$

The associated _curvature tensor_ is given by

$R^{V}(X,Y)Z\vcentcolon=\nabla^{V}_{X}\nabla^{V}_{Y}Z-\nabla^{V}_{Y}\nabla^{V}_{X}Z-\nabla^{V}_{[X,Y]}Z,\quad
X,Y,Z\in\Gamma TM,$

which is skewsymmetric: $R^{V}(X,Y)=-R^{V}(Y,X)$.

######  Definition 2.4.

Given a curve $\gamma:[a,b]\rightarrow M$ and
$W\in\Gamma\gamma^{*}\widehat{TM}$, the _covariant derivative_ along $\gamma$
is defined as

$D^{W}_{\gamma}:\Gamma\gamma^{*}TM\rightarrow\Gamma\gamma^{*}TM,$

which satisfies the following metric compatibility

$\frac{d}{dt}g_{W}(X,Y)=g_{W}(D^{W}_{\gamma}X,Y)+g_{W}(X,D^{W}_{\gamma}Y)+2C_{W}(D^{W}_{\gamma}W,X,Y),$

for any $X,Y\in\Gamma\gamma^{*}TM$. If $\dot{\gamma}(t)\neq 0$, then a vector
field $X\in\Gamma\gamma^{*}TM$ is said to _parallel_ with respect to $\gamma$
if $D^{\dot{\gamma}}_{\gamma}X=0$.

####  2.1.2. Legendre Transformation

We have a _nonlinear_ fiber preserving map
$\mathfrak{L}:\widehat{TM}\rightarrow\widehat{T^{*}M}$ given by

$\mathfrak{L}(\mathbf{v})(\mathbf{w})\vcentcolon=g_{\mathbf{v}}(\mathbf{v},\mathbf{w}),\quad\mathbf{v}\in
T_{p}M\setminus 0,\;\mathbf{w}\in T_{p}M.$

$\mathfrak{L}$ is called the _Legendre transformation_ associated to $(M,F)$.
It follows that $\mathfrak{L}$ is a $C^{\infty}$-diffeomorphism. We extend
$\mathfrak{L}$ to all of $TM$ by setting $\mathfrak{L}(0)=0$. The extension
$\mathfrak{L}:TM\rightarrow T^{*}M$ is then a fiber-preserving _homeo_
morphism.

###  2.2. Geodesics and Jacobi Fields

Let us denote the space of piecewise smooth paths $\gamma:[a,b]\rightarrow M$
as $\mathcal{P}=\mathcal{P}([a,b])$. We have two functionals defined on this
path space.

  * •

(Length Functional)

$\displaystyle L:\mathcal{P}$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle\gamma$ $\displaystyle\mapsto\int_{a}^{b}F(\dot{\gamma}(t))dt$

  * •

(Energy Functional)

$\displaystyle E:\mathcal{P}$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle\gamma$
$\displaystyle\mapsto\frac{1}{2}\int_{a}^{b}F(\dot{\gamma}(t))^{2}dt$

Note that $L$ is independent of the reparametrization of a curve, whereas $E$
is not. A geodesic is defined as a critical point of the energy functional in
the variational sense.

Let us recall the notion of variation of a given curve. Given a
$\gamma\in\mathcal{P}$, we identify the tangent space to $\mathcal{P}$ at
$\gamma$ as the infinite dimensional vector space of vector fields along
$\gamma$:

$T_{\gamma}\mathcal{P}\coloneqq\Gamma\gamma^{*}TM.$

Given a vector field $W\in\Gamma\gamma^{*}TM$, a _$W$ -variation_ of $\gamma$
is a piecewise smooth map $\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow
M$ satisfying

$\Lambda(0,t)=\gamma(t),\quad\left.\frac{\partial}{\partial
s}\right|_{s=0}\Lambda(s,t)=W(t),$

for all $(s,t)$ in the domain. We tacitly assume that each curve
$t\mapsto\Lambda_{s}(t)=\Lambda(s,t)$ has the same breakpoints as $\gamma$.
Lemma 2.1 from Sakai’s book [Sak96] can be easily generalized to the Finsler
setup to prove the existence of $W$-variation.

######  Definition 2.5.

A variation $\Lambda$ of a curve $\gamma$ joining $p=\gamma(a)$ to
$q=\gamma(b)$ is said to be _proper_ if the endpoints are kept fixed, i.e.,

$\Lambda(s,a)=p,\quad\Lambda(s,b)=q,\qquad-\epsilon<s<\epsilon.$

Similarly, a vector field $W\in\Gamma\gamma^{*}TM$ is said to be _proper_ if
$W(a)=0=W(b)$ holds.

We refer to [Oht21, Chapter 7] for the first and second variation formulas for
the Length functional (with respect to proper variations), and as for the
energy functional we refer to [JS15].

######  Definition 2.6.

A piecewise $C^{1}$ curve $\gamma:[a,b]\rightarrow M$ is called a _geodesic_
if it is a critical point of the energy functional with respect to proper
variations, i.e., for any proper variation $\Lambda$ of $\gamma$ we have
$\left.\frac{d}{ds}\right|_{s=0}E(\Lambda_{s})=0$.

As usual, a geodesic is locally distance minimizing, where we have the
following notion of Finsler distance.

######  Definition 2.7.

Given a Finsler manifold $(M,F)$, the distance between points $p,q\in M$ is
defined as

$d(p,q)=\inf\left\\{L(\gamma)\;\middle|\;\gamma\in\mathcal{P},\gamma(a)=p,\gamma(b)=q\right\\}.$

######  Remark 2.8.

In general, the distance function is asymmetric, although it induces the same
topology as that of the manifold. If $F$ is reversible, then $d$ is a
symmetric metric (see, e.g., [BCS00, pg. 151]).

######  Proposition 2.9.

Given a unit-speed curve $\gamma$ (i.e., $F(\dot{\gamma}(t))=1$ for all $t$),
the following are equivalent.

  * •

$\gamma$ is a geodesic.

  * •

$\gamma$ is a critical point of the length functional with respect to proper
variations.

  * •

$\gamma$ is locally distance minimizing, i.e., for any $a<t_{0}<b$, there
exists an open interval containing $t_{0}$, say,
$t_{0}\in(a_{t_{0}},b_{t_{0}})\subset[a,b]$, such that
$d(\gamma(s),\gamma(t))=t-s$ holds for any $a_{t_{0}}\leq s\leq t\leq
b_{t_{0}}$.

  * •

$\dot{\gamma}$ is a parallel vector field with respect to $\gamma$, i.e.,
$D^{\dot{\gamma}}_{\gamma}\dot{\gamma}=0$.

For proof of the above equivalences, we refer to [Oht21]. Geodesics are
precisely the solutions to an initial value problem known as the geodesic
equation (e.g., [Oht21, Equation 3.15]). As such, geodesics are always smooth,
and given a vector $\mathbf{v}\in T_{p}M$, we have a unique maximal geodesic
$\gamma_{\mathbf{v}}:[0,\ell]\rightarrow M$ satisfying

$\gamma(0)=p,\qquad\dot{\gamma}(0)=\mathbf{v}.$

Note that, unlike Riemannian geometry, due to the asymmetry of the Finsler
metric, a geodesic traversed in the reverse direction need not be a geodesic.
Even then, the following observation remains true.

######  Proposition 2.10.

Two distinct geodesics in a Finsler manifold always intersect transversally.

###### Proof.

Let $\gamma:[0,a]\rightarrow M$ and $\eta:[0,b]\rightarrow M$ be two distinct
geodesics meeting at $p=\gamma(a)=\eta(b)$. If possible, suppose
$\mathbf{v}=\dot{\gamma}(a)=\dot{\eta}(b)$. Then, the reversed curves
$\bar{\gamma}:[a,0]\rightarrow M,\bar{\eta}:[b,0]\rightarrow M$ are geodesics
for the reverse Finsler metric ([Oht21, Section 2.5]), with same initial
velocity $-\mathbf{v}$. Consequently, they are identical. But then
$\gamma,\eta$ are identical as well, which is a contradiction. ∎

By the Hopf-Rinow theorem ([Oht21, Theorem 3.21]), a Finsler manifold $(M,F)$
is forward (resp. backward) complete if and only if given any two points
$p,q\in M$ there exists a geodesic joining $p$ to $q$ (resp. $q$ to $p$). In
particular, for a forward (or backward) complete Finsler manifold, the domain
of definition of $\gamma_{\mathbf{v}}$ can be taken to be $[0,\infty)$.
Throughout this article, we shall assume $(M,F)$ to be a forward complete
Finsler manifold, unless otherwise mentioned.

######  Definition 2.11.

The _exponential map_

$\exp:TM\rightarrow M$

at a point $p\in M$ is defined as
$\exp_{p}(\mathbf{v})=\gamma_{\mathbf{v}}(1)$ for any $\mathbf{v}\in T_{p}M$,
where $\gamma_{\mathbf{v}}:[0,\infty)\rightarrow M$ is the unique geodesic
starting at $p$ in the direction of $\mathbf{v}$.

It follows from the theory of ordinary differential equation that the
exponential map is smooth on $\widehat{TM}$, but only $C^{1}$ on the whole
tangent bundle. In fact, the exponential is $C^{2}$ on $TM$ precisely when the
Finsler metric comes from a Riemannian metric, in which case the exponential
map is $C^{\infty}$ on $TM$ as well.

######  Definition 2.12.

A geodesic $\gamma:[0,\ell]\rightarrow M$ is said to be a _global distance
minimizer_ (or simply a _minimizer_) if
$d(\gamma(0),\gamma(\ell))=\ell=L(\gamma)$.

We have the following easy observation.

######  Proposition 2.13.

If $\gamma:[0,\ell]\rightarrow M$ is a minimizer, then for any $0\leq a\leq
b\leq\ell$ we have $\gamma|_{[a,b]}$ is a minimizer, and in particular
$d(\gamma(a),\gamma(b))=b-a$.

Any piecewise smooth curve $\gamma$ joining $p$ to $q$, and satisfying
$L(\gamma)=d(p,q)$ is, up to a reparametrization, a minimizing geodesic from
$p$ to $q$, and in particular $\gamma$ is smooth. A consequence of the
completeness assumption is that any two points $p,q\in M$ can be joined by a
minimizer, say, $\gamma:[0,\ell]\rightarrow M$ so that

$\gamma(0)=p,\qquad\gamma(\ell)=q,\qquad d(p,q)=\ell=L(\gamma).$

####  2.2.1. Jacobi Fields

######  Definition 2.14.

Given a unit-speed geodesic $\gamma:[a,b]\rightarrow M$, a variation
$\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow M$ of $\gamma$ is said to
be a _geodesic variation_ , if for each $-\epsilon<s<\epsilon$, the curve
$\Lambda_{s}:[a,b]\rightarrow M$ is a constant-speed geodesic.

######  Definition 2.15.

Let $\gamma:[a,b]\rightarrow M$ be a unit-speed geodesic. A vector field
$J\in\Gamma\gamma^{*}TM$ along $\gamma$ is called a _Jacobi field_ of $\gamma$
if there exists a geodesic variation
$\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow M$ of $\gamma$ satisfying

$\left.\frac{\partial}{\partial s}\right|_{s=0}\Lambda(s,t)=J(t),\quad
t\in[a,b].$

######  Proposition 2.16.

Given a unit-speed geodesic $\gamma:[a,b]\rightarrow M$, a vector field
$J\in\Gamma\gamma^{*}TM$ is a Jacobi field if and only if it satisfies the
Jacobi equation

$D^{\dot{\gamma}}_{\gamma}D^{\dot{\gamma}}_{\gamma}J-R^{\dot{\gamma}}(\dot{\gamma},J)\dot{\gamma}=0.$

Since the Jacobi equation is a second order ODE, given the initial data,
$\mathbf{u},\mathbf{v}\in T_{\gamma(a)}M$, there exists a unique Jacobi field
$J$ along $\gamma$ satisfying, $J(a)=\mathbf{u}$ and
$D^{\dot{\gamma}}_{\gamma}J(a)=\mathbf{v}$. In particular, the collection of
all Jacobi fields along $\gamma$ forms a vector space of dimension $2\dim M$.

######  Proposition 2.17.

Let $\mathbf{v}\in T_{p}M$ with $F(\mathbf{v})=1$, and
$\gamma(t)=\exp_{p}(t\mathbf{v})$ be the geodesic in the direction of
$\mathbf{v}$. Then, the kernel of the map

$d(\exp_{p})|_{t\mathbf{v}}:T_{t\mathbf{v}}(T_{p}M)=T_{p}M\rightarrow
T_{\gamma(t)}M$

is naturally isomorphic to the vector space

$\left\\{J\;\middle|\;\text{$J$ is a Jacobi field along $\gamma$, satisfying
$J(0)=0=J(t)$.}\right\\}.$

##  3\. Submanifolds in Finsler Manifold

In this section, we generalize the notions from the previous section.

###  3.1. Normal Cone Bundles

Given a submanifold $N$, it is natural to ask for a notion of a normal bundle
to it.

######  Definition 3.1.

Given a submanifold $N\subset M$, the set

$\nu_{p}=\nu_{p}(N)=\left\\{\mathbf{v}\in
T_{p}M\;|\;g_{\mathbf{v}}(\mathbf{v},\mathbf{w})=0\;\forall\mathbf{w}\in
T_{p}N\right\\}$

is called the _normal cone_ of $N$ at $p\in N$. The set $\nu=\nu(N)=\cup_{p\in
N}\nu_{p}(N)$ is called the _normal cone bundle_ of $N$.

It should be noted that $\nu(N)$ is _not_ a vector bundle in general; in fact,
it is a cone bundle. Nevertheless, this is the obvious replacement for the
notion of a normal bundle as in the Riemannian setting. We shall denote
$\hat{\nu}_{p}\vcentcolon=\nu_{p}\setminus\\{0\\}$, and
$\hat{\nu}\vcentcolon=\cup_{p\in N}\hat{\nu}_{p}$ is then the _slit cone
bundle_. The Legendre transformation $\mathfrak{L}$ maps $\nu$
homeomorphically onto the annihilator bundle $\nu^{*}\subset T^{*}M|_{N}$ of
$TN$ defined as

$\nu^{*}|_{p}\vcentcolon=\left\\{\omega\in
T_{p}^{*}M\;\middle|\;\omega|_{T_{p}N}=0\right\\},\quad p\in N.$

It follows that $\hat{\nu}$ is a smooth submanifold of $TM$ of dimension $\dim
N+\operatorname{codim}N=\dim M$, whereas $\nu$ is only a topological
submanifold. Note that $\nu$ is a closed subset of $TM$.

######  Definition 3.2.

The _unit cone bundle_ of $N$ is denoted as $S(\nu)=\cup_{p\in N}S(\nu_{p})$,
where

$S(\nu_{p})\vcentcolon=\left\\{\mathbf{v}\in\nu_{p}\;\middle|\;F_{p}(\mathbf{v})=1\right\\}.$

######  Remark 3.3.

If $N$ is a compact submanifold, then $S(\nu)$ is again a compact submanifold
of $TM$ of dimension $\dim M-1$.

######  Definition 3.4.

Given $N\subset M$, the _normal exponential map_

$\exp^{\nu}:\nu(N)\rightarrow M$

is defined as the restriction of the exponential map to the cone bundle
$\nu(N)$.

As $\hat{\nu}$ is a smooth submanifold of $TM$, the restricted map
$\exp^{\nu}|_{\hat{\nu}}$ is smooth.

###  3.2. $N$-Geodesics and $N$-Jacobi Fields

Given a submanifold $N\subset M$, we have the subspace of piecewise smooth
paths

$\mathcal{P}_{N}=\mathcal{P}_{N}([a,b])=\\{\gamma\in\mathcal{P}([a,b])\;|\;\gamma(a)\in
N\\}$

starting at $N$. For any $\gamma\in\mathcal{P}_{N}$ we say $\gamma$ is a curve
joining $N$ to $\gamma(b)$.

######  Definition 3.5.

Given $q\in M$, the distance from $N$ to $q$ is defined as

$d(N,q)\vcentcolon=\inf\left\\{L(\gamma)\;\middle|\;\gamma\in\mathcal{P}_{N},\;\gamma(b)=q\right\\}.$

We identify the tangent space of $\mathcal{P}_{N}$ at a given
$\gamma\in\mathcal{P}_{N}$ as the infinite dimensional vector space

$T_{\gamma}\mathcal{P}_{N}\vcentcolon=\left\\{W\in\Gamma\gamma^{*}TM\;\middle|\;W(a)\in
T_{\gamma(a)}N\right\\}.$

Given $W\in T_{\gamma}\mathcal{P}_{N}$, a _$W$ -variation_ is a piecewise
smooth map $\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow M$ satisfying

$\Lambda(0,t)=\gamma(t),\quad\left.\frac{\partial}{\partial
s}\right|_{s=0}\Lambda(s,t)=W(t),\quad\Lambda(s,a)\in N,$

for all $(s,t)$ in the domain of the definition. The variation is said to be
_proper_ if it further satisfies $\Lambda(s,b)=\gamma(b)$ for all
$-\epsilon<s<\epsilon$. Clearly, proper variations correspond to those $W\in
T_{\gamma}\mathcal{P}_{N}$ that satisfy $W(b)=0$. This leads to the first and
second variational formulas for the restricted length and energy functionals:
$L|_{\mathcal{P}_{N}}$ and $E|_{\mathcal{P}_{N}}$. We refer to [JS15] for the
relevant formulas.

######  Definition 3.6.

A piecewise $C^{1}$ curve $\gamma:[a,b]\rightarrow M$ in $\mathcal{P}(N)$ is
called an _$N$ -geodesic_ if $\gamma$ is a critical point of the restricted
energy functional $E|_{\mathcal{P}(N)}$ with respect to proper variations,
i.e., for any proper variation $\Lambda$ of $\gamma$ we have
$\left.\frac{d}{ds}\right|_{s=0}E(\Lambda_{s})=0$

######  Proposition 3.7.

Given a unit-speed curve $\gamma:[a,b]\rightarrow M$ in $\mathcal{P}(N)$, the
following are equivalent.

  * •

$\gamma$ is an $N$-geodesic

  * •

$\gamma$ is a critical point of the restricted length functional
$L|_{\mathcal{P}(N)}$ with respect to proper variations.

  * •

$\gamma(t)=\exp^{\nu}(t\mathbf{v})$, where $p=\gamma(0)$ and
$\mathbf{v}=\dot{\gamma}(a)\in S(\nu_{p})$.

For proof of the above equivalences, we refer to [Zha17]. Next, we define the
notion of global distance minimizer from $N$.

######  Definition 3.8.

An $N$-geodesic $\gamma:[0,\ell]\to M$ is called an _$N$ -segment_ joining $N$
to $\gamma(\ell)$ if $d(N,\gamma(\ell))=L(\gamma)=\ell$.

######  Proposition 3.9.

Suppose $(M,F)$ is a forward complete Finsler manifold, and $N$ is a closed
submanifold of $M$. Then, for any $q\in M$ there exists an $N$-segment joining
$N$ to $q$.

Similar to 2.13, we have the following result.

######  Proposition 3.10.

An $N$-geodesic $\gamma:[0,\ell]\rightarrow M$ is an $N$-segment if and only
if $d(N,\gamma(t))=t$ for all $0\leq t\leq\ell$.

###### Proof.

Suppose $\gamma$ is an $N$-segment. Then, $\gamma$ is in particular a
minimizer joining $\gamma(0)$ to $\gamma(\ell)$. If possible, say, there
exists some $0<T<\ell$ so that $d(N,\gamma(T))<T=L(\gamma|_{[0,T]})$. Then,
there exists some $N$-segment $\eta:[0,T_{0}]\rightarrow M$ joining $N$ to
$q=\gamma(T)=\eta(T_{0})$, so that
$L(\eta)=d(N,\gamma(T))<L(\gamma|_{[0,T]})$. Consider the _broken_ curve
$\zeta$ obtained by concatenating $\eta$ and $\gamma|_{[T,\ell]}$. It follows
that

$L(\zeta)=L(\eta)+L(\gamma|_{[T,\ell]})<L(\gamma|_{[0,T]})+L(\gamma|_{[T,\ell]})=L(\gamma)=d(N,\gamma(\ell)),$

which contradicts the fact that $\zeta$ is a curve joining $N$ to
$\gamma(\ell)$. The converse statement is obvious. ∎

####  3.2.1. $N$-Jacobi Fields

######  Definition 3.11.

Given a unit-speed $N$-geodesic $\gamma:[a,b]\rightarrow M$, a variation
$\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow M$ is said to be an _$N$
-geodesic variation_ if for each $-\epsilon<s<\epsilon$, the curve
$\Lambda_{s}:[a,b]\rightarrow M$ is a constant speed $N$-geodesic.

######  Definition 3.12.

Let $\gamma:[a,b]\rightarrow M$ be a unit-speed $N$-geodesic. A vector field
$J\in\Gamma\gamma^{*}TM$ is called an $N$-Jacobi field of $\gamma$ if there
exists an $N$-geodesic variation
$\Lambda:(-\epsilon,\epsilon)\times[a,b]\rightarrow M$ satisfying

$\left.\frac{\partial}{\partial s}\right|_{s=0}\Lambda(s,t)=J(t),\quad
t\in[a,b].$

######  Proposition 3.13.

[JS15] Given a unit-speed $N$-geodesic $\gamma:[a,b]\rightarrow M$, a vector
field $J\in\Gamma\gamma^{*}TM$ is an $N$-Jacobi field if and only if it
satisfies

$D^{\dot{\gamma}}_{\gamma}D^{\dot{\gamma}}_{\gamma}J-R^{\dot{\gamma}}(\dot{\gamma},J)\dot{\gamma}=0,\quad
J(a)\in T_{\gamma(a)}N,\quad\pi
D^{\dot{\gamma}}_{\gamma}J(a)=\pi\nabla^{\dot{\gamma}}_{J}\dot{\gamma}(a),$

where $\pi$ is the first component projection of the canonical splitting
$T_{\gamma(a)}M=T_{\gamma(a)}N\oplus\left(T_{\gamma(a)}N\right)^{\perp_{g_{\dot{\gamma}(a)}}}$.

From the second variation formula of the restricted energy functional
$E|_{\mathcal{P}_{N}}$, we get the index form.

######  Definition 3.14.

Let $\gamma:[a,b]\rightarrow M$ be a unit-speed $N$-geodesic. Then, the _index
form_ $I_{\gamma}$ is defined for $X,Y\in T_{\gamma}\mathcal{P}_{N}$ with
$X(b)=0=Y(b)$ as

$\mathcal{I}_{\gamma}(X,Y)\coloneqq\int_{a}^{b}\left[g_{\dot{\gamma}}(D^{\dot{\gamma}}_{\gamma}X,D^{\dot{\gamma}}_{\gamma}Y)-g_{\dot{\gamma}}\left(R^{\dot{\gamma}}(\dot{\gamma},X)Y,\dot{\gamma}\right)\right]-g_{\dot{\gamma}(a)}\left(\pi^{\perp}\nabla^{\dot{\gamma}(a)}_{X}Y,\dot{\gamma}(a)\right),$

where $\pi^{\perp}$ is the second component projection of the canonical
splitting
$T_{\gamma(a)}M=T_{\gamma(a)}N\oplus\left(T_{\gamma(a)}N\right)^{\perp_{g_{\dot{\gamma}(a)}}}$.

It follows that $I_{\gamma}$ is a symmetric $2$-form, and the kernel of the
index form $I_{\gamma}$ consists of precisely the $N$-Jacobi fields along
$\gamma$.

###  3.3. Cut Locus of a Submanifold

######  Definition 3.15.

The _cut locus of a point_ $p\in M$ is the set consisting of all $q\in M$ 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 $\mathrm{Cu}(p)$.

Generalizing the notion of a cut locus of a point to an arbitrary submanifold
(or, more generally, any subset), we have the following definition.

######  Definition 3.16.

Given a submanifold $N\subset M$, the _cut locus_ of $N$ consists of points
$q\in M$ such that there exists an $N$-segment joining $N$ to $q$, whose
extension fails to be an $N$-segment.

######  Proposition 3.17.

$q\in\mathrm{Cu}(N)$ if and only if for _any_ $N$-segment $\gamma$ joining $N$
to $q$, an extension of $\gamma$ fails to be an $N$-segment.

###### Proof.

Suppose $q\in\mathrm{Cu}(N)$. We then have an $N$-segment
$\gamma:[0,\ell]\rightarrow M$ with $\ell=d(N,q)$ and $q=\gamma(\ell)$, whose
extension fails to be an $N$-segment. If possible, let
$\eta:[0,\ell]\rightarrow M$ be another $N$-segment joining $N$ to
$q=\eta(\ell)$, which remains an $N$-segment when extended to
$\eta:[0,\ell+\epsilon]\rightarrow M$ for some $\epsilon>0$. Since
$\gamma,\eta$ are distinct, we have $\dot{\gamma}(\ell)\neq\dot{\eta}(\ell)$
(2.10). Thus, we have a _broken_ curve $\zeta$ obtained from concatenating
$\gamma$ and $\eta|_{[\ell,\ell+\epsilon]}$. Now,
$L(\zeta)=L(\gamma)+L(\eta|_{[\ell,\ell+\epsilon]})=\ell+\epsilon=d(N,\eta(\ell+\epsilon))$,
i.e., $\zeta$ achieves the minimal distance. But then $\zeta$ must be a smooth
curve, which is a contradiction.

The converse statement is obvious. ∎

######  Definition 3.18.

Given $\mathbf{v}\in S(\nu)$, the _cut time_ of $\mathbf{v}$ is defined as

$\rho(\mathbf{v})\vcentcolon=\sup\left\\{t\;\middle|\;d(N,\gamma_{\mathbf{v}}(t\mathbf{v}))=t\right\\},$

where $\gamma_{\mathbf{v}}(t)=\exp^{\nu}(t\mathbf{v})$ is the unique
$N$-geodesic with initial velocity $\mathbf{v}$.

Note that we allow $\rho$ to take the value $\infty$, although if $M$ is
compact then $\rho(\mathbf{v})<\infty$ for any $\mathbf{v}\in\nu$. Indeed, we
shall see in 4.7 that the map $\rho:\nu\rightarrow[0,\infty]$ is continuous,
and consequently, $\rho$ is finite if $N$ is closed. It follows from [AJ19]
that $\rho$ is always strictly positive.

######  Definition 3.19.

Given a submanifold $N\subset M$, the _tangent cut locus_ of $N$ is defined as

$\widetilde{\mathrm{Cu}}(N)\vcentcolon=\left\\{\rho(\mathbf{v})\mathbf{v}\;\middle|\;\mathbf{v}\in
S(\nu),\;\rho(\mathbf{v})\neq\infty\right\\}\subset\nu.$

It follows from 3.16 that
$\mathrm{Cu}(N)=\exp^{\nu}(\widetilde{\mathrm{Cu}}(N))$.

###  3.4. Focal Locus of a Submanifold

######  Definition 3.20.

Given a submanifold $N\subset M$, a vector $\mathbf{v}\in\hat{\nu}$ is said to
be a _tangent focal point_ of $N$ if

$d(\exp^{\nu})|_{\mathbf{v}}:T_{\mathbf{v}}\hat{\nu}\rightarrow
T_{\exp^{\nu}(\mathbf{v})}M$

is degenerate, i.e., if $\mathbf{v}$ is a critical point of
$\exp^{\nu}|_{\hat{\nu}}$. The nullity of the map is known as the _index_ of
the tangent focal point $\mathbf{v}$. The collection of all such vectors is
called the _tangent focal locus_ of $N$.

######  Definition 3.21.

Given $N\subset M$ and $\mathbf{v}\in S(\nu)$, we define the _(first) focal
time_ as

$\lambda(\mathbf{v})=\inf\left\\{t\;\middle|\;\text{$d(\exp^{\nu})|_{t\mathbf{v}}$
is degenerate}\right\\}.$

We set $\lambda(\mathbf{v})=\infty$ if there are no focal points of $N$ in the
direction of $\mathbf{v}$.

######  Definition 3.22.

The _focal locus_ of $N$ is defined as the image of the tangent focal locus
under the $\exp^{\nu}$ map. In other words, the focal locus of $N$ consists of
the critical values of the (restricted) normal exponential map
$\exp^{\nu}|_{\hat{\nu}}$. The _first focal locus_ of $N$ is defined as the
set $\left\\{\exp^{\nu}(\lambda(\mathbf{v})\mathbf{v})\mid\mathbf{v}\in
S(\nu),\lambda(\mathbf{v})\neq\infty\right\\}$.

In the literature, the focal locus of a submanifold (of a Riemannian manifold)
is also known as the conjugate locus. In this article, we reserve the term
conjugate locus for when $N$ is a point.

######  Proposition 3.23.

Let $\mathbf{v}\in S(\nu)$ and $\gamma$ be the unit-speed $N$-geodesic given
as $\gamma(t)=\exp^{\nu}(t\mathbf{v})$. Then for any $t>0$, the kernel of the
map

$d\left(\exp^{\nu}|_{\hat{\nu}}\right)\big{|}_{t\mathbf{v}}:T_{t\mathbf{v}}\hat{\nu}\rightarrow
T_{\gamma(t)}M$

is naturally isomorphic to the vector space

$\left\\{J\;\middle|\;\text{$J$ is an $N$-Jacobi field along $\gamma$,
satisfying $J(t)=0$}\right\\}.$

##  4\. Main Results

We start from an important characterization of the cut locus in terms of focal
locus. A result of Basu and Prasad [BP21] may be generalized to prove that the
separating set is dense in the cut locus (4.8). We will also see that the
complement of the cut locus and submanifold strongly deforms to the
submanifold and the cut locus, respectively. We will prove the existence of
$N$-geodesic loop which is a generalization of the result of [ISS12].

###  4.1. Characterization of Cut Locus

In this section, we discuss two ways one can characterize the cut locus of a
submanifold, which are well-known in the Riemannian setting. Recall first the
definition.

######  Definition 4.1.

Given a submanifold $N\subset M$, a point $p\in M$ is said to be a _separating
point_ of $N$ if there exists two distinct $N$-segments joining $N$ to $p$.
The collection of all separating points of $N$ is called the _separating set_
(or the _Maxwell strata_) of $N$, denoted $\mathrm{Se}(N)$,

######  Proposition 4.2.

Let $N$ be a submanifold of $(M,F)$. Then,
$\mathrm{Se}(N)\subset\mathrm{Cu}(N)$.

###### Proof.

Let $q\in\mathrm{Se}(N)$, and take two $N$-segments
$\gamma_{1},\gamma_{2}:[0,\ell]\to M$ joining, respectively, $p_{1},p_{2}\in
N$ to $q$. If possible, suppose $q\not\in\mathrm{Cu}(N)$. Then, a sufficiently
small extension of $\gamma_{2}$, say, $\gamma_{2}:[0,\ell+\epsilon]\to M$
remains an $N$-segment (3.17), and hence,
$d(N,\gamma_{2}(\ell+\epsilon))=\ell+\epsilon$.

$\gamma_{1}$
---
$\gamma_{2}$
---
$N$
---
$p_{1}$
---
$p_{2}$
---
$\gamma_{1}(\ell)=q=\gamma_{2}(\ell)$
---
$\gamma_{2}(\ell+\epsilon)$
---
$\eta$
---
Figure 1. Constructing the broken curve $\eta$.

On the other hand, by 2.10,
$\dot{\gamma}_{1}(\ell)\neq\dot{\gamma}_{2}(\ell)$, and so we have a broken
curve $\eta:[0,\ell+\epsilon]\to M$ obtained from concatenating $\gamma_{1}$
and $\gamma_{2}|_{[\ell,\ell+\epsilon]}$ (Figure 1). In particular, from the
triangle inequality, we get

$d(p_{1},\eta(\ell+\epsilon))<d(p_{1},\eta(\ell))+d(\eta(\ell),\eta(\ell+\epsilon))=\ell+\epsilon.$

Consequently,

$d(N,\gamma_{2}(\ell+\epsilon))=d(N,\eta(\ell+\epsilon))\leq
d(p_{1},\eta(\ell+\epsilon))<\ell+\epsilon.$

This is a contradiction and so, $q\in\mathrm{Cu}(N)$. ∎

In fact, for any point on an $N$-segment not in the cut locus, we have a
uniqueness.

######  Proposition 4.3.

Let $N$ be a submanifold of a Finsler manifold $(M,F)$. Let
$q\in\mathrm{Cu}(N)$ and $\gamma:[0,\ell]\to M$ be an $N$-segment joining
$p\in N$ to $q$. Then, for any $0<t_{0}<\ell$, $\gamma|_{[0,t_{0}]}$ is the
unique $N$-segment joining $N$ to $\gamma(t_{0})$.

###### Proof.

Suppose, if possible, there exists some $N$-segment $\eta:[0,t_{0}]\to M$
joining $N$ to $q_{0}=\gamma(t_{0})$, which is distinct from
$\gamma|_{[0,t_{0}]}$. Then $q_{0}\in\mathrm{Se}(N)\subset\mathrm{Cu}(N)$ by
4.2. On the other hand, $\gamma$ remains an $N$-segment beyond
$\gamma(t_{0})$, and so $q_{0}\not\in\mathrm{Cu}(N)$ by 3.17. This is a
contradiction. ∎

Let us now recall the following useful result, which is known for the case
when $N$ is a point ([Oht21, Prop 7.16, pg. 89]).

######  Lemma 4.4.

Let $\gamma:[0,\ell]\to M$ be a unit speed $N$-geodesic. If $\gamma(T)$ is a
focal point of $N$ along $\gamma$ for some $T\in(0,\ell)$, then
$d(N,\gamma(t))<t$ for all $t\in(T,\ell]$. In particular, if $\gamma|_{[0,T]}$
is an $N$-segment, then $\gamma(T)\in\mathrm{Cu}(N)$.

###### Proof.

Without loss of generality, we may assume that $\gamma(T)$ is the first focal
point of $N$ along $\gamma$. Let $J_{1}$ be a non-zero $N$-Jacobi field along
$\gamma$ such that $J_{1}(T)=0$ and $J_{1}(0)\in T_{p}N$, where $p=\gamma(0)$.
We extend $J_{1}$ to $[0,\ell]$ by $0$. By a similar argument as in [Sak96,
Lemma 4.9, pg. 61], we can find an $\epsilon>0$ so that
$\gamma|_{[T-\epsilon,T+\epsilon]}$ has no pair of conjugate points. Let us
consider the (unique) Jacobi field $J_{2}$ along
$\gamma|_{[T-\epsilon,T+\epsilon]}$ satisfying
$J_{2}(T-\epsilon)=J_{1}(T-\epsilon)$ and $J_{2}(T+\epsilon)=0$. Since
$\gamma|_{[T-\epsilon,T+\epsilon]}$ has no pair of conjugate points,
$J_{2}\neq 0$ on $[T-\epsilon,T+\epsilon)$. Next, consider the piecewise
smooth vector field $J$ along $\gamma$ defined as follows

$J(t)=\begin{cases}J_{1}(t),&t\in[0,T-\epsilon]\\\
J_{2}(t),&t\in[T-\epsilon,T+\epsilon]\\\ 0,&t\in[T+\epsilon,\ell].\end{cases}$

Since $J\neq J_{1}$ on $[T-\epsilon,T+\epsilon]$, we have the index form
$I(J,J)<I(J_{1},J_{1})$ ([Pet06, Lemma 4.2]). On the other hand
$I(J_{1},J_{1})=0$, as $J_{1}$ is an $N$-Jacobi field on $[0,T]$ and $0$ on
$[T,\ell]$.

Now, suppose $\gamma_{s}:[0,T]\rightarrow M$ is a $J$-variation of $\gamma$,
so that $\gamma_{0}=\gamma$. Since $\gamma$ is a geodesic, we have
$E^{\prime}(0)=\frac{d}{ds}|_{s=0}E(\gamma_{s})=0$, and then it follows that
$E^{\prime\prime}(0)=\frac{d^{2}}{ds^{2}}|_{s=0}E(\gamma_{s})=I(J,J)<I(J_{1},J_{1})=0$.
Hence, for any $s\neq 0$ sufficiently near 0, we have
$E(\gamma_{s})<E(\gamma_{0})$. But then $\gamma$ cannot be an $N$-segment, as
it is not a local minima of the energy functional. In particular, $\gamma$
fails to be distance minimizing beyond $T$. ∎

Recall that we have $\rho$ as the cut time (3.18), and $\lambda$ as the focal
time (3.21) map. Then, as an immediate corollary, we have the following.

######  Corollary 4.5.

$\rho(\mathbf{v})\leq\lambda(\mathbf{v})$ for all $\mathbf{v}\in S(\nu)$.

The next result is well-known in the Riemannian setting ([BP21]).

######  Theorem 4.6.

Let $N$ be a closed submanifold of a Finsler manifold $(M,F)$. Then,
$q\in\mathrm{Cu}(N)$ if and only if $\gamma:[0,\ell]\to M$ is an $N$-segment
with $q=\gamma(\ell)$ and at least one of the following holds true.

  1. (1)

There exists two distinct $N$-segments joining $N$ to $q$, i.e,
$q\in\mathrm{Se}(N)$, or

  2. (2)

$q$ is the first focal point of $N$ along $\gamma$.

###### Proof.

Suppose $q\in\mathrm{Se}(N)$. Then by 4.2, we have $q\in\mathrm{Cu}(N)$. On
the other hand, suppose $q$ is the first focal point of $N$ along $\gamma$,
and then by 4.4, any extension of $\gamma$ fails to be distance minimizing
from $N$. As $\gamma$ is an $N$-segment, we have $q\in\mathrm{Cu}(N)$.

In order to prove the converse, let
$q\in\mathrm{Cu}(N)\setminus\mathrm{Se}(N)$. Then, there exists a unique
$N$-segment $\gamma:[0,\ell]\to M$ joining $p=\gamma(0)\in N$ to
$q=\gamma(\ell)$, where $\ell=d(N,q)$, so that any extension of $\gamma$ fails
to be distance minimizing from $N$. Say, $\mathbf{v}=\dot{\gamma}(0)$, and
then $\gamma(t)=\exp^{\nu}(t\mathbf{v})$. Let
$q_{i}=\gamma\left(\ell+\frac{1}{i}\right)$. Then, $q_{i}\to q$ and
$d(N,q_{i})<L\left(\gamma|_{\left[0,\ell+\frac{1}{i}\right]}\right)$. Let
$\gamma_{i}:[0,\ell_{i}]\to M$ be $N$-segments joining $p_{i}=\gamma_{i}(0)\in
N$ to $q_{i}=\gamma_{i}(\ell_{i})$, where $\ell_{i}=d(N,q_{i})$ (see Figure
2). Note that $F_{p_{i}}(\mathbf{v}_{i})=1$ where
$\mathbf{v}_{i}=\dot{\gamma}_{i}(0)\in\nu_{p_{i}}$, i.e., $(p_{i},v_{i})\in
S(\nu)$, the unit normal cone bundle. As
$\gamma|_{\left[0,\ell+\frac{1}{i}\right]}$ is not an $N$-segment,
$\gamma_{i}$ is distinct from $\gamma$ and in particular,
$\mathbf{v}_{i}\neq\mathbf{v}$. Since $N$ is compact, $S(\nu)$ is compact by
3.3. Passing to a subsequence if necessary, we may assume
$(p_{i},\mathbf{v}_{i})\rightarrow(p^{\prime},\mathbf{v}^{\prime})\in S(\nu)$.

$p$
---
$\gamma$
---
$\gamma(\ell)=q$
---
$N$
---
$\mathbf{v}$
---
$\gamma(\ell+\frac{1}{i})=q_{i}$
---
$p_{i}$
---
$\mathbf{v}_{i}$
---
$\gamma_{i}$
---
$\gamma\left(\ell_{i}\right)$
---
Figure 2. Characterization of $\mathrm{Cu}(N)$

Now, consider the $N$-geodesic $\eta(t)=\exp^{\nu}(t\mathbf{v}^{\prime})$. We
have

$\ell=d(N,q)=\lim d(N,q_{i})=\lim d(p_{i},q_{i})=\lim\ell_{i}.$

Then, it follows from the continuity of the exponential map that,

$q=\lim
q_{i}=\lim\gamma_{i}(\ell_{i})=\lim\exp^{\nu}(\ell_{i}\mathbf{v}_{i})=\exp^{\nu}(\ell\mathbf{v}^{\prime})=\eta(\ell).$

Thus, $\eta$ is an $N$-segment joining $p^{\prime}$ to $q$. Since
$q\not\in\mathrm{Se}(N)$, we must have $\gamma=\eta$, and in particular,
$(p_{i},\mathbf{v}_{i})\to(p^{\prime},\mathbf{v}^{\prime})=(p,\mathbf{v})$. On
the other hand

$\exp^{\nu}(\ell_{i}\mathbf{v}_{i})=\gamma_{i}(\ell_{i})=q_{i}=\exp^{\nu}\left(\left(\ell+\frac{1}{i}\right)\mathbf{v}\right).$

As $\mathbf{v}_{i}\neq\mathbf{v}$, we have
$\ell_{i}\mathbf{v}\neq\left(\ell+\frac{1}{i}\right)\mathbf{v}$. Then, the
normal exponential map fails to be injective in any neighborhood of
$\ell\mathbf{v}$, since both sequences
$\ell_{i}\mathbf{v}_{i}\rightarrow\mathbf{v}$ and
$\left(\ell+\frac{1}{i}\right)\mathbf{v}\rightarrow\mathbf{v}$. As a
consequence of the inverse function theorem, we then have
$d(\exp^{\nu})|_{\ell\mathbf{v}}$ is degenerate, i.e.,
$q=\exp^{\nu}(\ell\mathbf{v})$ is a focal point of $N$ along $\gamma$.

As $\gamma$ is an $N$-segment joining $N$ to $q$, it follows from 4.4 that
$\gamma(t)$ is not a focal point of $N$ along $\gamma$ for $0\leq t<T$. Thus,
$q$ is the first focal point of $N$ along $\gamma$. This completes the proof.
∎

###  4.2. Continuity of the $\rho$ and $\lambda$ maps

Recall the cut time map $\rho$ from 3.18. We have the following result.

######  Theorem 4.7.

The map $\rho:S(\nu)\to[0,\infty]$ is continuous.

###### Proof.

Consider a converging sequence $(p_{i},\mathbf{v}_{i})\to(p,\mathbf{v})$ in
$S(\nu)$. We show that $\rho(p_{i},\mathbf{v}_{i})\to\rho(p,\mathbf{v})$. Let
$\ell_{i}=\rho(p_{i},\mathbf{v}_{i})$, and consider the $N$-geodesics
$\gamma_{i}:[0,\ell_{i}]\to M$ joining $p_{i}=\gamma_{i}(0)$ to
$q_{i}=\gamma_{i}(\ell_{i})$, given by
$\gamma_{i}(t)=\exp^{\nu}(t\mathbf{v}_{i})$. From the definition of $\rho$, we
have $\gamma_{i}$ is an $N$-segment for each $i$. Since
$\rho(p_{i},\mathbf{v}_{i})\in[0,\infty]$, we must have some accumulation
point, say, $L_{0}\in[0,\infty]$ for the set
$\\{\rho(p_{i},\mathbf{v}_{i})\\}$. Consider the $N$-geodesic
$\gamma:[0,L_{0}]\to M$ defined as $\gamma(t)=\exp^{\nu}(t\mathbf{v})$.
Passing to subsequence, we have
$\mathbf{v}_{i}=\dot{\gamma}_{i}(0)\to\mathbf{v}=\dot{\gamma}(0)$, and
$\ell_{i}=\rho(p_{i},\mathbf{v}_{i})\to L_{0}$. But then,

$d(N,\gamma(L_{0}))=d\left(N,\exp^{\nu}(L_{i}\mathbf{v})\right)=\lim
d(N,\exp^{\nu}(\ell_{i}\mathbf{v}_{i}))=\lim
d(p_{i},\gamma_{i}(\ell_{i}))=\lim\ell_{i}=L_{0},$

and thus $\gamma$ is an $N$-segment. By definition of $\rho$, we have
$L_{0}\leq\rho(p,\mathbf{v})$. If $L_{0}=\infty$, then
$\rho(p,\mathbf{v})=L_{0}$, and we are done.

Suppose $L_{0}<\infty$, and so for infinitely many $(p_{i},\mathbf{v}_{i})$,
$q_{i}=\exp^{\nu}(\ell_{i}\mathbf{v}_{i})$ is a cut point of $N$. It follows
from 4.6 that there are infinitely many $(p_{i},\mathbf{v}_{i})$ which satisfy
one of the following.

  1. (a)

$q_{i}=\gamma_{i}(\ell_{i})=\exp^{\nu}(\ell_{i}\mathbf{v}_{i})$ is the first
focal point of $N$ along $\gamma_{i}$.

  2. (b)

There exists some $N$-segemnt ending at $q_{i}$ distinct from $\gamma_{i}$.

If (a) holds for infinitely many $(p_{i},\mathbf{v}_{i})$, then
$\ker\left(d(\exp^{\nu})|_{\ell_{i}\mathbf{v}_{i}}\right)\neq 0$. Since
$\lim\ell_{i}=L_{0}$ is finite, passing to a subsequence, we may assume that
there exists some $\epsilon>0$ so that
$L_{0}-\epsilon<\ell_{i}<L_{0}+\epsilon$ for all $i$. Consider the subset

$\widetilde{\nu}\coloneqq\\{(p,\mathbf{v})\in\nu\;|\;L_{0}-\epsilon\leq
F_{p}(\mathbf{v})\leq L_{0}+\epsilon\\},$

which is clearly a compact submanifold of $\hat{\nu}$ of codimension $0$.
Furthermore, $\ell_{i}\mathbf{v}_{i}\in\widetilde{\nu}$, and
$T_{\ell_{i}\mathbf{v}_{i}}\hat{\nu}=T_{\ell_{i}\mathbf{v}_{i}}\widetilde{\nu}$.
Let us now fix some auxiliary Riemannian metric on $\widetilde{\nu}$, and
choose unit vectors
$\mathbf{w}_{i}\in\ker\left(d(\exp^{\nu})|_{\ell_{i}\mathbf{v}_{i}}\right)$.
Since $\mathbf{w}_{i}$ is in the unit sphere bundle $S(T\widetilde{\nu})$ of a
compact manifold $\widetilde{\nu}$, passing to a further subsequence, we may
assume $\mathbf{w}_{i}\to\mathbf{w}\in S(T\widetilde{\nu})$. Then $0=\lim
d(\exp^{\nu})|_{\ell_{i}\mathbf{v}_{i}}(\mathbf{w}_{i})=d(\exp^{\nu})|_{L_{0}\mathbf{v}}(\mathbf{w})$.
As $\mathbf{w}\neq 0$, $q=\gamma(T)=\exp^{\nu}(L_{0}\mathbf{v})$ is a focal
point of $N$ along $\gamma$. Hence, any extension of $\gamma$ beyond $L_{0}$
fails to be distance minimizing (4.4), and we get $\rho(p,\mathbf{v})=L_{0}$.

If (b) holds for infinitely many $(p_{i},\mathbf{v}_{i})$, there exists
$(p_{i}^{\prime},\mathbf{v}_{i}^{\prime})\in S(\nu)$ so that the geodesic
$\eta_{i}:[0,\ell_{i}]\to M$ defined by
$\eta_{i}(t)=\exp^{\nu}(t\mathbf{v}_{i}^{\prime})$ is distinct from
$\gamma_{i}$, and is an $N$-segment satisfying $q_{i}=\eta_{i}(\ell_{i})$.
Passing to a subsequence, we may assume
$(p_{i}^{\prime},\mathbf{v}_{i}^{\prime})\to(p^{\prime},\mathbf{v}^{\prime})$.
The $N$-geodesic $\gamma^{\prime}:[0,L_{0}]\to M$ defined by
$\gamma^{\prime}(t)=\exp^{\nu}(t\mathbf{v}^{\prime})$ satisfies
$\gamma^{\prime}(L_{0})=\exp^{\nu}(L_{0}\mathbf{v}^{\prime})=\lim\exp^{\nu}(\ell_{i}\mathbf{v}_{i}^{\prime})=\lim
q_{i}=q$, and so it is an $N$-segment. If $v^{\prime}\neq v$, then we have
$q\in\mathrm{Se}(N)\subset\mathrm{Cu}(N)$ (4.2), and we get
$\rho(p,\mathbf{v})=L_{0}$. Next, suppose $\mathbf{v}^{\prime}=\mathbf{v}$ and
so, $\gamma^{\prime}=\gamma$. But then $\exp^{\nu}$ fails to be injective in
any neighborhood of $L_{0}\mathbf{v}$, since we have
$(p_{i}^{\prime},\mathbf{v}_{i}^{\prime})\neq(p_{i},\mathbf{v}_{i})\Rightarrow\ell_{i}\mathbf{v}_{i}^{\prime}\neq\ell_{i}\mathbf{v}_{i}$,
both of which are getting mapped to $q_{i}$ under the exponential map. In
particular, $d(\exp^{\nu})|_{L_{0}\mathbf{v}}$ is non-degenerate and thus
$q=\exp^{\nu}(L_{0}\mathbf{v})$ is a focal point of $N$ along $\gamma$. Then
by 4.4, we get $\rho(p,\mathbf{v})=L_{0}$.

Thus, for any accumulation point $L_{0}$ of $\\{\rho(p_{i},\mathbf{v}_{i})\\}$
we have $L_{0}=\rho(p,\mathbf{v})$. In particular,
$\rho(p_{i},\mathbf{v}_{i})\to\rho(p,\mathbf{v})$, which completes the proof.
∎

We can now prove the following.

######  Theorem 4.8.

Let $N$ be a submanifold of a Finsler manifold $(M,F)$. Then, the separating
set of $N$ is dense in the cut locus of $N$. Furthermore, if $N$ is closed,
then the cut locus is a closed set, and we have
$\mathrm{Cu}(N)=\overline{\mathrm{Se}(N)}$.

###### Proof.

Let $q\in\mathrm{Cu}(N)\setminus\mathrm{Se}(N)$. Then, there exists a unique
$N$-segment $\gamma:[0,\ell]\to M$ joining $p=\gamma(0)\in N$ to
$q=\gamma(\ell)$, and furthermore by 4.6, $q$ is the first focal point of $N$
along $\gamma$. We have $q=\exp^{\nu}(\ell\mathbf{v})$, where
$\mathbf{v}=\dot{\gamma}(0)\in\nu_{p}$, and $\ell=\rho(\mathbf{v})$.

Fix some auxiliary metric on $TM$. Then, for some $\delta>0$ sufficiently
small, consider a chart $V(\mathbf{v},\delta)$ of radius $\delta$ around
$\mathbf{v}$ in $S(\nu)$, which is homeomorphic to an open Euclidean
$(n-1)$-ball (3.3), where $n=\dim M$. Next, define the set

$U(\mathbf{v},\delta)=\left\\{\rho(\mathbf{w})\mathbf{w}\;\middle|\;\mathbf{w}\in
V(\mathbf{v},\delta)\right\\}.$

As $\rho$ is finite and continuous (4.7), we have a surjective continuous map
$\Phi:V(\mathbf{v},\delta)\rightarrow U(\mathbf{v},\delta)$ defined as
$\Phi(\mathbf{w})=\rho(\mathbf{w})\mathbf{w}$ (see Figure 3). Clearly $\Phi$
is injective as well. Hence, by the invariance of domain, we have
$U(\mathbf{v},\delta)$ is an open Euclidean $(n-1)$-ball. Taking a decreasing
sequence $\delta_{k}\rightarrow 0$, we then have a decreasing sequence of
$(n-1)$-balls $U_{k}\coloneqq U(\mathbf{v},\delta_{k})$. Note that
$\mathbf{u}=\ell\mathbf{v}\in U_{k}$ as $\ell=\rho(\mathbf{v})$, and
furthermore $\cap_{k}U_{k}=\left\\{\mathbf{u}\right\\}$. We now have a
dichotomy.

  1. (a)

For some $\delta>0$ small, $\exp^{\nu}|_{U(\mathbf{v},\delta)}$ is injective,
and consequently, again by an appeal to the invariance of domain,
$W\vcentcolon=\exp^{\nu}\left(U_{k_{0}}\right)$ is an open Euclidean
$(n-1)$-ball in $M$ for some $k_{0}$ large.

  2. (b)

For any $\delta>0$, $\exp^{\nu}|_{U(\mathbf{v},\delta)}$ is not injective.

If (b) appears, then for each $k$ we can pick
$\mathbf{u}_{k}^{1}\neq\mathbf{u}_{k}^{2}\in U_{k}$, so that
$q_{k}=\exp^{\nu}(\mathbf{u}_{k}^{1})=\exp^{\nu}(\mathbf{u}_{k}^{2})$. Note
that $q_{k}\in\mathrm{Se}(N)$. As $\cap_{k}U_{k}=\left\\{\mathbf{u}\right\\}$,
we have $\mathbf{u}_{k}^{1}\rightarrow u$ and $\mathbf{u}_{k}^{2}\rightarrow
u$. Hence, it follows from the continuity of $\exp^{\nu}$ that
$q_{k}\rightarrow q=\exp^{\nu}(u)$. As $q_{k}\in\mathrm{Se}(N)$, $q$ is in the
closure of $\mathrm{Se}(N)$, completing the proof.

If (a) appears, then we shall show that it will lead to a contradiction.
Consider an open $n$-ball $B(q,\epsilon)$ around the point $q\in M$. Since $W$
is an open Euclidean $(n-1)$-ball around $q$, we must have $B(q,\epsilon)$
intersects the compliment $W$. Also, by [AJ19], for $\epsilon$ sufficiently
small $B(q,\epsilon)\cap N=\emptyset$ as well. Taking
$\epsilon_{i}=\frac{1}{i}$, we now pick points $q_{i}\in
B(q,\epsilon_{i})\setminus(W\cup N)$. Observe that $q_{i}\rightarrow q$. Let
us consider $N$-segments $\gamma_{i}:[0,\ell_{i}]\rightarrow M$ joining
$p_{i}=\gamma_{i}(0)$ to $q_{i}$, with initial velocity
$\mathbf{v}_{i}=\dot{\gamma}_{i}(0)\in S(\nu)$.

$V(\mathbf{v},\delta)$
---
$S(\nu)$
---
$U(\mathbf{v},\delta)$
---
$\mathbf{v}$
---
$\ell_{i}\mathbf{v}_{i}$
---
$\mathbf{v}_{i}$
---
$\exp^{\nu}$
---
$q$
---
$B(q,\varepsilon)$
---
$W$
---
$q_{i}$
---
$p_{i}$
---
$\gamma_{i}$
---
$\mathbf{v}_{i}=\dot{\gamma}_{i}(0)$
---
$\mathbf{v}=\dot{\gamma}(0)$
---
$\gamma$
---
$p$
---
$N$
---
$\Phi$
---
$\rho(\mathbf{v})\mathbf{v}$
---
Figure 3. The open sets in $\nu$ and in $M$

As $S(\nu)$ is compact, passing to a subsequence if necessary, we have
$(p_{i},\mathbf{v}_{i})\rightarrow(p^{\prime},\mathbf{v}^{\prime})$. We also
have,

$\ell=d(N,q)=\lim d(N,q_{i})=\lim\ell_{i}.$

Consequently, the $N$-geodesic $\eta:[0,\ell]\rightarrow M$ given by
$\eta(t)=\exp^{\nu}(t\mathbf{v}^{\prime})$ is in fact an $N$-segment joining
$N$ to $q$. We claim that $\gamma$ and $\eta$ are distinct. If this were not
the case, we must have $(p,\mathbf{v})=(p^{\prime},\mathbf{v}^{\prime})$. But
as $\mathbf{v}_{i}\rightarrow\mathbf{v}$ in $S(\nu)$, and as
$V_{k_{0}}\vcentcolon=V\left(\mathbf{v},\frac{1}{k_{0}}\right)=\Phi^{-1}(U_{k_{0}})$
is an open neighborhood of $\mathbf{v}$ in $S(\nu)$, we must have
$\mathbf{v}_{i}\in V_{k_{0}}$ for all $i$ sufficiently large. Consequently,
$\Phi(\mathbf{v}_{i})=\ell_{i}\mathbf{v}_{i}\in U_{k_{0}}$ for all $i$ large.
But that would imply $q_{i}=\exp^{\nu}(\ell_{i}\mathbf{v}_{i})\in W$, which is
a contradiction. Thus, we have two distinct $N$-segments $\gamma$ and $\eta$
joining $N$ to $q$, making $q\in\mathrm{Se}(N)$, which is again a
contradiction. Therefore, (a) cannot appear.

Now, suppose $N$ is closed. Then $S(\nu)$ is compact (3.3). It follows from
4.7, that we have a continuous map

$\displaystyle\Phi:S(\nu)$ $\displaystyle\rightarrow\nu$
$\displaystyle\mathbf{v}$ $\displaystyle\mapsto\rho(\mathbf{v})\mathbf{v}$

Then, $\widetilde{\mathrm{Cu}}(N)=\Phi(S(\nu))$ is a compact set, and hence,
$\mathrm{Cu}(N)=\exp^{\nu}\left(\widetilde{\mathrm{Cu}}(N)\right)$ is a
compact set. The proof then follows. ∎

###  4.3. The Topology of the Complement of Cut Locus

Let us consider the subset

$\mathcal{I}(N)=\left\\{t\mathbf{v}\;\middle|\;0\leq
t<\rho(\mathbf{v}),\mathbf{v}\in S(\nu)\right\\}\subset\nu,$

and denote $\hat{\mathcal{I}}(N)=\mathcal{I}(N)\cap\hat{\nu}$. It is clear
that $\mathcal{I}(N)\cap\widetilde{\mathrm{Cu}}(N)=\emptyset$, and
$\exp^{\nu}(\mathcal{I}(N))\cap\mathrm{Cu}(N)=\emptyset$. Observe that
$\mathcal{I}(N)$ is _star-shapped_ , i.e., for any
$\mathbf{v}\in\mathcal{I}(N)$ we have $t\mathbf{v}\in\mathcal{I}$ for $0\leq
t\leq 1$.

######  Theorem 4.9.

The normal exponential map $\exp^{\nu}$ induces a

  * •

$C^{1}$-diffoemorphism of $\mathcal{I}(N)$ onto $M\setminus\mathrm{Cu}(N)$

  * •

$C^{\infty}$ diffeomorphism of $\hat{\mathcal{I}}(N)$ onto
$M\setminus(N\cup\mathrm{Cu}(N))$.

###### Proof.

Let $q\in M\setminus\left(N\cup\mathrm{Cu}(N)\right)$. Then, it follows from
4.2, that there exists a unique $N$-segment $\gamma:[0,\ell]\rightarrow M$
joining $N$ to $q=\gamma(\ell)$, where $\ell=d(N,q)$. We have
$\gamma(t)=\exp^{\nu}(t\mathbf{v})$, where $\mathbf{v}=\dot{\gamma}(0)\in
S(\nu)$, and $\ell\mathbf{v}\in\hat{\mathcal{I}}(N)$. In particular, we have a
map $\Phi:M\setminus\mathrm{Cu}(N)\rightarrow\mathcal{I}(N)$ defined as
follows.

$\Phi(q)=\begin{cases}\ell\mathbf{v},&q\not\in N\cup\mathrm{Cu}(N),\text{ and
}q=\exp^{\nu}(\ell\mathbf{v})\\\ 0\in T_{q}N,&q\in N\end{cases}$

By construction, $\Phi$ is the required inverse map to $\exp^{\nu}$. We show
that $\Phi$ is $C^{1}$ on $M\setminus\mathrm{Cu}(N)$, and is $C^{\infty}$ when
restricted to $M\setminus(N\cup\mathrm{Cu}(N))$.

Let $q\not\in N\cup\mathrm{Cu}(N)$, and
$\mathbf{u}=\ell\mathbf{v}=\Phi(q)\in\hat{\mathcal{I}}(N)$. It follows from
4.4 that $q$ is not a focal point of $N$ along the $N$-geodesic
$\gamma(t)=\exp^{\nu}(t\mathbf{v})$. In particular,
$d\left(\exp^{\nu}\right)|_{\mathbf{u}}$ is nondegenerate, and hence, it is a
local diffeomorphism. By the implicit function theorem, $\Phi$ is then
$C^{\infty}$ in a neighborhood of $q$. Thus, $\Phi$ is $C^{\infty}$ on
$M\setminus\left(N\cup\mathrm{Cu}(N)\right)$. Since $\exp^{\nu}$ is $C^{1}$ on
the $0$ section of $\nu$, same argument as above gives that $\Phi$ is $C^{1}$
on $M\setminus\mathrm{Cu}(N)$. This completes the proof. ∎

As a consequence, we now have the following.

######  Theorem 4.10.

Let $N$ be a closed submanifold of $(M,F)$.

  1. (1)

$M\setminus\mathrm{Cu}(N)$ strongly deformation retracts onto $N$.

  2. (2)

$M\setminus N$ strongly deformation retracts onto $\mathrm{Cu}(N)$.

###### Proof.

Consider the linear homotopy $L_{s}:TM\rightarrow TM$ given by
$L_{s}(\mathbf{v})=(1-s)\mathbf{v}$, for $0\leq s\leq 1$. By construction,
$\mathcal{I}(N)$ is star-shaped, and thus, $L_{s}$ restricts to a homotopy of
$\nu$ as well. Indeed,

$L_{0}|_{\mathcal{I}(N)}=\mathrm{Id}_{\mathcal{I}(N)},\quad
L_{1}|_{\mathcal{I}(N)}=\pi|_{\mathcal{I}(N)},$

where $\pi:\nu\rightarrow N$ is the projection. Next, we define the map
$h:I\times(M\setminus\mathrm{Cu}(N))\rightarrow(M\setminus\mathrm{Cu}(N))$ by

$h(s,q)\coloneqq L_{s}\circ\left(\exp^{\nu}\right)^{-1}(q).$

It follows from 4.9, that $h$ is a well-defined homotopy. Clearly,
$h_{0}=h(0,\\_)$ is identity on $M\setminus\mathrm{Cu}(N)$, whereas
$h_{1}=h(1,\\_)$ is contained in $N$. Furthermore, $h_{s}=h(s,\\_)$ is
identity on $N$ for all $0\leq s\leq 1$, and
$h_{1}\circ\iota_{N}=\mathrm{Id}_{N}$, where $\iota_{N}:N\hookrightarrow M$ is
the inclusion. Thus, $M\setminus\mathrm{Cu}(N)$ strongly deformation retracts
onto $N$, which proves (1).

In order to prove (2) consider the homotopy $H:I\times(M\setminus
N)\rightarrow M\setminus N$ defined as

$H(s,q)\coloneqq\begin{cases}\exp^{\nu}\left[\left(s\rho\left(\frac{\mathbf{v}}{F(\mathbf{v})}\right)+(1-s)F(\mathbf{v})\right)\frac{\mathbf{v}}{F(\mathbf{v})}\right],&q\in
M\setminus\left(N\cup\mathrm{Cu}(N)\right),\\\
&\mathbf{v}\coloneqq\left(\exp^{\nu}\right)^{-1}(q)\\\\[10.00002pt]
q,&q\in\mathrm{Cu}(N).\end{cases}$

Continuity of $H$ follows from 4.7 and 4.9. Clearly, $H_{0}=H(0,\\_)$ is
identity on $M\setminus N$, whereas $H_{1}=H(1,\\_)$ is contained in
$\mathrm{Cu}(N)$. Furthermore, for all $0\leq s\leq 1$, we have
$H_{s}=H(s,\\_)$ is identity on $\mathrm{Cu}(N)$, and
$H_{1}\circ\iota_{\mathrm{Cu}(N)}=\mathrm{Id}|_{\mathrm{Cu}(N)}$, where
$\iota_{\mathrm{Cu}(N)}:\mathrm{Cu}(N)\hookrightarrow M$ is the inclusion.
Thus, $M\setminus N$ strongly deformation retracts onto $\mathrm{Cu}(N)$. ∎

######  Observation 4.11.

Suppose $N$ is a closed submanifold of $M$. We make a few easy observations.

  1. (1)

$M=\exp^{\nu}\left(\mathcal{I}(N)\cup\widetilde{\mathrm{Cu}}(N)\right)$.

  2. (2)

$M$ is compact if and only if $\rho$ is finite. Indeed, if $M$ is compact,
then continuity of $\rho$ (4.7) implies that $\rho$ is finite. For the
converse, first, consider the unit normal disc bundle of $N$

$D(\nu)\coloneqq\left\\{t\mathbf{v}\;\middle|\;0\leq t\leq 1,\;\mathbf{v}\in
S(\nu)\right\\},$

which is compact as $N$ is so. Since $\rho$ is finite and continuous, we have
a continuous injective map $f:D(\nu)\rightarrow\nu$ defined as

$f(\mathbf{v})=\begin{cases}\rho\left(\frac{\mathbf{v}}{F(\mathbf{v})}\right)\mathbf{v},&\mathbf{v}\neq
0\\\ 0,&\mathbf{v}=0\end{cases}.$

Note that $\operatorname{im}f=\mathcal{I}(N)\cup\mathrm{Cu}(N)$, which is
compact. But then $M$ is compact by the previous observation.

  3. (3)

If $\lambda<\infty$, then $M$ is compact, since $\rho\leq\lambda$ by 4.5. The
converse may not be true, even for the Riemannian case: consider any compact
Riemannian manifold $M$ with flat curvature, then any submanifold $N\subset M$
cannot have any focal locus and hence $\lambda=\infty$.

###  4.4. Distance Function from $N$

######  Theorem 4.12.

Let $N$ be a closed submanifold, and denote by $f:M\rightarrow\mathbb{R}$ the
distance squared function $f(q)=d(N,q)^{2}$. Then,

  1. (1)

$f$ is $C^{1}$ precisely on $M\setminus\mathrm{Se}(N)$, and $f$ is
$C^{\infty}$ on $M\setminus(N\cup\mathrm{Cu}(N))$.

  2. (2)

For $q\not\in\mathrm{Se}(N)$ and $X\in T_{q}M$, we have $df(X)=2\ell
g_{\dot{\gamma}(\ell)}(\dot{\gamma}(\ell),X)$, where
$\gamma:[0,\ell]\rightarrow M$ is the unique $N$-segment joining $N$ to
$q=\gamma(\ell)$ and $\ell=d(N,q)$.

  3. (3)

For $q\in\mathrm{Se}(N)$, $f$ fails to be differentiable precisely along those
$X\in T_{q}M$ for which there exists an $N$-segment
$\gamma:[0,\ell]\rightarrow M$ joining $N$ to $q$ with $\dot{\gamma}(\ell)=X$.

The proof is essentially the same as the one given in [ST16], where the
authors proved that the distance function from an arbitrary closed set $N$ in
a Finsler manifold is $C^{1}$ precisely in the compliment of
$N\cup\mathrm{Cu}(N)$. Recall the following.

######  Lemma 4.13.

[ST16, Prop 2.2] Consider the following setup.

  * •

$\gamma:[0,\ell]\rightarrow M$ is an $N$-segment, joining $p=\gamma(0)$ to
$q=\gamma(\ell)$, where $\ell=d(N,q)$.

  * •

$q_{i}\rightarrow q$ is a converging sequence in $M$.

  * •

$\gamma_{i}:[0,\ell_{i}]\rightarrow M$ is an $N$-segment, joining
$p_{i}=\gamma_{i}(0)$ to $q_{i}=\gamma_{i}(\ell_{i})$, where
$\ell_{i}=d(N,q_{i})$, and $\mathbf{v}_{i}=\dot{\gamma}_{i}(0)$.

  * •

$\sigma_{i}:[0,T_{i}]\rightarrow M$ is an $N$-segment, joining
$q=\sigma_{i}(0)$ to $q_{i}=\sigma_{i}(T_{i})$, where $T_{i}=d(q,q_{i})$, and
$\mathbf{w}_{i}=\dot{\sigma}_{i}(0)$.

Suppose, the limit $\mathbf{w}=\lim_{i}\mathbf{w}_{i}$ exists, and
$\gamma_{i}$ converges uniformly to $\gamma$. Then, the limit

$\lim_{i\rightarrow\infty}\frac{d(N,q_{i})-d(N,q)}{d(q,q_{i})}$

exists, and equals
$g_{\dot{\gamma}(\ell)}\left(\dot{\gamma}(\ell),\mathbf{w}\right)$.

###### Proof of 4.12.

We have from 4.9, $\exp^{\nu}$ restricts to a $C^{\infty}$-diffeomorphism
$\hat{\mathcal{I}}(N)\rightarrow M\setminus(N\cup\mathrm{Cu}(N))$. For each
$q\in M\setminus(N\cup\mathrm{Cu}(N))$, we have the unique $N$-segment
$\gamma_{\mathbf{v}}$ joining $N$ to $q$, where
$\mathbf{v}=(\exp^{\nu})^{-1}(q)$, and furthermore
$d(N,q)=L(\gamma_{\mathbf{v}})=F(\mathbf{v})$. Thus, we have the identity

$f|_{M\setminus(N\cup\mathrm{Cu}(N))}=\left(F\circ(\exp^{\nu})^{-1}\right)^{2},$

which proves the smoothness of $f$ on the compliment of $N\cup\mathrm{Cu}(N)$.

Now, let $q\not\in\mathrm{Se}(N)$. Then, there exists a unique $N$-segment
$\gamma:[0,\ell]\rightarrow M$ joining $p=\gamma(0)\in N$ to $q=\gamma(\ell)$,
where $\ell=d(N,q)$ and $\mathbf{v}=\dot{\gamma}(0)\in S(\nu)$. Clearly,
$f(q)=\ell^{2}=(L(\gamma))^{2}$. Let $X\in T_{q}M$, and consider a smooth
curve $\sigma:(-\epsilon,\epsilon)\rightarrow M$ such that
$\sigma(0)=q,\dot{\sigma}(0)=X$. Let $s_{i}\rightarrow 0$ be a sequence in the
domain of $\sigma$. Suppose, $\gamma_{i}:[0,\ell_{i}]\rightarrow M$ are
$N$-segments, joining $p_{i}=\gamma_{i}(0)\in N$ to
$q_{i}=\sigma(s_{i})=\gamma_{i}(\ell_{i})$, where
$\ell_{i}=d(N,q_{i})=L(\gamma_{i})$. As $s_{i}\rightarrow 0$, we have
$q_{i}=\sigma(s_{i})\rightarrow\sigma(0)=q$, and $\ell_{i}\rightarrow\ell$.
Set $\mathbf{v}_{i}=\dot{\gamma}(0)\in S(\nu)$. It follows from the
compactness of $S(\nu)$ and the uniqueness of the $N$-segment $\gamma$ joining
$N$ to $q$, that $(p_{i},\mathbf{v}_{i})\rightarrow(p,\mathbf{v})$, and
consequently $\gamma_{i}\rightarrow\gamma$ uniformly. In particular,
$\dot{\gamma}(\ell)=\lim_{i}\dot{\gamma}_{i}(\ell_{i})$. Then, by an
application of the chain rule, it follows from 4.13 that the limit

$\lim_{i\rightarrow\infty}\frac{f(\sigma(s_{i}))-f(\sigma(0))}{s_{i}}=\lim_{i\rightarrow\infty}\frac{d(N,q_{i})^{2}-d(N,q)^{2}}{s_{i}}$

exists, and equals $2\ell g_{\dot{\gamma}(\ell)}(\dot{\gamma}(\ell),X)$. A
careful examination of the proof of 4.13, reveals that $f$ fails to be
differentiable at points of $\mathrm{Se}(N)$ precisely along those directions
which is the terminal velocity of some $N$-segment. This concludes the proof.
∎

###  4.5. Existence of $N$-Geodesic Loops

In this section, we generalize a result from [ISS12] and [Xu15]. The proof is
the same vein, with suitable modifications needed to deal with the asymmetric
nature of Finsler distance.

######  Theorem 4.14.

Let $N$ be a closed submanifold of $(M,F)$. Then one of the following is true.

  * •

There are points in $\mathrm{Cu}(N)$ which are also focal points of $N$.

  * •

For each $q\in M$, there exists at least two $N$-geodesics joining $N$ to $q$,
one of which crosses the cut locus $\mathrm{Cu}(N)$. For $q\in N$, we allow
the constant curve at $q$ to be considered as an $N$-geodesic.

For any $q\in M$, consider the function

$\displaystyle\mathcal{M}=\mathcal{M}_{q}:M$
$\displaystyle\rightarrow[0,\infty)$ $\displaystyle x$ $\displaystyle\mapsto
d(N,x)+d(x,q)$

Since $N$ is closed, we have $\mathrm{Cu}(N)$ is closed (4.8), and
consequently, there exists some $x_{0}\in\mathrm{Cu}(N)$ so that
$\mathcal{M}|_{\mathrm{Cu}(N)}$ attains a global minima at $x_{0}$, i.e.,

$\mathcal{M}(x_{0})=\min_{x\in\mathrm{Cu}(N)}\mathcal{M}(x).$

We have the following lemma, which is quite interesting in itself.

######  Lemma 4.15.

With the above notation, further assume that $x_{0}$ is not a focal point of
$N$. We then have the following dichotomy.

  1. (1)

$q\not\in\mathrm{Cu}(x_{0})$. Then, there exists exactly two $N$-segments,
say, $\gamma_{1},\gamma_{2}$, joining $N$ to $q$, and a unique minimizer, say
$\eta$, joining $x_{0}$ to $q$. Furthermore, exactly one of
$\gamma_{1}\cup\eta$ and $\gamma_{2}\cup\eta$ is an $N$-geodesic joining $N$
to $q$, and intersecting $\mathrm{Cu}(N)$ at $x_{0}$.

  2. (2)

$q\in\mathrm{Cu}(x_{0})$. Then, there exists exactly two $N$-segments, say,
$\gamma_{1},\gamma_{2}$, joining $N$ to $x_{0}$, and at most two minimizers,
say, $\eta_{1},\eta_{2}$, joining $x_{0}$ to $q$. Furthermore, if
$\eta_{1},\eta_{2}$ are distinct, then exactly two of the four curves
$\left\\{\gamma_{i}\cup\eta_{j}\;\middle|\;i,j=1,2\right\\}$ are $N$-geodesics
joining $N$ to $q$ and crossing $\mathrm{Cu}(N)$. The other two are broken at
$x_{0}$. If $\eta_{1}=\eta_{2}=\eta$, then exactly one of
$\left\\{\gamma_{1}\cup\eta,\gamma_{2}\cup\eta\right\\}$ is an $N$-geodesic
joining $N$ to $q$ crossing $\mathrm{Cu}(N)$ at $x_{0}$.

  3. (2’)

For $q\in\mathrm{Cu}(x_{0})$, if we furthermore assume that $F$ is a
reversible Finsler metric, then exactly two minimizers exist from $x_{0}$ to
$q$, and the rest of (2) holds.

###### Proof.

Case (1): Suppose $q\not\in\mathrm{Cu}(x_{0})$. Then, it follows from 4.3 that
there exists a unique minimizer $\eta:[0,a]\rightarrow M$ joining $x_{0}$ to
$q$, where $a=d(x_{0},q)$. If possible, assume that we have at least three
$N$-segments joining $N$ to $x_{0}$, say,
$\gamma_{1},\gamma_{2},\gamma_{3}:[0,\ell]\rightarrow M$, where
$\ell=d(N,x_{0})$ (Figure 4). Since they are distinct, their terminal
velocities $\dot{\gamma}_{i}(\ell)$ are distinct as well (2.10). Consequently,
_at most_ one of $\gamma_{i}\cup\eta$ can be smooth at $x_{0}$. Without loss
of generality, assume that both $\gamma_{1}\cup\eta$ and $\gamma_{2}\cup\eta$
are broken at $x_{0}$.

$x_{0}$
---
$\mathrm{Cu}(N)$
---
$q\notin\mathrm{Cu}(x_{0})$
---
$\gamma_{1}$
---
$\gamma_{3}$
---
$\gamma_{2}$
---
$N$
---
$T$
---
Figure 4. Case (1)

Since $\gamma_{1}$ is in particular a minimizer, $\gamma_{1}|_{[t,\ell]}$ is a
minimizer for any $0<t<\ell$ (2.13). Then, for $y=\gamma_{1}(t)$, it follows
from the brokenness of $\gamma_{1}\cup\eta$ that

$d(y,q)<d(y,x_{0})+d(x_{0},q).$

Hence,

$\mathcal{M}(y)=d(N,y)+d(y,q)<d(N,y)+d(y,x_{0})+d(x_{0},q)=d(N,x_{0})+d(x_{0},q)=\mathcal{M}(x_{0}).$

Now, let $\zeta:[0,b]\rightarrow M$ be a minimizer joining $y$ to $q$, where
$b=d(y,q)$. We claim that $\zeta$ does not intersect $\mathrm{Cu}(N)$. If not,
we must have some $0<s<b$ so that $x_{0}^{\prime}=\zeta(s)\in\mathrm{Cu}(N)$
(see Figure 5).

$x_{0}$
---
$\mathrm{Cu}(N)$
---
$q\notin\mathrm{Cu}(x_{0})$
---
$\gamma_{1}$
---
$\gamma_{3}$
---
$\gamma_{2}$
---
$N$
---
$T$
---
$y$
---
$\zeta$
---
$\zeta$
---
$x_{0}^{\prime}$
---
Figure 5. Case (1) : Minimizer $\zeta$ joining $y$ to $q$, crossing
$\mathrm{Cu}(N)$

We have

$\displaystyle\mathcal{M}(x_{0})\leq\mathcal{M}(x_{0}^{\prime})$
$\displaystyle=d(N,x_{0}^{\prime})+d(x_{0}^{\prime},q)$ $\displaystyle\leq
d(N,y)+\underbrace{d(y,x_{0}^{\prime})+d(x_{0}^{\prime},q)}_{\text{$y,x_{0}^{\prime},q$
are on the minimizer $\zeta$}}$
$\displaystyle=d(N,y)+d(y,q)=\mathcal{M}(y)<\mathcal{M}(x_{0}),$

which is a contradiction. Thus, $\zeta$ does not intersect $\mathrm{Cu}(N)$.
Similarly, for any $y=\gamma_{2}(t)$, we have that any minimizer joining $y$
to $q$ does not intersect $\mathrm{Cu}(N)$ (Figure 6).

$x_{0}$
---
$\mathrm{Cu}(N)$
---
$q\notin\mathrm{Cu}(x_{0})$
---
$\gamma_{1}$
---
$\gamma_{3}$
---
$\gamma_{2}$
---
$N$
---
$T$
---
$y_{1j}$
---
$\zeta_{1j}$
---
Figure 6. Case (1) : Minimizer between $y_{ij}$ and $q$

Pick some sequence $t_{j}\rightarrow\ell$ and set
$y_{ij}=\gamma_{i}(t_{j}),i=1,2$. Then,
$y_{ij}\rightarrow\gamma_{i}(\ell)=x_{0}$ for $i=1,2$. Consider minimizers
$\zeta_{ij}$ joining $y_{ij}$ to $q$. Since $\zeta_{ij}$ does not intersect
$\mathrm{Cu}(N)$, it follows from 4.9 that there exists curves
$\tilde{\zeta}_{ij}=\left(\exp^{\nu}\right)^{-1}(\zeta_{ij})$ in
$\mathcal{I}(N)$ joining $\tilde{y}_{ij}=\left(\exp^{\nu}\right)^{-1}(y_{ij})$
to $\tilde{q}=\left(\exp^{\nu}\right)^{-1}(q)$. As $x_{0}$ is not a focal
point of $N$, it follows that $\exp^{\nu}$ is a local diffeomorphism near
$\ell\dot{\gamma}_{i}(0)\in\left(\exp^{\nu}\right)^{-1}(x_{0})$ for $i=1,2$.
In particular, passing to a subsequence if necessary, we have
$\tilde{y}_{ij}\rightarrow\ell\dot{\gamma}_{i}(0)$ for $i=1,2$. By an
Arzelà–Ascoli type argument ([Bus55, Theorem 5.16]), we then have (possibly a
subsequence of) $\tilde{\zeta}_{ij}$ converges uniformly to some
$\tilde{\zeta}_{i}$ in $\mathcal{I}(N)\cup\widetilde{\mathrm{Cu}}(N)$ for
$i=1,2$, which must join $\ell\dot{\gamma}_{i}(0)$ to $\tilde{q}$,
respectively. Suppose, $\zeta_{i}=\exp^{\nu}(\tilde{\zeta}_{i})$, which then
joins $x_{0}$ to $q$ for $i=1,2$. Note that

$L(\zeta_{i})=L\left(\exp^{\nu}\left(\tilde{\zeta}_{i}\right)\right)=L\left(\lim_{j}\exp^{\nu}(\tilde{\zeta}_{ij})\right)=L\left(\lim_{j}\zeta_{ij}\right)=\lim_{j}d(y_{ij},q)=d(x_{0},q),$

and hence, $\zeta_{i}$ is a minimizer joining $x_{0}$ to $q$. From uniqueness,
we have $\zeta_{1}=\eta=\zeta_{2}$, up to a reparametrization. We now show
that this leads to a contradiction.

Consider a reparametrization of $\tilde{\zeta}_{i}$ given as
$\tilde{\eta}_{i}:[0,a]\rightarrow\nu$, where $a=d(x_{0},q)$, so that
$\exp^{\nu}(\tilde{\eta}_{i}(t))=\eta(t)$ for $0\leq t\leq a$ and $i=1,2$. Set

$s_{0}:=\inf_{0\leq s\leq
a}\left\\{s\;\middle|\;\tilde{\eta}_{1}|_{[s,a]}=\tilde{\eta}_{2}|_{[}s,a]\right\\}.$

Since $\exp^{\nu}$ is injective on $\mathcal{I}(N)$ and
$\tilde{\eta}_{1}(a)=\tilde{q}=\tilde{\eta}_{2}(a)\in\mathcal{I}(N)$, we have
$s_{0}<a$. Again, from the hypothesis, $\exp^{\nu}$ is a local diffeomorphism
on every point of $\mathcal{I}(N)\cup\left(\exp^{\nu}\right)^{-1}(x_{0})$, and
hence in particular, injective in a neighborhood of
$\tilde{\eta}_{1}(s_{0})=\tilde{\eta}_{2}(s_{0})$. Consequently, by a standard
argument using the connectivity of the interval $[0,a]$, we get $s_{0}=0$. But
this implies that $\tilde{\eta}_{1}=\tilde{\eta}_{2}$ everywhere, which
contradicts that
$\tilde{\eta}_{1}(0)=\ell\dot{\gamma}_{1}(0)\neq\ell\dot{\gamma}_{2}(0)=\tilde{\eta}_{2}(0)$.

Thus, we conclude that there exists exactly two $N$-segments joining $N$ to
$x_{0}$. Suppose, $\gamma_{1},\gamma_{2}:[0,\ell]\rightarrow M$ are the
$N$-segments. We must have exactly one of $\gamma_{i}\cup\eta$ is smooth,
since otherwise by the above argument we will again reach a contradiction.
Without loss of generality, assume that $\gamma_{1}\cup\eta$ is smooth at
$x_{0}=\gamma_{1}(\ell)=\eta(0)$, i.e.,
$\dot{\gamma}_{1}(\ell)=\dot{\eta}(0)$. But then it follows from the
uniqueness of the solution to the geodesic equation that $\gamma_{1}\cup\eta$
is an $N$-geodesic. This proves (1).

Case (2): In order to prove (2), assume that $q\in\mathrm{Cu}(x_{0})$. Say,
$\eta:[0,a]\rightarrow M$ is a minimizer joining $x_{0}$ to $q$, where
$a=d(x_{0},q)$. We can choose some $0<a_{0}<a$, so that
$q_{0}=\eta(a_{0})\not\in\mathrm{Cu}(x_{0})$. Consider the function
$\mathcal{M}_{0}=\mathcal{M}_{q_{0}}$ defined as
$\mathcal{M}_{0}(x)\vcentcolon=d(N,x)+d(x,q_{0})$ for $x\in M$. We show that
$\mathcal{M}_{0}(x_{0})=\min_{x\in\mathrm{Cu}(N)}\mathcal{M}_{0}(x)$. If not,
suppose there exists some $x_{1}\in\mathrm{Cu}(N)$ such that
$\mathcal{M}_{0}(x_{1})<\mathcal{M}_{0}(x_{0})$. Note that

$\displaystyle\mathcal{M}(x_{1})$ $\displaystyle=d(N,x_{1})+d(x_{1},q)$
$\displaystyle\leq d(N,x_{1})+d(x_{1},q_{0})+d(q_{0},q)$
$\displaystyle=\mathcal{M}_{0}(x_{1})+d(q_{0},q)$
$\displaystyle<\mathcal{M}_{0}(x_{0})+d(q_{0},q)$
$\displaystyle=d(N,x_{0})+\underbrace{d(x_{0},q_{0})+d(q_{0},q)}_{\text{$x_{0},q_{0},q$
are on the minimizer $\eta$}}$ $\displaystyle=d(N,x_{0})+d(x_{0},q)$
$\displaystyle=\mathcal{M}(x_{0}),$

which is a contradiction as $\mathcal{M}$ achieves a global minima value on
$\mathrm{Cu}(N)$ at $x_{0}$. Hence, $\mathcal{M}_{0}$ also achieves a global
minima on $\mathrm{Cu}(N)$ at $x_{0}$. But now we can argue as in (1), for
$q_{0}$ and $\mathcal{M}_{0}$ instead of $q$ and $\mathcal{M}$ respectively.
In particular, we have exactly two $N$-segments joining $N$ to $x_{0}$, say,
$\gamma_{1},\gamma_{2}$.

Now, we have a minimizer joining $x_{0}$ to $q$. Suppose we have at least
three distinct minimizers from $x_{0}$ to $q$. Then, we have at least one
geodesic, say, $\eta$ from $x_{0}$ to $q$ such that both $\gamma_{1}\cup\eta$
and $\gamma_{2}\cup\eta$ are broken at $x_{0}$. But then picking some
$q_{0}\neq x_{0},q$ on $\eta$, and then arguing as in (1) for
$\mathcal{M}_{0}$ and $q_{0}$ as above, we again reach a contradiction. Hence,
there exists at most two minimizers, say, $\eta_{1},\eta_{2}$, from $x_{0}$ to
$q$. If $\eta_{1},\eta_{2}$ are distinct, then same argument gives that two
(and hence, exactly two) of the curves
$\left\\{\gamma_{i}\cup\eta_{j},\;i,j=1,2\right\\}$ are smooth, which in turn
gives two $N$-geodesics, both crossing $\mathrm{Cu}(N)$ at $x_{0}$. The other
two curves are broken at $x_{0}$. If $\eta_{1}=\eta_{2}=\eta$, then we are in
the same situation as (1), and the conclusion follows. This completes the
proof of (2).

Case (2’): Lastly, let us assume that $F$ is a reversible Finsler metric so
that the distance is now symmetric. In particular,
$q\in\mathrm{Cu}(x_{0})\Rightarrow x_{0}\in\mathrm{Cu}(q)$. If possible, let
$\eta$ be the unique minimizer joining $x_{0}$ to $q$, and without loss of
generality, $\gamma_{1}\cup\eta$ be smooth at $x_{0}$. Then, for any point
$y\neq x_{0}$ on $\gamma_{1}$, the minimizer joining $y$ to $q$ cannot be
contained in $\gamma_{1}\cup\eta$, otherwise this would extend the reversed
minimizer $\bar{\eta}$, contradicting $x_{0}\in\mathrm{Cu}(N)$. Consequently,
$d(y,q)<d(y,x_{0})+d(x_{0},q)$. Since $\gamma_{2}\cup\eta$ is broken at
$x_{0}$, for any $y\neq x_{0}$ on $\gamma_{2}$, we again have
$d(y,q)<d(y,x_{0})+d(x_{0},q)$. But now we are in the same situation as in
case (1), and arguing similarly we reach a contradiction. Thus, $\eta$ cannot
be the unique minimizer joining $x_{0}$ to $q$. Arguing as in (2), we conclude
that there are exactly two such minimizers, and this finishes the proof. ∎

###### Proof of 4.14.

Suppose $\mathrm{Cu}(N)$ has no focal points of $N$ so that we can appeal to
4.15. If $q\in\mathrm{Cu}(N)$, then by 4.6, there exists at least two
$N$-geodesics (in fact, $N$-segments) joining $N$ to $q$. Suppose
$q\not\in\mathrm{Cu}(N)$, so that we have a (unique) $N$-segment joining $N$
to $q$, which clearly does not intersect $\mathrm{Cu}(N)$. Note that if $q\in
N$, we consider the constant curve at $q$ as an $N$-segment. Also, from 4.15
we get at least one $N$-geodesic joining $N$ to $q$, which crosses
$\mathrm{Cu}(N)$, and thus distinct from the $N$-segment we already have. This
concludes the claim. ∎

Following [Xu15], we have the following generalization.

######  Lemma 4.16.

Suppose $x_{0}$ is a _local_ minima of the function
$\mathcal{M}|_{\mathrm{Cu}(N)}$. If $x_{0}$ is not a focal point of $N$, then
there exists at least two $N$-geodesics joining $N$ to $q$, one of which
crosses $\mathrm{Cu}(N)$ at $x_{0}$.

###### Proof.

Let $\eta:[0,a]\rightarrow M$ be a minimizer joining $x_{0}$ to $q$, where
$a=d(x_{0},q)$. For any $z=\eta(t)$, we have

$\mathcal{M}_{z}(x_{0})=d(N,x_{0})+d(x_{0},z)=d(N,x_{0})+d(x_{0},q)-d(z,q)=\mathcal{M}_{q}(x_{0})-d(z,q).$

Then, for $x\in\mathrm{Cu}(N)$ sufficiently close to $x_{0}$ we see,

$\displaystyle\mathcal{M}_{z}(x)$ $\displaystyle=d(N,x)+d(x,z)$
$\displaystyle\geq d(N,x)+d(x,q)-d(z,q)$
$\displaystyle=\mathcal{M}_{q}(x)-d(z,q)$
$\displaystyle\geq\mathcal{M}_{q}(x_{0})-d(z,q)$
$\displaystyle=\mathcal{M}_{z}(x).$

In particular, $x_{0}$ is again a local minima of
$\mathcal{M}_{z}|_{\mathrm{Cu}(N)}$. We show that for some $z=\eta(t)$,
$\mathcal{M}_{z}|_{\mathrm{Cu}(N)}$ attains a global minima at $x_{0}$. In
fact, it suffices to show that $\mathcal{M}_{z}|_{\mathrm{Cu}(N)}$ attains its
global minima sufficiently close to $x_{0}$ for some $z$, whence, $x_{0}$
being a local minima will imply it is a global minima as well.

Fix some $t_{j}\rightarrow 0$ and denote $q_{j}=\eta(t_{j})$, so that
$q_{j}\rightarrow x_{0}$. If possible, suppose, for some open neighborhood
$U\subset\mathrm{Cu}(N)$ of $x_{0}$, we have that
$\mathcal{M}_{q_{j}}|_{\mathrm{Cu}(N)}$ attains a global minima, say,
$z_{j}\in\mathrm{Cu}(N)\setminus U$. As $\mathrm{Cu}(N)$ is compact, passing
to a subsequence, we assume that $z_{j}\rightarrow
z_{0}\in\mathrm{Cu}(N)\setminus U$. Now, for any $y\in\mathrm{Cu}(N)$ we have

$d(N,z_{j})+d(z_{j},q_{j})=\mathcal{M}_{q_{j}}(z_{j})\leq\mathcal{M}_{q_{j}}(z_{j})\leq\mathcal{M}_{q_{j}}(y)=d(N,q_{j})+d(q_{j},y).$

Taking $j\rightarrow\infty$ we have,

$d(N,z_{0})+d(z_{0},x_{0})\leq
d(N,x_{0})+d(x_{0},y)\Rightarrow\mathcal{M}_{x_{0}}(z_{0})\leq\mathcal{M}_{x_{0}}(y),\quad\forall
y\in\mathrm{Cu}(N).$

In particular, taking $y=x_{0}$ we get

$\displaystyle\mathcal{M}_{x_{0}}(z_{0})\leq\mathcal{M}_{x_{0}}(x_{0})$
$\displaystyle\Rightarrow$ $\displaystyle d(N,z_{0})+d(z_{0},x_{0})\leq
d(N,x_{0})+0\leq d(N,z_{0})+d(z_{0},x_{0})$ $\displaystyle\Rightarrow$
$\displaystyle d(N,x_{0})=d(N,z_{0})+d(z_{0},x_{0}).$

In particular, $z_{0}$ must be on an $N$-segment joining $N$ to $x_{0}$. But
$z_{0}\in\mathrm{Cu}(N)$, and hence any $N$-segment connecting $N$ to $z_{0}$
cannot be an $N$-segment after $z_{0}$. In particular $z_{0}=x_{0}$, which
contradicts $z_{0}\not\in U$. Thus, we must have for some
$q_{j_{0}}=\eta(t_{j_{0}})$, $\mathcal{M}_{q_{j_{0}}}|_{\mathrm{Cu}(N)}$
attains a global minima at $x_{0}$.

Now, we can apply 4.15 for $\mathcal{M}_{q_{j_{0}}}$, and conclude the proof.
∎

Let us now state the following result, originally due to Klingenberg in the
Riemannian case for a point, and later generalized in [Xu15].

######  Theorem 4.17.

Let $N$ be a closed submanifold of $(M,F)$, where $F$ is _reversible_. Suppose
$x_{0}\in\mathrm{Cu}(N)$ is a local minima of the function $x\mapsto d(N,x)$
on $\mathrm{Cu}(N)$. Then one of the following holds.

  * •

$x_{0}$ is a focal point of $N$.

  * •

There exists a unit-speed $N$-geodesic loop, crossing the $\mathrm{Cu}(N)$ at
$x_{0}$, at the midpoint

###### Proof.

Suppose $x_{0}$ is not a focal point of $N$. Pick some $q\in N$ so that
$d(q,x_{0})=d(N,x_{0})$. Since $F$ is reversible, we have

$\mathcal{M}_{q}(x_{0})=d(N,x_{0})+d(x_{0},q)=d(N,x_{0})+d(q,x_{0})=2d(N,x_{0}).$

Then, for $x\in\mathrm{Cu}(N)$ sufficiently close to $x_{0}$, we have from 2.8
that

$\displaystyle\mathcal{M}_{q}(x)=d(N,x)+d(x,q)$ $\displaystyle=d(N,x)+d(q,x)$
$\displaystyle\geq d(N,x)+d(N,x)-\underbrace{d(N,q)}_{0}=2d(N,x)$
$\displaystyle\geq 2d(N,x_{0})=\mathcal{M}_{q}(x_{0}),$

i.e., $x_{0}$ is a local minima of $\mathcal{M}_{q}|_{\mathrm{Cu}(N)}$. By
4.16, we then have an $N$-geodesic joining $N$ to $q\in N$ (i.e., an
$N$-geodesic loop), which intersects $\mathrm{Cu}(N)$ at $x_{0}$ at time
$d(N,x_{0})$, which is the midpoint. ∎

## Acknowledgments

The authors would like to express their gratitude to S. Basu and J. Itoh for
many fruitful discussions and suggestions in the preparation of this article.
They would also like to thank ICTS for the wonderful hosting and a research
friendly environment during the workshop Dualities in Topology and Geometry,
where the majority 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 ICTS postdoctoral fellowship.

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