# The Morse index theorem in the case of two variable
endpoints in conic Finsler manifolds 00footnotetext:  2020 Mathematics Subject
Classification. Primary 53C22, 53C50, 53C60. Key words and phrases. pseudo-
Finsler manifold, Morse index form, Jacobi fields, focal points.

Guangcun Lu111 E-mail<EMAIL_ADDRESS>2, 2023(first), July 28,
2023(third revised).

Abstract In this note, we prove the Morse index theorem for a geodesic
connecting two submanifolds in a $C^{7}$ manifold with a $C^{6}$ (conic)
pseudo-Finsler metric provided that the fundamental tensor is positive
definite along velocity curve of the geodesic.

##  1 Introduction, preliminaries and statements of results

1.1. Introduction. Ioan Radu Peter proved the Morse index theorem where the
ends are submanifolds in Finsler geometry in [25]. In this paper we generalize
it to (conic) pseudo-Finsler manifolds with lower smoothness. Instead of the
Cartan connection used in [25] we employ the Chern connection to introduce the
Morse index form, $P$-Jacobi field and the normal second fundamental form (or
shape operator). In addition, our proof method is different from that of [25].
For example, we do not use the (conic) pseudo-Finsler version (Proposition
3.2) of the so-called index lemma [25, Lemma 4.2] in Finsler geometry. The
Morse index theorem proved in [9, §31] by R.A. Hermann plays a key role in our
proof.

1.2. Preliminaries for Finsler geometry. The theory of geodesics and Jacobi
fields on smooth pseudo-Finsler manifolds was developed by Javaloyes and
collaborators in [11, 12, 13, 14, 15, 16, 18]. The related notions and results
may be stated for pseudo-Finsler manifolds with lower smoothness.

Let $M$ be a $n$-dimensional, connected $C^{7}$ submanifold of
${\mathbb{R}}^{N}$. Its tangent bundle $TM$ is a $C^{6}$ manifold of dimension
$2n$. Let $\pi:TM\rightarrow M$ be the natural projection and let $A\subset
TM\setminus 0_{TM}$ be an open subset of $TM$ which is conic, that is, such
that $\pi(A)=M$ and $\lambda v\in A$, for every $v\in A$ and $\lambda>0$. A
$C^{6}$ (conic) pseudo-Finsler metric (with domain $A$) is a $C^{6}$ function
$L:A\rightarrow{\mathbb{R}}$ satisfying the following conditions:

(i)

$L(\lambda v)=\lambda^{2}L(v)$ for any $v\in A$ and $\lambda>0$,

(ii)

for every $v\in A$, the fundamental tensor $g_{v}$ of $L$ at $v$ defined by

$T_{\pi(v)}M\times T_{\pi(v)}M\ni(u,v)\mapsto
g_{v}(u,w):=\frac{1}{2}\frac{\partial^{2}}{\partial t\partial
s}L(v+tu+sw)|_{t=s=0}$

is nondegenerate.

The pair $(M,L)$ is called a $C^{6}$ (conic) pseudo-Finsler manifold (with
domain $A$). In particular, if each $g_{v}$ in (ii) is positive definite, we
call the square root $F=\sqrt{L}$ a $C^{6}$ conic Finsler metric (cf. [14]).
By [15, Proposition 2.3] $g_{v}(v,v)=L(v)$ and $g_{\lambda v}=g_{v}$ for any
$v\in A$ and $\lambda>0$. The Cartan tensor of $L$ is the trilinear form

$C_{v}(w_{1},w_{2},w_{3})=\frac{1}{4}\left.\frac{\partial^{3}}{\partial
s_{3}\partial s_{2}\partial
s_{1}}L\left(v+\sum_{i=1}^{3}s_{i}w_{i}\right)\right|_{s_{1}=s_{2}=s_{3}=0}$
(1.1)

for $v\in A$ and $w_{1},w_{2},w_{3}\in T_{\pi(v)}M$. By homogeneity,
$C_{v}(w_{1},w_{2},w_{3})=0$ if $w_{i}=v$ for some $i$.

For an open subset $\Omega\subset M$ and an integer $0\leq\ell\leq 6$ let
$\Gamma^{\ell}(T\Omega)$ be the space of $C^{\ell}$ vector fields on $\Omega$.
$V\in\Gamma^{\ell}(T\Omega)$ is called $A$-admissible if $V(x)\in A$ for every
$x\in\Omega$. Denote by $\Gamma^{\ell}_{A}(T\Omega)$ the set of all
$A$-admissible $V\in\Gamma^{\ell}(T\Omega)$.

Let $(\Omega,x^{i})$ be a coordinate chart on $M$ and let $(x^{i},y^{i})$ be
the canonical coordinates on $\pi^{-1}(\Omega)\subset TM$ associated with it,
i.e., $v=v^{i}\frac{\partial}{\partial
x^{i}}\big{|}_{\pi(v)}=y^{i}(v)\frac{\partial}{\partial
x^{i}}\big{|}_{\pi(v)}$ for $v\in\pi^{-1}(\Omega)$. Define

$\displaystyle T\Omega\cap A\ni v\mapsto
g_{ij}(v)=g_{v}\left(\frac{\partial}{\partial
x^{i}}\Big{|}_{\pi(v)},\frac{\partial}{\partial
x^{j}}\Big{|}_{\pi(v)}\right),$ $\displaystyle T\Omega\cap A\ni v\mapsto
C_{ijk}(v)=C_{v}\left(\frac{\partial}{\partial
x^{i}}\Big{|}_{\pi(v)},\frac{\partial}{\partial
x^{j}}\Big{|}_{\pi(v)},\frac{\partial}{\partial
x^{k}}\Big{|}_{\pi(v)}\right).$

They are $C^{4}$ and $C^{3}$ functions, respectively. For $v\in T\Omega\cap A$
let $(g^{ij}(v))=(g_{ij}(v))^{-1}$ and

$\displaystyle\gamma^{i}_{jm}(v):=\frac{1}{2}\sum_{s}{g}^{is}(v)\left(\frac{\partial{g}_{sj}}{\partial{x^{m}}}(v)+\frac{\partial{g}_{ms}}{\partial{x^{j}}}(v)-\frac{\partial{g}_{jm}}{\partial{x^{s}}}(v)\right),$
$\displaystyle{C}^{i}_{jm}(v):=\sum_{l}{g}^{il}(v){C}_{ljm}(v),$
$\displaystyle
N^{i}_{j}(v):=\sum_{m}\gamma^{i}_{jm}(x,y)v^{m}-\sum_{m,r,s}\mathcal{C}^{i}_{jm}(v)\gamma^{m}_{rs}(v)v^{r}v^{s},$
$\displaystyle\Gamma^{i}_{jk}(v):=\gamma^{i}_{jk}(v)-\sum_{l,r}g^{li}(v)\left(C_{jlr}(v)N^{r}_{k}(v)-C_{jkr}(v)N^{r}_{l}(v)+C_{lkr}(v)N^{r}_{j}(v)\right).$

These are $C^{3}$. Moreover $N^{j}_{i}(v)=\sum_{k}\Gamma^{j}_{ik}(v)v^{k}$ and
the Chern connection $\nabla$ (on the pulled-back tangent bundle
$\pi^{\ast}TM$) is given by

$\nabla_{\partial_{x^{i}}}\partial_{x^{j}}(v)=\sum_{m}\Gamma^{i}_{jm}(v)\partial_{x^{m}},\quad
i,j\in\\{1,\cdots,n\\}.$ (1.2)

The trilinear map $R_{v}$ from $T_{\pi(v)}M\times T_{\pi(v)}M\times
T_{\pi(v)}M$ to $T_{\pi(v)}M$ given by

$R_{v}(\xi,\eta)\zeta=\sum_{i,j,k,l}\xi^{k}\eta^{l}\zeta^{j}R_{j\,\,kl}^{\,\,\,i}(v)\partial_{x^{i}}|_{\pi(v)}$
(1.3)

for $v\in T\Omega\cap(TM\setminus 0_{TM})$ defines the Chern curvature tensor
(or $hh$-curvature tensor [1, (3.3.2) & Exercise 3.9.6]) $R_{V}$ on
$\Omega\subset M$, where

$R^{i}_{jkl}=\frac{\delta\Gamma^{i}_{jl}}{\delta
x^{k}}-\frac{\delta\Gamma^{i}_{jk}}{\delta
x^{l}}+\sum_{h}\Gamma^{i}_{hk}\Gamma^{h}_{jl}-\sum_{h}\Gamma^{i}_{hl}\Gamma^{h}_{jk}\quad\hbox{and}\quad\frac{\delta}{\delta
x^{k}}=\frac{\partial}{\partial
x^{k}}-\sum_{i}N^{i}_{k}\frac{\partial}{\partial y^{i}}.$

Let $0\leq\ell\leq 3$, $s\geq 1$, $V\in\Gamma^{\ell}_{A}(T\Omega)$. Define
$\nabla^{V}:\Gamma^{s}(T\Omega)\times\Gamma^{s}(T\Omega)\to\Gamma^{\min\\{\ell,s-1\\}}(T\Omega)$
by

$\nabla^{V}_{f\partial_{x^{i}}}(g\partial_{x^{j}})(x)=f(x)g(x)\sum_{m}\Gamma^{i}_{jm}(V(x))\partial_{x^{m}}+f(x)\partial_{x^{i}}g(x)\partial_{x^{j}}$
(1.4)

for $f,g\in C^{s}(\Omega)$ and $i,j\in\\{1,\cdots,n\\}$. (This depends only on
$V(x)$ as in the proof of [26, Lemma 3.12].) It satisfies the following
properties for $f\in C^{s}(\Omega)$ and $X,Y,Z\in\Gamma^{s}(T\Omega)$:

(i)

$\nabla^{V}_{X}(fY)=f\nabla^{V}_{X}Y+X(f)Y$ and
$\nabla^{V}_{fX}Y=f\nabla^{V}_{X}Y$,

(ii)

$\nabla^{V}_{X}Y-\nabla^{V}_{Y}X=[X,Y]$,

(iii)

$X(g_{V}(Y,Z))=g_{V}(\nabla^{V}_{X}Y,Z)+g_{V}(Y,\nabla^{V}_{X}Z)+2C_{V}(\nabla^{V}_{X}Y,Y,Z)$,
where $g_{V}$ and $C_{V}$ are, respectively, the fundamental tensor and the
Cartan tensor of $L$ evaluated on the vector field $V$.

$\nabla^{V}$ is called the Chern connection of $(M,L)$ associated with $V$.
For a $C^{1}$ function $f:A\to\mathbb{R}$, the vertical derivative of $f$ at
$v\in A$ in direction $u\in TM$ with $\pi(u)=\pi(v)$ was defined in [18] by

$\partial^{\nu}f_{v}(u)=\left.\frac{d}{dt}\right|_{t=0}f(v+tu).$

Define the vertical gradient $(\nabla^{\nu}f)_{v}$ at $v\in A$, and the
horizontal gradient $(\nabla^{\rm h}f)_{v}$ at $v\in A$ by

$\displaystyle\partial^{\nu}f_{v}(u)=g_{v}((\nabla^{\nu}f)_{v},u)\;\;\forall
u\in\pi^{-1}(\pi(v)),$ (1.5) $\displaystyle
u(f(V))-\partial^{\nu}f_{v}(\nabla^{V}_{u}V)=g_{v}((\nabla^{\rm
h}f)_{v},u)\;\;\forall u\in\pi^{-1}(\pi(v)),$ (1.6)

respectively, where $V$ is a local $A$-admissible $C^{1}$ extension of $v$.
See [18, (3),(4),(5)].

If $1\leq\ell\leq 3$ and $s\geq 2$, the connection $\nabla^{V}$ in (1.4) has
the curvature tensor $R^{V}$ defined by

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

for vector fields $V\in\Gamma^{\ell}_{A}(T\Omega)$ and
$X,Y,Z\in\Gamma^{s}(T\Omega)$. Clearly, $R^{V}(X,Y)Z$ is
$C^{\min\\{\ell-1,s-2\\}}$, and it was proved in [13, Theorem 2.1] that

$R_{V}(X,Y)Z=R^{V}(X,Y)Z-P_{V}(Y,Z,\nabla^{V}_{X}V)+P_{V}(X,Z,\nabla^{V}_{Y}V),$
(1.7)

where $P_{V}(X,Y,Z)=\frac{\partial}{\partial t}(\nabla_{X}^{V+tZ})|_{t=0}$ and
so $P_{V}(X,Y,V)=0$ ([13, (1)]). The Chern tensor $P_{V}(X,Y,Z)$ is symmetric
in $X$ and $Y$, and $P_{V}(X,Y,Z)|_{x}$ depends only on $V(x)$. By [13, Lemma
1.2]

$P_{v}(v,v,u)=0,\quad\forall v\in T\Omega\cap A,\;\forall u\in T_{\pi(v)}M.$
(1.8)

For $v\in(T_{x}M)\cap A$ and $u,w\in T_{x}M$, by [16, Proposition 3.1] we have

$\displaystyle g_{v}(R_{v}(v,u)v,w)$ $\displaystyle=$ $\displaystyle
g_{v}(R_{v}(v,w)v,u),$ (1.9) $\displaystyle g_{v}(R_{v}(u,v)v,w)$
$\displaystyle=$ $\displaystyle-g_{v}(R_{v}(u,v)w,v).$ (1.10)

For a $C^{7}$ connected submanifold $P\subset M$ of dimension $k<n$, call

$\displaystyle TP^{\bot}=\\{v\in A~{}|~{}\pi(v)\in P,\;g_{v}(v,w)=0\;\forall
w\in T_{\pi(v)}P\\}$ (1.11)

the normal bundle of $P$ in $(M,L)$ though $TP^{\bot}$ is not necessarily a
fiber bundle over $P$. By [15, Lemma 3.3] a nonempty $TP^{\perp}$ is an
$n$-dimensional submanifold of $TM$, $P_{0}=\pi(TP^{\bot})$ is open in $P$,
and the map $\pi:TP^{\perp}\rightarrow P_{0}$ is a submersion. For $v\in
TP^{\bot}$ with $\pi(v)=p$, suppose that $g_{v}|_{T_{p}P\times T_{p}P}$ is
nondegenerate. (This clearly holds if $P$ is a point.) Then there exists a
splitting

$T_{p}M=T_{p}P\oplus(T_{p}P)^{\perp}_{v},$ (1.12)

where $(T_{p}P)^{\perp}_{v}=\\{u\in T_{p}M\,|\,g_{v}(u,w)=0\;\forall w\in
T_{p}P\\}$. Clearly, $v\in(T_{p}P)^{\perp}_{v}$, and each $u\in T_{p}M$ has a
decomposition ${\rm tan}^{P}_{v}(u)+{\rm nor}^{P}_{v}(u)$, where ${\rm
tan}^{P}_{v}(u)\in T_{p}P$ and ${\rm nor}^{P}_{v}(u)\in(T_{p}P)^{\perp}_{v}$.
Let $V$ be local $C^{6}$ admissible extensions of $v$, and for $u,w\in T_{p}P$
let $U$ and $W$ be local $C^{6}$ extensions of $u$ and $w$ to $M$ in such a
way that $U$ and $W$ are tangent to $P$ along $P$. Then

$\displaystyle S^{P}_{v}:T_{p}P\times
T_{p}P\to(T_{p}P)^{\perp}_{v},\;(u,w)\mapsto{\rm
nor}^{P}_{v}\left((\nabla^{V}_{U}W)(p)\right),$
$\displaystyle\tilde{S}^{P}_{v}:T_{p}P\to T_{p}P,\;u\mapsto{\rm
tan}^{P}_{v}\left((\nabla^{V}_{U}V)(p)\right)$

are well-defined (cf. [15, subsection 3.1] and [11, §3.1]). They are called
the second fundamental form of $P$ in the direction $v$ and the normal second
fundamental form (or shape operator) of $P$ in the direction $v$,
respectively. $S^{P}_{v}$ is bilinear and symmetric, $\tilde{S}^{P}_{v}$ is
linear, and $g_{v}({S}^{P}_{v}(u,w),v)=-g_{v}(\tilde{S}^{P}_{v}(u),w)$. Hence
$\tilde{S}^{P}_{v}$ is symmetric.

For a curve $c\in W^{1,2}([a,b],M)$ let $W^{1,2}(c^{\ast}TM)$ and
$L^{2}(c^{\ast}TM)$ denote the spaces of all $W^{1,2}$ and $L^{2}$ vector
fields along $c$, respectively. Then $\dot{c}\in L^{2}(c^{\ast}TM)$. Let
$(x^{i},y^{i})$ be the canonical coordinates around $\dot{c}(t)\in TM$. Write
$\dot{c}(t)=\dot{c}^{i}(t)\partial_{x^{i}}|_{c(t)}$ and
$\zeta(t)=\zeta^{i}(t)\partial_{x^{i}}|_{c(t)}$ for a vector field $\zeta$
along $c$. Call $\xi\in C^{0}(c^{\ast}TM)$ $A$-admissible if $\xi(t)\in A$ for
all $t\in[a,b]$. The covariant derivative of $\zeta\in W^{1,2}(c^{\ast}TM)$
along $c$ (with this $A$-admissible $\xi$ as reference vector) is defined by

$D^{\xi}_{\dot{c}}\zeta(t):=\sum_{m}\bigl{(}\dot{\zeta}^{m}(t)+\sum_{i,j}\zeta^{i}(t)\dot{c}^{j}(t)\Gamma_{ij}^{m}(c(t),\xi(t))\bigr{)}\partial_{x^{m}}|_{c(t)}.$
(1.13)

Clearly, $D^{\xi}_{\dot{c}}$ belongs to $L^{2}(c^{\ast}TM)$, and sits in
$C^{\min\\{1,r\\}}(c^{\ast}TM)$ provided that $c$ is of class $C^{r+1}$,
$\zeta\in C^{r+1}(c^{\ast}TM)$ and $\xi\in C^{r}(c^{\ast}TM)$ for some $0\leq
r\leq 6$; $D^{\xi}_{\dot{c}}\zeta(t)$ depends only on $\xi(t)$, $\dot{c}(t)$
and behavior of $\zeta$ near $t$. Moreover
$D^{\xi}_{\dot{c}}\zeta(t)=\nabla^{\tilde{\xi}}_{\dot{c}}\tilde{\zeta}(c(t))$
if $\dot{c}(t)\neq 0$ and $\tilde{\xi}$ (resp. $\tilde{\zeta}$) is any
$A$-admissible extension (resp. any extension) of $\xi$ (resp. $\zeta$) near
$c(t)$. When the above $\xi$ belongs to $W^{1,2}(c^{\ast}TM)$,
$D^{\xi}_{\dot{c}}$ is  almost $g_{\xi}$-compatible, that is, for any
$\eta,\zeta\in W^{1,2}(c^{\ast}TM)$ we have

$\frac{d}{dt}g_{\xi}(\zeta,\eta)=g_{\xi}\bigl{(}D^{\xi}_{\dot{c}}\zeta,\eta\bigr{)}+g_{\xi}\bigl{(}\zeta,D^{\xi}_{\dot{c}}\eta\bigr{)}+2{C}_{\xi}\bigl{(}D^{\xi}_{\dot{c}}\xi,\zeta,\eta\bigr{)}\quad{\rm
a.e.}$ (1.14)

(cf. [12, (4)]). Moreover, if $c$, $\xi$ and $\zeta$ are $C^{3}$, $C^{1}$ and
$C^{2}$, respectively, then $D^{\xi}_{\dot{c}}\zeta$ is $C^{1}$ and
$D^{\xi}_{\dot{c}}D^{\xi}_{\dot{c}}\zeta$ is well-defined and is $C^{0}$. A
$C^{1}$ curve $\gamma:[a,b]\to M$ is said to be $A$-admissible if
$\dot{\gamma}\in C^{0}(\gamma^{\ast}TM)$ is $A$-admissible. A $W^{1,2}$-vector
field $X$ along such a curve $\gamma$ is called parallel if
$D^{\dot{\gamma}}_{\dot{\gamma}}X=0$. A $C^{2}$ $A$-admissible curve $\gamma$
in $(M,L)$ is called an $L$-geodesic if $\dot{\gamma}$ is parallel along
$\gamma$, i.e., $D^{\dot{\gamma}}_{\dot{\gamma}}\dot{\gamma}(t)\equiv 0$. $L$
is always constant along a geodesic $\gamma$. In particular, if
$L(\dot{\gamma}(t))\equiv 0$ then $\gamma$ is called a lightlike geodesic.

Remark A. $L$-geodesics must first be $A$-admissible, and so constant curves
in $(M,L)$ cannot be geodesics. For an $L$-geodesic $\gamma$ in $M$, each
$\dot{\gamma}(t)$ belongs to $A$ and therefore $g_{\dot{\gamma}(t)}$ must be
nondegenerate. By [15, Proposition 2.3]
$g_{\dot{\gamma}(t)}(\dot{\gamma}(t),\dot{\gamma}(t))=L(\dot{\gamma}(t))\equiv
L(\dot{\gamma}(0))$. If $g_{\dot{\gamma}(t_{0})}$ is positive definite for
some $t_{0}\in[0,\tau]$, so is each $g_{\dot{\gamma}(t)}$ because $t\mapsto
g_{\dot{\gamma}(t)}$ is continuous; see the arguments below (2.4).

Observe that in local coordinates the geodesic equation has the form

$\ddot{\gamma}^{k}+\sum_{i,j=1}^{n}\dot{\gamma}^{i}\dot{\gamma}^{j}(\Gamma^{k}_{ij}\circ\dot{\gamma})=0,\quad
k\in\\{1,\dots,n\\}.$

Since $\Gamma^{k}_{ij}$ is $C^{3}$, the geodesics are actually $C^{4}$ by the
ordinary differential equation theory.

As usual, using a $C^{6}$ Riemannian metric $h$ on $M$ and its exponential map
$\exp$ may determine a Riemannian-Hilbert metric on $W^{1,2}([0,\tau];M)$
given by

$\langle\xi,\eta\rangle_{1}=\int^{\tau}_{0}h(\xi(t),\eta(t))dt+\int^{\tau}_{0}h(\nabla^{h}_{\dot{\gamma}}\xi(t),\nabla^{h}_{\dot{\gamma}}\xi(t))dt$
(1.15)

for $\xi,\eta\in T_{\gamma}W^{1,2}([0,\tau];M)=W^{1,2}(\gamma^{\ast}TM)$
(where the $L^{2}$ covariant derivative along $\gamma$ associated with the
Levi-Civita connection $\nabla^{h}$ of the metric $h$ is defined as (1.13)).

Let $P$ be as above, and let $Q$ be another $C^{7}$ connected submanifold in
$M$ of dimension less than $n$. Consider the submanifold of
$W^{1,2}([0,\tau];M)$,

$W^{1,2}([0,\tau];M,P,Q):=\\{\gamma\in W^{1,2}([0,\tau];M)\,|\,\gamma(0)\in
P,\,\gamma(\tau)\in Q\\}.$

Its tangent space at $\gamma\in W^{1,2}([0,\tau];M,P,Q)$ is

$W^{1,2}_{P\times Q}(\gamma^{\ast}M):=\\{\xi\in
W^{1,2}(\gamma^{\ast}TM)\,|\,\xi(0)\in T_{\gamma(0)}P,\;\xi(\tau)\in
T_{\gamma(\tau)}Q\\}.$

Let $C^{1}([0,\tau];M,P,Q)=\\{\gamma\in C^{1}([0,\tau];M)\,|\,\gamma(0)\in
P,\;\gamma(\tau)\in Q\\}$ and let

$C^{1}_{A}([0,\tau];M,P,Q)$

consist of all $A$-admissible curves in $C^{1}([0,\tau];M,P,Q)$. The latter is
an open subset of the Banach manifold $C^{1}([0,\tau];M,P,Q)$. Define the
energy functional $\mathcal{E}_{P,Q}:C^{1}_{A}([0,\tau];M,P,Q)\to{\mathbb{R}}$
by

$\mathcal{E}_{P,Q}(\gamma)=\frac{1}{2}\int^{\tau}_{0}L(\dot{\gamma}(t))dt.$
(1.16)

It is $C^{2}$, and $\gamma\in C^{1}_{A}([0,\tau];M,P,Q)$ is a critical point
of $\mathcal{E}_{P,Q}$ if and only if $\gamma$ is an $L$-geodesic satisfying
the boundary condition

$\left\\{\begin{array}[]{ll}&g_{\dot{\gamma}(0)}(u,\dot{\gamma}(0))=0\quad\forall
u\in T_{\gamma(0)}P,\\\
&g_{\dot{\gamma}(\tau)}(v,\dot{\gamma}(\tau))=0\quad\forall v\in
T_{\gamma(\tau)}Q\end{array}\right.$ (1.17)

(cf. [2, Chap.1, §1], [4, Proposition 2.1] and [15, Prop. 3.1, Cor.3.7]. Note
that $dL(\dot{\gamma}(t))\equiv 0$). The curve $\gamma$ satisfying (1.17) is
said to be $g_{\dot{\gamma}}$-orthogonal (or perpendicular) to $P$ and $Q$.
Clearly, (1.17) implies that $\dot{\gamma}(0)\in TP^{\bot}\subset A$ and
$\dot{\gamma}(\tau)\in TQ^{\bot}\subset A$.

For an $L$-geodesic $\gamma:[0,\tau]\to M$ satisfying (1.17), from now on we
also suppose that both $g_{\dot{\gamma}(0)}|_{T_{\gamma(0)}P\times
T_{\gamma(0)}P}$ and $g_{\dot{\gamma}(\tau)}|_{T_{\gamma(\tau)}Q\times
T_{\gamma(\tau)}Q}$ are nondegenerate. (As noted in [15, Remark 3.9], these
hold if $P$ (resp. $Q$) is a hypersurface of $M$ and $L(\dot{\gamma}(0))\neq
0$ (resp. $L(\dot{\gamma}(\tau))\neq 0$).) Then the normal second fundamental
forms $\tilde{S}^{P}_{\dot{\gamma}(0)}$ and
$\tilde{S}^{Q}_{\dot{\gamma}(\tau)}$ are well-defined. Suppose that $\gamma\in
C^{1}_{A}([0,\tau];M,P,Q)$ is a geodesic as above, hence $C^{4}$. Let
$\Lambda:[0,\tau]\times(-\varepsilon,\varepsilon)\to
M,\;(t,s)\mapsto\gamma_{s}(t)$ be a $C^{4}$ variation of $\gamma_{0}=\gamma$
and $W=\partial_{s}\Lambda|_{s=0}$. By [15, §3] we have the following relation
between the Hessian of $\mathcal{E}_{P,Q}$ at $\gamma\in
C^{1}_{A}([0,\tau];M,P,Q)$ and the second variation
$\frac{d^{2}}{ds^{2}}\mathcal{E}_{P,Q}(\gamma_{s})|_{s=0}$:

$\displaystyle D^{2}\mathcal{E}_{P,Q}(\gamma)[W,W]$ $\displaystyle=$
$\displaystyle\frac{d^{2}}{ds^{2}}\mathcal{E}_{P,Q}(\gamma_{s})|_{s=0}=\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},W)\dot{\gamma},W)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}W,D_{\dot{\gamma}}^{\dot{\gamma}}W)\right)dt$
(1.18)
$\displaystyle+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(W(0)),W(0))-g_{\dot{\gamma}(\tau)}(\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(W(\tau)),W(\tau)).$

(Here we use the equality
$R^{\gamma}(\dot{\gamma},V)\dot{\gamma}=R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma}$
in [15, page 66].) If $V\in T_{\gamma}C^{1}_{A}([0,\tau];M,P,Q)$ comes from
another $C^{4}$ variation of $\gamma_{0}=\gamma$, as usual we derive from
(1.9) and symmetry of $\tilde{S}^{P}_{v}$ that

$\displaystyle
D^{2}\mathcal{E}_{P,Q}(\gamma)[V,W]=\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},W)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,D_{\dot{\gamma}}^{\dot{\gamma}}W)\right)dt$
$\displaystyle\qquad+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0))-g_{\dot{\gamma}(\tau)}(\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(V(\tau)),W(\tau)).$
(1.19)

According to (1.13), it is clear that the right side of (1) actually defines a
continuous symmetric bilinear form ${\bf I}^{\gamma}_{P,Q}$ on
$W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$, called as the $(P,Q)$-index form of
$\gamma$. When $V$ is also $C^{2}$, by the proof of [15, Proposition 3.11]

$\displaystyle{\bf I}^{\gamma}_{P,Q}(V,W)$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},W)-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}V,W)\right)dt$
(1.20) $\displaystyle
g_{\dot{\gamma}(0)}(D_{\dot{\gamma}}^{\dot{\gamma}}V(0)-\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0))$
$\displaystyle-
g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}V(\tau)-\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(V(\tau)),W(\tau)).$

Then $V\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$ belongs to ${\rm Ker}({\bf
I}^{\gamma}_{P,Q})$ if and only if it is $C^{4}$ and satisfies

$\displaystyle\left.\begin{array}[]{ll}D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}V-R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma}=0,\\\
{\rm
tan}^{P}_{\dot{\gamma}(0)}\big{(}(D_{\dot{\gamma}}^{\dot{\gamma}}V)(0)\big{)}=\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),\\\
{\rm
tan}^{Q}_{\dot{\gamma}}\big{(}(D_{\dot{\gamma}}^{\dot{\gamma}}V)(\tau)\big{)}=\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(V(\tau))\end{array}\right\\}$
(1.24)

because $\gamma$ is $C^{4}$ and $R^{i}_{jkl}$ are $C^{2}$. To prove the
implication to the right, we may construct $C^{4}$ parallel orthonormal frame
fields $E_{1},\cdots,E_{n}$ along $\gamma$ as done below (2.4), and reduce the
expected conclusions to the standard result in differential equations.
Elements in $\mathscr{J}^{P,Q}_{\gamma}:={\rm Ker}({\bf I}^{\gamma}_{P,Q})$
are called $(P,Q)$-Jacobi field along $\gamma$.

Recall that $g_{v}(v,v)=L(v)$ ([15, Proposition 2.3]). Suppose
$L\circ\dot{\gamma}\neq 0$. Then there exists a
$\gamma^{\ast}TM=\mathbb{R}\dot{\gamma}\oplus\dot{\gamma}^{\bot}$, where
$(\dot{\gamma}^{\bot})_{t}=\\{v\in
T_{\gamma(t)}M\,|\,g_{\dot{\gamma}(t)}(\dot{\gamma}(t),v)=0\\}$. Therefore
each $X\in\gamma^{\ast}TM$ has a decomposition
$\textsf{tan}_{\gamma}(X)+\textsf{nor}_{\gamma}(X)$, where

$\displaystyle\textsf{tan}_{\gamma}(X):=\frac{g_{\dot{\gamma}}(X,\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\dot{\gamma}\quad\hbox{and}\quad\textsf{nor}_{\gamma}(X):=X-\frac{g_{\dot{\gamma}}(X,\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\dot{\gamma}.$
(1.25)

Clearly, $W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$ is invariant for operations
$\textsf{tan}_{\gamma}$ and $\textsf{nor}_{\gamma}$. Let

$\displaystyle W^{1,2}_{P\times
Q}(\gamma^{\ast}TM)^{\bot}:=\\{\textsf{nor}_{\gamma}(X)\,|\,X\in
W^{1,2}_{P\times Q}(\gamma^{\ast}TM)\\}.$

Denote by ${\bf I}^{\gamma,\bot}_{P,Q}$ be the restriction of ${\bf
I}^{\gamma}_{P,Q}$ to $W^{1,2}_{P\times Q}(\gamma^{\ast}TM)^{\bot}$. We have

Proposition B. Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic satisfying
(1.17). Suppose $L\circ\dot{\gamma}\neq 0$. Then

(i)

${\bf I}^{\gamma}_{P,Q}(\textsf{nor}_{\gamma}(V),\textsf{tan}_{\gamma}(W))=0$
for all $V,W\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$.

(ii)

${\rm Ker}({\bf I}^{\gamma,\bot}_{P,Q})={\rm Ker}({\bf I}^{\gamma}_{P,Q})$,
and ${\rm Index}({\bf I}^{\gamma,\bot}_{P,Q})={\rm Index}({\bf
I}^{\gamma}_{P,Q})$ if $g_{\dot{\gamma}(t)}$ is positive definite for some
(and so all) $t$ in $[0,\tau]$, where ${\rm Index}({\bf
I}^{\gamma,\bot}_{P,Q})$ and ${\rm Index}({\bf I}^{\gamma}_{P,Q})$ are Morse
indexes of ${\bf I}^{\gamma,\bot}_{P,Q}$ and ${\bf I}^{\gamma}_{P,Q}$,
respectively.

Its proof will be given at the end of Section 2.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic. (It is $C^{4}$.) By [15,
Definition 3.12], a $C^{2}$ vector field $J$ along $\gamma$ is said to be a
Jacobi field if it satisfies the so-called Jacobi equation

$D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}J-R_{\dot{\gamma}}(\dot{\gamma},J)\dot{\gamma}=0.$
(1.26)

(Jacobi fields along $\gamma$ must be $C^{4}$ because $\gamma$ is $C^{4}$ and
$R^{i}_{jkl}$ are $C^{2}$.) The set $\mathscr{J}_{\gamma}$ of all Jacobi
fields along $\gamma$ is a $2n$-dimensional vector space ([15, Lemma 3.14]).
For $0\leq t_{1}<t_{2}\leq\tau$ if there exists a nonzero Jacobi field $J$
along $\gamma|_{[t_{1},t_{2}]}$ such that $J$ vanishes at $\gamma(t_{1})$ and
$\gamma(t_{2})$, then $\gamma(t_{1})$ and $\gamma(t_{2})$ are said to be
mutually conjugate along $\gamma|_{[t_{1},t_{2}]}$.

Let $P\subset M$ be as above, and let $\gamma:[0,\tau]\to M$ be an
$L$-geodesic such that $\gamma(0)\in P$, $\dot{\gamma}(0)$ is
$g_{\dot{\gamma}(0)}$-orthogonal to $P$, and that
$g_{\dot{\gamma}(0)}|_{T_{\gamma(0)}P\times T_{\gamma(0)}P}$ is nondegenerate.
A Jacobi field $J$ along $\gamma$ is called $P$-Jacobi if

$J(0)\in T_{\gamma(0)}P\quad\hbox{and}\quad{\rm
tan}^{P}_{\dot{\gamma}(0)}\big{(}(D_{\dot{\gamma}}^{\dot{\gamma}}J)(0)\big{)}=\tilde{S}^{P}_{\dot{\gamma}(0)}(J(0)).$
(1.27)

An instant $t_{0}\in(0,\tau]$ is called $P$-focal if there exists a non-null
$P$-Jacobi field $J$ such that $J(t_{0})=0$; and $\gamma(t_{0})$ is said to be
a $P$-focal point along $\gamma$. The dimension of the space
$\mathscr{J}^{P}_{\gamma}$ of all $P$-Jacobi fields along $\gamma$ is equal to
$n=\dim M$. The dimension $\mu^{P}(t_{0})$ of

$\mathscr{J}^{P}_{\gamma}(t_{0}):=\\{J\in\mathscr{J}^{P}_{\gamma}\,|\,J(t_{0})=0\\}$

is called the (geometrical) multiplicity of $\gamma(t_{0})$. For convenience
we understand $\mu^{P}(t_{0})=0$ if $\gamma(t_{0})$ is not a $P$-focal point
along $\gamma$. Then the claim below (1.20) implies that for any
$t\in(0,\tau]$,

${\rm Ker}({\bf
I}^{\gamma_{t}}_{P,\gamma(t)})=\mathscr{J}^{P}_{\gamma_{t}}(t)\quad\hbox{with
$\gamma_{t}=\gamma|_{[0,t]}$}.$ (1.28)

In particular, if $g_{\dot{\gamma}(t)}$ is positive definite for some
$t\in[0,\tau]$, this and Proposition B(ii) give

${\rm Ker}({\bf I}^{\gamma,\bot}_{P,q})={\rm Ker}({\bf
I}^{\gamma}_{P,q})=\mathscr{J}^{P}_{\gamma}(\tau)\quad\hbox{with
$q=\gamma(\tau)$}.$ (1.29)

1.3. Main results. Recall that the (Morse) index of a bilinear and symmetric
form $I$ on a vector space $E$ is the maximum number of linearly independent
elements of $E$ on which the function $I$ is negative. Here is our main
result.

######  Theorem 1.1 (Morse index theorem).

Let $P,Q$ be two $C^{7}$ submanifolds in $M$ of dimension less than $n=\dim
M$, and let $\gamma:[0,\tau]\to M$ be an $L$-geodesic which is perpendicular
to $P$ at $\gamma(0)$, and $q=\gamma(\tau)$. Suppose that
$g_{\dot{\gamma}(0)}$ is positive definite (and so each $g_{\dot{\gamma}(t)}$
is positive definite and $L\circ\dot{\gamma}$ is a positive constant by Remark
A below (1.14)). Then

(i)

$(0,\tau]$ contains only a finite number of instants $t$ such that $\gamma(t)$
are $P$-focal points along $\gamma|_{[0,t]}$.

(ii)

There is a partition $0=t_{0}<\cdots<t_{m}=\tau$ such that $\gamma((0,t_{1}])$
contains no $P$-focal points and that for any two instants $a<b$ in each
interval $[t_{i},t_{i+1}]$, $i=0,\cdots,m-1$, points $\gamma(a)$ and
$\gamma(b)$ are not mutually conjugate along $\gamma|_{[a,b]}$.

(iii)

${\rm Index}({\bf
I}^{\gamma}_{P,q})=\sum\limits_{t_{0}\in(0,\tau)}\mu^{P}(t_{0}).$ (1.30)

Moreover, if $\gamma$ is also perpendicular to $Q$ at $q=\gamma(\tau)$ and
$\\{X(\tau)\,|\,X\in\mathscr{J}^{P}_{\gamma}\\}\supseteq
T_{\dot{\gamma}(\tau)}Q$ (the latter may be satisfied if $\gamma(\tau)$ is not
a $P$-focal point), then

$\displaystyle{\rm Index}({\bf I}^{\gamma}_{P,Q})={\rm Index}({\bf
I}^{\gamma}_{P,q})+{\rm Index}(\mathcal{A}_{\gamma}),$ (1.31)
$\displaystyle{\rm Index}({\bf
I}^{\gamma,\perp}_{P,q})=\sum\limits_{t_{0}\in(0,\tau)}\mu^{P}(t_{0}),$ (1.32)
$\displaystyle{\rm Index}({\bf I}^{\gamma,\perp}_{P,Q})={\rm Index}({\bf
I}^{\gamma,\perp}_{P,q})+{\rm Index}(\mathcal{A}_{\gamma}),$ (1.33)

where $\mathcal{A}_{\gamma}$ is the bilinear symmetric form on
$\mathscr{J}^{P}_{\gamma}$ defined by

$\mathcal{A}_{\gamma}(J_{1},J_{2})=g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{1}(\tau)+\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(J_{1}(\tau)),J_{2}(\tau)).$

######  Remark 1.2.

For a smooth Finsler manifold $(M,F)$, Peter [25] used the Cartan connection
to define the Morse index form, $P$-Jacobi field and the shape operator in the
horizontal subspaces of $T(TM\setminus\\{0\\})$ and proved (1.32)-(1.33). In
this case it holds naturally that $g_{\dot{\gamma}(t)}$ is positive definite
for all $t$; see Corollary 1.3. Different from our method outlined below
(2.2), Peter’s proof is to follow the line of Milnor [23] and Piccione and
Tausk [24]. We won’t discuss whether Peter’s method is still effective under
our assumption of lower smoothness about $(M,F)$ though parallel vector fields
and geodesics are the same for the Cartan and Chern connections.

When $L$ is equal to the square of a conic Finsler metric on $M$, since
$L$-geodesics must be nonconstant curves, it naturally holds that
$L\circ\dot{\gamma}>0$ and $g_{\dot{\gamma}(t)}$ is positive definite for each
$t\in[0,\tau]$. Hence we have:

######  Corollary 1.3.

Let $M$ and $P,Q$ be as in Theorem 1.1, and let $F$ be a $C^{6}$ conic Finsler
metric on $M$ with domain $A$. For a nonconstant $F$-geodesic
$\gamma:[0,\tau]\to M$ which is perpendicular to $P$ at $\gamma(0)$, and
$q=\gamma(\tau)$, that is, it is a critical point of the functional
$\mathcal{E}_{P,q}:C^{1}_{A}([0,\tau];M,P,q)\to{\mathbb{R}}$ defined by (1.16)
with $L=F^{2}$, suppose that ${\bf I}^{\gamma}_{P,q}$ and ${\bf
I}^{\gamma}_{P,Q}$ with $L=F^{2}$ are defined as above. Then the conclusions
(i)-(ii) and (1.30) in Theorem 1.1 hold true; moreover, if $\gamma$ is also
perpendicular to $Q$ at $q=\gamma(\tau)$ and
$\\{X(\tau)\,|\,X\in\mathscr{J}^{P}_{\gamma}\\}\supseteq
T_{\dot{\gamma}(\tau)}Q$ (the latter may be satisfied if $\gamma(\tau)$ is not
a $P$-focal point), then (1.31), (1.32) and (1.33) also hold true.

After proving Theorem 1.1 in next section we shall give some related
conclusions, consequences and examples in Section 3.

##  2 Proof of Theorem 1.1

Denote by $PC^{2}(\gamma^{\ast}TM)$ the space of continuous and piecewise
$C^{2}$ sections of the bundle $\gamma^{\ast}TM$ and by

$PC^{2}_{P\times Q}(\gamma^{\ast}TM)=\\{\xi\in
PC^{2}(\gamma^{\ast}TM)\,|\,\xi(0)\in T_{\gamma(0)}P,\;\xi(\tau)\in
T_{\gamma(\tau)}Q\\}.$

Let $\tilde{\bf I}^{\gamma}_{P,Q}$ be the restriction of ${\bf
I}^{\gamma}_{P,Q}$ to $PC^{2}_{P\times Q}(\gamma^{\ast}TM)$. Since
$PC^{2}(\gamma^{\ast}TM)$ is dense in $W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$
and ${\rm Ker}({\bf I}^{\gamma}_{P,Q})\subset PC^{2}(\gamma^{\ast}TM)$ we get

${\rm Ker}(\tilde{\bf I}^{\gamma}_{P,Q})={\rm Ker}({\bf
I}^{\gamma}_{P,Q})\quad\hbox{and}\quad{\rm Index}(\tilde{\bf
I}^{\gamma}_{P,Q})={\rm Index}({\bf I}^{\gamma}_{P,Q}).$ (2.1)

Therefore (1.30) is equivalent to the equality

${\rm Index}(\tilde{\bf
I}^{\gamma}_{P,q})=\sum\limits_{t_{0}\in(0,\tau)}\mu^{P}(t_{0}).$ (2.2)

Our proof will be completed in four steps. Firstly, we shall use the Morse
index theorem by Hermann (Theorem 2.2) and a result by Hartman and Wintner [8,
page 77] to prove the conclusions (i)-(ii) and Proposition 2.1. The latter
shows that the index of the restriction of $\tilde{\bf I}^{\gamma}_{P,q}$ to a
subspace $PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}$ of $PC^{2}_{P\times
q}(\gamma^{\ast}TM)$ is equal to the right side of (2.2). Next, we prove that
(2.2) easily follows from Proposition 2.1 and Proposition 2.9. The latter
claims that $\tilde{\bf I}^{\gamma}_{P,q}$ and the restriction of $\tilde{\bf
I}^{\gamma}_{P,q}$ to a finite dimensional subspace $PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{00}$ of $PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}$ have
the same nullity and index. In Step 3 we shall prove Proposition 2.9 through
several lemmas which are closely related to [15]. Finally, we use the same
ideas as those of [25, Theorem 1.2] to complete the proofs of the second part
in (iii). The so-called index lemma ([25, Lemma 4.2]) in Finsler geometry
played a key role in the proof of [25, Theorem 1.1]. Because of using Theorem
2.2 we avoid to use the pseudo-Finsler version (Proposition 3.2) of [25, Lemma
4.2].

Step 1(Prove the conclusions (i)-(ii) and the following proposition).

######  Proposition 2.1.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic which is perpendicular to $P$ at
$\gamma(0)$, $q=\gamma(\tau)$ and let $g_{\dot{\gamma}(0)}$ (and so each
$g_{\dot{\gamma}(t)}$) be positive definite. Then the index of the restriction
of $\tilde{\bf I}^{\gamma}_{P,q}$ to

$PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}:=\\{V\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)\,|\,{\rm
tan}^{P}_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V)(0)=\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0))\\}.$
(2.3)

is equal to

$\sum\limits_{t_{0}\in(0,\tau)}\mu^{P}(t_{0}).$

If $P$ is a point $p=\gamma(0)$, then $PC^{2}_{p\times
q}(\gamma^{\ast}TM)_{0}=PC^{2}_{p\times q}(\gamma^{\ast}TM)$ and Proposition
2.1 gives (2.2) and hence (1.30).

Proposition 2.1 will be derived from the following Morse index theorem proved
in [9, §31] by R.A. Hermann.

######  Theorem 2.2 (Morse index theorem).

Let $E$ be a finite-dimensional real vector space with inner product
$\langle\cdot,\cdot\rangle$, $E_{0}\subset E$ a subspace,
$\mathcal{Q}:E_{0}\times E_{0}\to\mathbb{R}$ a bilinear, symmetric form.222 In
[9, §31], $E$, $E_{0}$, $\mathcal{Q}$ and $\tau$ were denoted by $V$, $W$, $Q$
and $a$, respectively.  Denote by $\Omega([0,\tau],E)$ the space of
continuous, piecewise $C^{2}$ curves $v:[0,\tau]\to E$ satisfying the
following conditions:

$v(0)\in E_{0},\quad\langle\dot{v}(0),w\rangle=-\mathcal{Q}(v(0),w)\;\forall
w\in E_{0},\quad v(\tau)=0.$

It is a real vector space. Given a continuous333It was not required in [9,
§31] that $t\mapsto R_{t}$ is continuous. However, this condition is necessary
in order to ensure that the integrations of $\langle R_{t}(v(t)),v(t)\rangle$
appeared on [9, page 403] and other places are well-defined. map $[0,\tau]\ni
t\mapsto{R}_{t}\in\mathscr{L}_{s}(E)$, where $\mathscr{L}_{s}(E)$ consists of
linear transformations $L:E\to E$ satisfying $\langle L(u),v\rangle=\langle
u,L(v)\rangle$ for any $u,v\in E$, consider the quadratic form

$\mathcal{I}(v)=-\mathcal{Q}(v(0),v(0))+\int^{\tau}_{0}\left[\langle\dot{v}(t),\dot{v}(t)\rangle-\langle{R}_{t}(v(t)),v(t)\rangle\right]dt$

for $v\in\Omega([0,\tau],E)$, and the differential operator

$\mathcal{J}=\frac{d^{2}}{dt^{2}}+{R}_{t}.$

Define the Morse index of $\mathcal{I}$ as the maximum number of linearly
independent elements of $\Omega([0,\tau],E)$ on which the function
$\mathcal{I}$ is negative. A point $a\in(0,\tau]$ is said to be a focal point
for the operator $\mathcal{J}$ and boundary condition $(E_{0},\mathcal{Q})$ if
there is a nontrivial $C^{2}$ curve $u:[0,a]\to E$ satisfying

$\displaystyle\mathcal{J}(u)=0,\quad u(0)\in
E_{0},\quad\langle\dot{u}(0),w\rangle=-\mathcal{Q}(u(0),w)\;\forall w\in
E_{0},\quad u(a)=0.$

(Clearly, $u$ can uniquely extend a $C^{2}$ curve $\tilde{u}:[0,\tau]\to E$
satisfying $\mathcal{J}(\tilde{u})=0$ by the continuation theorem for
solutions of ordinary differential equations.) The multiplicity of such a
focal point is equal to the dimension of the linear space of all curves
satisfying these conditions (hence, it is always finite and no greater than
the dimension of $E$). Then the index of $\mathcal{I}$ is finite and equal to
the sum of multiplicities of the focal points contained in the open interval
$(0,\tau)$. It is also equal to the maximal number of linearly independent
elements of $\Omega([0,\tau],E)$ that are $C^{2}$ and are eigenfunctions of
the differential operator $\mathcal{J}$ for positive eigenvalues.

######  Corollary 2.3 ([9, Corollary of Lemma 31.8]).

Under the assumptions of Theorem 2.2 the interval $[0,\tau]$ contains no focal
points if and only if $\mathcal{I}(u)>0$ for all curves
$u\in\Omega([0,\tau],E)\setminus\\{0\\}$.

######  Lemma 2.4 ([9, Lemmas 31.5 and 31.6]).

Let the assumptions of Theorem 2.2 be satisfied. If $\varepsilon\in(0,\tau]$
is sufficiently small, there are no focal points on the interval
$[0,\varepsilon]$ for $\mathcal{J}$ and $(E_{0},\mathcal{Q})$. Moreover, if
$t_{0}\in(0,\tau]$ is a focal point and $\varepsilon>0$ is small enough, then
the interval $[t_{0}-\varepsilon,t_{0}+\varepsilon]\cap[0,\tau]$ contains no
other focal point. Consequently, $[0,\tau]$ contains only a finite number of
focal points for $\mathcal{J}$ and $(E_{0},\mathcal{Q})$.

######  Definition 2.5 ([9, page 411, Definition]).

(Let $R_{t}$ be as in Theorem 2.2.) Reals $a<b$ in $[0,\tau]$ are said to be
mutually conjugate if there is a $C^{2}$ curve $v(t)$, not identically zero,
satisfying $\ddot{v}(t)+R_{t}(v(t))=0\;\forall t\in[a,b]$ and $v(a)=v(b)=0$.

Clearly, a point $a\in(0,\tau]$ is a focal point for the operator
$\mathcal{J}$ and boundary condition $(E_{0},\mathcal{Q})=(\\{0\\},0)$ if and
only if $0$ and $a$ are mutually conjugate. If $t_{0}\in(0,\tau]$ and $0$ are
mutually conjugate, then Lemma 2.4 implies that for a sufficiently small
$\varepsilon>0$, $0$ and each point $t\neq t_{0}$ in the interval
$[t_{0}-\varepsilon,t_{0}+\varepsilon]\cap[0,\tau]$ are not mutually
conjugate. For an interval $I\subset[0,\tau]$, if no nontrivial solution $u\in
C^{2}(I,E)$ of $\mathcal{J}u=0$ vanishes twice in $I$, we say $\mathcal{J}$ to
be disconjugate on $I$. Let $0\leq a<b\leq\tau$. According to (IV) in [8, page
77] by Hartman and Wintner, if there exists a continuous differentiable map
$[a,b]\ni t\mapsto V(t)\in\mathscr{L}_{s}(E)$ such that
$-V^{\prime}(t)+\frac{1}{4}(V(t))^{2}+R_{t}\leq 0$ for all $t\in[a,b]$,
(hereafter for $P\in\mathscr{L}_{s}(E)$ we write $P\leq 0$ if $\langle
Pu,u\rangle\leq 0$ for all $u\in E$), then $\mathcal{J}$ is disconjugate on
$[a,b]$. Given $t_{0}\in[0,\tau]$ let $V(t)=(t-t_{0})C{\rm id}_{E}$, where
$C>0$. Then

$-V^{\prime}(t)+\frac{1}{4}(V(t))^{2}+R_{t}=-C{\rm
id}_{E}+\frac{1}{4}(t-t_{0})^{2}C^{2}{\rm id}_{E}+R_{t}.$

Since $t\mapsto R_{t}$ is continuous, we may choose a large $C>0$ and a small
$\varepsilon>0$ such that

$-C{\rm id}_{E}+\frac{1}{4}(t-t_{0})^{2}C^{2}{\rm id}_{E}+R_{t}\leq
0,\quad\forall t\in I:=[0,\tau]\cap[t_{0}-\varepsilon,t_{0}+\varepsilon].$

It follows that each point in $[0,\tau]$ has a neighborhood such that
$\mathcal{J}$ is disconjugate on this neighborhood. This and Lemma 2.4 imply

######  Lemma 2.6.

Under the assumptions of Theorem 2.2, there is a partition
$0=t_{0}<\cdots<t_{m}=\tau$ such that there are no focal points on the
interval $[0,t_{1}]$ for $\mathcal{J}$ and $(E_{0},\mathcal{Q})$, and that
$\mathcal{J}$ is disconjugate on each interval $[t_{i},t_{i+1}]$,
$i=0,1,\cdots,m-1$.

######  Lemma 2.7 ([9, Lemma 31.9]).

Under the assumptions of Theorem 2.2, let reals $a<b$ in $[0,\tau]$ be such
that $[a,b]$ contains no pair of mutually conjugate points. Suppose that
$u(t)$ and $v(t)$, $a\leq t\leq b$, are continuous curves such that $u$ is
piecewise $C^{2}$ and $v$ is $C^{2}$. Moreover, $v$ satisfies

$\ddot{v}(t)+R_{t}(v(t))=0\;\forall t\in[a,b],\quad u(a)=v(a),\quad
u(b)=v(b).$

Then

$\int^{b}_{a}\left[\langle\dot{v}(t),\dot{v}(t)\rangle-\langle{R}_{t}(v(t)),v(t)\rangle\right]dt\leq\int^{b}_{a}\left[\langle\dot{u}(t),\dot{u}(t)\rangle-\langle{R}_{t}(u(t)),u(t)\rangle\right]dt.$

Equality holds only if $u(t)=v(t)$ for all $t\in[a,b]$.

Since $\tilde{\bf I}^{\gamma}_{P,q}$ is the restriction of ${\bf
I}^{\gamma}_{P,q}$ to $PC^{2}_{P\times q}(\gamma^{\ast}TM)$, by (1.18)

$\displaystyle\tilde{\bf I}^{\gamma}_{P,q}(V,W)$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},W)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,D_{\dot{\gamma}}^{\dot{\gamma}}W)\right)dt$
(2.4)
$\displaystyle+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0)).$

Note that $\tilde{\bf I}^{\gamma}_{P,q}$ has still the kernel as in (1.29),
that is, ${\rm Ker}(\tilde{\bf
I}^{\gamma}_{P,q})=\mathscr{J}^{P}_{\gamma}(\tau)$.

Because of the splitting (1.12) with $p=\gamma(0)$ and $v=\dot{\gamma}(0)$, we
may use the positive definiteness of $g_{\dot{\gamma}(0)}$ to construct a
normalized orthonormal basis for $(T_{\gamma(0)}M,g_{\dot{\gamma}(0)})$,
$e_{1},\cdots,e_{n}$, that is, a basis satisfying
$g_{\dot{\gamma}(0)}(e_{i},e_{j})=\delta_{ij}$, where $\delta_{ij}$ is the
Kronecker’s delta for $i,j=1,\cdots,n$. Clearly, we can also require that
$e_{1},\cdots,e_{k}$ is a normalized orthonormal basis for
$(T_{\gamma(0)}P,g_{\dot{\gamma}(0)}|_{T_{\gamma(0)}P\times T_{\gamma(0)}P})$
and that $e_{k+1},\cdots,e_{n}$ be a normalized orthonormal basis for
$(T_{\gamma(0)}P)^{\perp}_{\dot{\gamma}(0)}$ and $g^{\bot}_{\dot{\gamma}(0)}$.
Since $\gamma$ is $C^{4}$ and $\Gamma^{m}_{ij}$ is $C^{3}$, as in [15, page
74] we may construct a $C^{4}$ parallel orthonormal frame fields
$E_{1},\cdots,E_{n}$ along $\gamma$ starting at $e_{1},\cdots,e_{n}$.
(Therefore $g_{\dot{\gamma}(t)}(E_{i}(t),E_{j}(t))=\delta_{ij}$ because
$t\mapsto g_{\dot{\gamma}(t)}(E_{i}(t),E_{j}(t))$ is constant.) Write
$V=\sum_{i}v^{i}E_{i}$ and $W=\sum_{i}w^{i}E_{i}$, and denote by
$\langle\cdot,\cdot\rangle_{\mathbb{R}^{n}}$ the standard Euclidean metric.
Then

$D_{\dot{\gamma}}^{\dot{\gamma}}V(t)=\sum_{i}({v}^{i})^{\prime}(t)E_{i},\quad
D_{\dot{\gamma}}^{\dot{\gamma}}W(t)=\sum_{i}({w}^{i})^{\prime}(t)E_{i}$

and therefore (2.4) becomes into

$\displaystyle\tilde{\bf I}^{\gamma}_{P,q}(V,W)$ $\displaystyle=$
$\displaystyle\int^{\tau}_{0}\langle\dot{v},\dot{w}\rangle_{{\mathbb{R}}^{n}}dt-\int_{0}^{\tau}\langle\mathfrak{R}_{t}v(t),w(t)\rangle_{{\mathbb{R}}^{n}}dt$
(2.5)
$\displaystyle+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0)),$

where $v(t)=(v^{1}(t),\cdots,v^{n}(t))$, $w(t)=(w^{1}(t),\cdots,w^{n}(t))$ and
$\mathfrak{R}_{t}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is given by

$\mathfrak{R}_{t}x=\left(\sum_{i}\mathfrak{R}_{t}^{i1}x^{i},\cdots,\sum_{i}\mathfrak{R}_{t}^{in}x^{i}\right)$

with
$\mathfrak{R}_{t}^{ij}=-g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},E_{i})\dot{\gamma},E_{j})(t)$.
Clearly, $t\mapsto\mathfrak{R}_{t}^{ij}$ is $C^{2}$ because $R^{i}_{jkl}$,
$\dot{\gamma}(t)$ and $E_{i}(t)$ are $C^{2}$, $C^{3}$ and $C^{4}$,
respectively. By (1.9), the matrix $(\mathfrak{R}_{t}^{ij})$ is also
symmetric, i.e.,

$g_{\dot{\gamma}(t)}(R_{\dot{\gamma}(t)}(\dot{\gamma}(t),E_{i}(t))\dot{\gamma}(t),E_{j}(t))=g_{\dot{\gamma}(t)}(R_{\dot{\gamma}(t)}(\dot{\gamma}(t),E_{j}(t))\dot{\gamma}(t),E_{i}(t)),\quad\forall
i,j.$

Moreover, since $W(0),V(0)$ and $\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0))$ belong
to $T_{\gamma(0)}P$, we deduce that $v(0)$ and $w(0)$ belong to the subspace
of ${\mathbb{R}}^{n}$,
${\mathbb{R}}^{k}\equiv\\{(x^{1},\cdots,x^{n})\in{\mathbb{R}}^{n}\,|\,x^{k+1}=\cdots
x^{n}=0\\}$, and

$\displaystyle\mathfrak{Q}({v}(0),{w}(0)):=-\sum^{k}_{i=1}\sum^{k}_{j=1}g_{\dot{\gamma}(0)}\left(\tilde{S}^{P}_{\dot{\gamma}(0)}(e_{i}),e_{j}\right)v^{i}(0)w^{j}(0)=-g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0)).$

Clearly, $\mathfrak{Q}$ is a symmetric bilinear form on ${\mathbb{R}}^{k}$ and
(2.5) becomes into

$\displaystyle\tilde{\bf
I}^{\gamma}_{P,q}(V,W)=-\mathfrak{Q}({v}(0),{w}(0))+\int^{\tau}_{0}\langle\dot{v},\dot{w}\rangle_{{\mathbb{R}}^{n}}dt-\int_{0}^{\tau}\langle\mathfrak{R}_{t}v(t),w(t)\rangle_{{\mathbb{R}}^{n}}dt.$
(2.6)

For a $C^{2}$ vector field $J$ along $\gamma$, we can write
$J(t)=\sum^{n}_{i=1}v^{i}(t)E_{i}(t)$. Then $J$ satisfies the Jacobi equation
(1.26) if and only if $v(t)=(v^{1}(t),\cdots,v^{n}(t))$ satisfies

$\ddot{v}(t)+\mathfrak{R}_{t}v(t)=0.$ (2.7)

When $J$ is also $P$-Jacobi, the boundary condition (1.27) becomes

$v(0)\in\mathbb{R}^{k}\quad\hbox{and}\quad\quad\langle\dot{v}(0),w\rangle=-\mathfrak{Q}({v}(0),{w})\;\forall
w\in\mathbb{R}^{k}.$ (2.8)

Thus $J$ is a $P$-Jacobi field along $\gamma$ if and only if $v$ satisfies
(2.7) and (2.8).

In Theorem 2.2 let us take

$E=\mathbb{R}^{n}$,
$\quad\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{\mathbb{R}^{n}}$,
$\quad E_{0}=\mathbb{R}^{k}$, $\quad\mathcal{Q}=\mathfrak{Q}\quad$ and
$\quad\mathcal{J}=\frac{d^{2}}{dt^{2}}+\mathfrak{R}_{t}$

and therefore the corresponding index form

$\displaystyle\mathcal{I}(v)=-\mathfrak{Q}(v(0),v(0))+\int^{\tau}_{0}\left[\langle\dot{v}(t),\dot{v}(t)\rangle_{{\mathbb{R}}^{n}}-\langle{\mathfrak{R}}_{t}(v(t)),v(t)\rangle_{{\mathbb{R}}^{n}}\right]dt.$
(2.9)

Then (2.6) implies that

$\displaystyle\mathcal{I}(v)=\tilde{\bf
I}^{\gamma}_{P,q}\left(\sum^{n}_{i=1}v^{i}E_{i},\sum^{n}_{i=1}v^{i}E_{i}\right),\quad\forall
v\in\Omega([0,\tau],{\mathbb{R}}^{n}).$ (2.10)

The arguments in last paragraph show:

######  Proposition 2.8.

$a\in(0,\tau]$ is a focal point for the operator
$\mathcal{J}=\frac{d^{2}}{dt^{2}}+\mathfrak{R}_{t}$ and boundary condition
$(E_{0},\mathcal{Q})=({\mathbb{R}}^{k},\mathfrak{Q})$ if and only if
$\gamma(a)$ is a $P$-focal point along $\gamma$; and the multiplicity of $a$
as a focal point for $\mathcal{J}$ and $({\mathbb{R}}^{k},\mathfrak{Q})$ is
equal to that of $\gamma(a)$ as a $P$-focal point, $\mu^{P}(a)$. Moreover

$\Omega([0,\tau],{\mathbb{R}}^{n})\ni(v^{1},\cdots,v^{n})\to\sum^{n}_{i=1}v^{i}E_{i}\in
PC^{2}(\gamma^{\ast}TM)_{0}$

is a linear isomorphism.

###### Proofs of the conclusions (i)-(ii) and Proposition 2.1.

Because of Proposition 2.8, the conclusions (i) and (ii) follow from Lemma 2.4
and 2.6, respectively.

By Theorem 2.2 the index of the form $\mathcal{I}$ in (2.9) is finite and
equal to the sum of multiplicities of the focal points contained in the open
interval $(0,\tau)$. These and the first two claims in Proposition 2.8 lead to

${\rm Index}(\mathcal{I})=\sum\limits_{t_{0}\in(0,\tau)}\mu^{P}(t_{0}).$

From (2.10), the definition of index and the third claim in Proposition 2.8 it
follows that ${\rm Index}(\mathcal{I})$ is equal to the index of the
restriction of $\tilde{\bf I}^{\gamma}_{P,q}$ to $PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{0}$. The desired conclusion in Proposition 2.1 is
obtained. (Note: Proposition B in §1.2 is not used above.) ∎

Step 2. Let $0=t_{0}<\cdots<t_{m}=\tau$ be a partition as in (ii). By
Corollary 2.3 we have

$\displaystyle\tilde{\bf I}^{\gamma_{0}}_{P,\gamma(t_{1})}(V,V)$
$\displaystyle=$
$\displaystyle\int_{0}^{t_{1}}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},V)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,D_{\dot{\gamma}}^{\dot{\gamma}}V)\right)dt$
(2.12)
$\displaystyle+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),V(0))>0,$
$\displaystyle\quad\forall V\in
PC^{2}_{P\times\gamma(t_{1})}(\gamma_{0}^{\ast}TM)_{0}\setminus\\{0\\},$

where $PC^{2}_{P\times\gamma(t_{1})}(\gamma_{0}^{\ast}TM)_{0}$ is as in (2.3).
Similarly, for each $i=1,\cdots,m-1$, Lemma 2.7 implies

$\displaystyle\tilde{\bf
I}^{\gamma_{i}}_{\gamma(t_{i}),\gamma(t_{i+1})}(V,V)=\int_{t_{i}}^{t_{i+1}}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},V)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,D_{\dot{\gamma}}^{\dot{\gamma}}V)\right)dt>0,$
(2.13) $\displaystyle\quad\forall V\in
PC^{2}(\gamma_{i}^{\ast}TM)\setminus\\{0\\}\;\hbox{satisfying}\;V(t_{i})=0\;\hbox{and}\;V(t_{i+1})=0.$
(2.14)

Let $PC^{2}_{P\times q}(\gamma^{\ast}TM)_{00}$ consist of $V\in
PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}$ such that $V|_{[t_{0},t_{1}]}$ is a
$P$-Jacobi field along $\gamma_{0}$ and that $V|_{[t_{i},t_{i+1}]}$ ($1\leq
i\leq m-1$) are Jacobi fields. It is a vector subspace of $PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{0}$ of finite dimension. We shall prove:

######  Proposition 2.9.

Under the assumptions of Proposition 2.1 let $\dim P>0$. Then the null space
of $\tilde{\bf I}^{\gamma}_{P,q}|_{PC^{2}_{P\times q}(\gamma^{\ast}TM)_{00}}$
coincides with the null space of $\tilde{\bf I}^{\gamma}_{P,q}$, which is
given by $\\{Y\in\mathscr{J}^{P}_{\gamma}\,|\,Y(\tau)=0\\}$. Moreover, the
index of $\tilde{\bf I}^{\gamma}_{P,q}|_{PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{00}}$ is equal to the index of $\tilde{\bf
I}^{\gamma}_{P,q}$, and hence finite.

We first admit it and give:

###### Proof of (2.2).

As remarked below Proposition 2.1 we may assume $\dim P>0$. By the definition
of the Morse index of a bilinear and symmetric form on a vector space we
directly deduce

${\rm Index}\left(\tilde{\bf I}^{\gamma}_{P,q}|_{PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{00}}\right)\leq{\rm Index}\left(\tilde{\bf
I}^{\gamma}_{P,q}|_{PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}}\right)\leq{\rm
Index}\left(\tilde{\bf I}^{\gamma}_{P,q}\right)\leq{\rm Index}\left({\bf
I}^{\gamma}_{P,q}\right).$

The final inequality is actually an equality since $PC^{2}_{P\times
q}(\gamma^{\ast}TM)$ is dense in $W^{1,2}_{P\times q}(\gamma^{\ast}TM)$. (2.2)
and hence (1.30) follows from these and Propositions 2.1 and 2.9 immediately.
∎

Spep 3(Prove Proposition 2.9). To this end we need some lemmas. By Remark A,
$L(\dot{\gamma}(t))\equiv L(\dot{\gamma}(0))$ is a positive constant. We may
define $\textsf{tan}_{\gamma}(X)$ and $\textsf{nor}_{\gamma}(X)$ as in (1.25).
By the proofs of [15, Lemmas 3.16,3.17(ii), Prop.3.18] we immediately obtain
(i), (ii) and (iii) of the following lemma, respectively.

######  Lemma 2.10.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic (and hence $C^{4}$) satisfying
$L\circ\dot{\gamma}\neq 0$. Then:

  1. (i)

$D_{\dot{\gamma}}^{\dot{\gamma}}\big{(}\textsf{tan}_{\gamma}(J)\big{)}=\textsf{tan}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}J)$
and
$D_{\dot{\gamma}}^{\dot{\gamma}}\big{(}\textsf{nor}_{\gamma}(J)\big{)}=\textsf{nor}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}J)$
for $J\in W^{1,2}(\gamma^{\ast}TM)$.

  2. (ii)

A $C^{2}$ vector field $J$ along $\gamma$ is a Jacobi field if and only if
$\textsf{nor}_{\gamma}(J)$ and $\textsf{tan}_{\gamma}(J)$ are Jacobi fields.

  3. (iii)

If $J_{1}$ and $J_{2}$ are $C^{2}$ Jacobi fields along $\gamma$, then the
function
$g_{\dot{\gamma}}(J_{1},D_{\dot{\gamma}}^{\dot{\gamma}}J_{2})-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{1},J_{2})$
is constant.

######  Lemma 2.11.

Under the assumptions of Lemma 2.10, let $PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{0}$ be as in (2.3). Then

$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V(0),W(0))=g_{\dot{\gamma}}(V(0),D_{\dot{\gamma}}^{\dot{\gamma}}W(0)),\quad\forall
V,W\in PC^{2}_{P\times q}(\gamma^{\ast}TM)_{0}.$

Moreover, if $V,W\in\mathscr{J}^{P}_{\gamma}$, then
$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V(t),W(t))=g_{\dot{\gamma}}(V(t),D_{\dot{\gamma}}^{\dot{\gamma}}W(t))$
for all $t\in[0,\tau]$.

###### Proof.

Since $V(0),W(0)\in T_{\dot{\gamma}(0)}P$ and
$\tilde{S}^{P}_{\dot{\gamma}(0)}$ is symmetric we have

$\displaystyle g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V(0),W(0))$
$\displaystyle=$ $\displaystyle g_{\dot{\gamma}}({\rm
tan}^{P}_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V(0)),W(0))$
$\displaystyle=$ $\displaystyle
g_{\dot{\gamma}}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0))$ $\displaystyle=$
$\displaystyle g_{\dot{\gamma}}(V(0),\tilde{S}^{P}_{\dot{\gamma}(0)}(W(0))$
$\displaystyle=$ $\displaystyle g_{\dot{\gamma}}(V(0),{\rm
tan}^{P}_{\dot{\gamma}(0)}(D_{\dot{\gamma}}^{\dot{\gamma}}W(0)))=g_{\dot{\gamma}}(V(0),D_{\dot{\gamma}}^{\dot{\gamma}}W(0)).$

The second claim follows from the first one and Lemma 2.10(iii). ∎

By Remark A, $L\circ\dot{\gamma}(t)$ is a positive constant. Note that
$(\dot{\gamma}^{\bot})_{0}=\\{v\in
T_{\gamma(0)}M\,|\,g_{\dot{\gamma}(0)}(\dot{\gamma}(0),v)=0\\}\supseteq
T_{\gamma(0)}P$. We may choose a normalized orthonormal basis for
$(\dot{\gamma}^{\bot})_{0}$ and $g_{\dot{\gamma}(0)}$, $e_{1},\cdots,e_{n-1}$,
such that $e_{1},\cdots,e_{k}$ is a $g_{\dot{\gamma}(0)}$-orthonormal basis
for $T_{\gamma(0)}P$. Therefore
$e_{1},\cdots,e_{n-1},e_{n}=\dot{\gamma}(0)/\sqrt{L\circ\dot{\gamma}}$ is a
$g_{\dot{\gamma}(0)}$-orthonormal basis for $T_{\gamma(0)}M$.

######  Lemma 2.12.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic which starts from $P$
perpendicularly. Suppose that $g_{\dot{\gamma}(0)}$ (and so each
$g_{\dot{\gamma}(t)}$) is positive definite, and that $e_{1},\cdots,e_{n}$ are
as in the above paragraph. Then there exists a basis of
$\mathscr{J}^{P}_{\gamma}$, $J_{1},\cdots,J_{n}$ satisfying the following
conditions:

$\displaystyle J_{i}(0)=e_{i}\quad\hbox{and}\quad
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=\tilde{S}^{P}_{\dot{\gamma}(0)}(e_{i}),\quad
i=1,\cdots,k,$ (2.15) $\displaystyle J_{i}(0)=0\quad\hbox{and}\quad
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=e_{i},\quad i=k+1,\cdots,n-1,$ (2.16)
$\displaystyle J_{n}(t)=t\dot{\gamma}(t)\;\forall t\;\hbox{and
so}\;D_{\dot{\gamma}}^{\dot{\gamma}}J_{n}(0)=\dot{\gamma}(0)=e_{n},$ (2.17)
$\displaystyle\textsf{nor}_{\gamma}(J_{i})=J_{i},\;i=1,\cdots,n-1.$ (2.18)

###### Proof.

Since for any $v,w\in T_{\gamma(0)}P$ there exists a unique $C^{4}$ Jacobi
field $J$ along $\gamma$ such that $J(0)=v$ and
$D_{\dot{\gamma}}^{\dot{\gamma}}J(0)=w$ (cf. [15, Lemma 3.14]), we have
$C^{4}$ Jacobi fields $J_{i}$ along $\gamma$ satisfying (2.15) and (2.16).
Because $\tilde{S}^{P}_{\dot{\gamma}(0)}(e_{i})\in T_{\gamma(0)}P$ for
$i=1,\cdots,k$, and $e_{i}$ is $g_{\dot{\gamma}(0)}$-orthonormal to
$T_{\gamma(0)}P$ for $i=k+1,\cdots,n$, that is,
$(T_{\gamma(0)}P)^{\perp}_{\dot{\gamma}(0)}={\rm
Span}(\\{e_{k+1},\cdots,e_{n}\\})$, from (2.15)-(2.16) we deduce

$\displaystyle{\rm
tan}_{\dot{\gamma}(0)}\left((D_{\dot{\gamma}}^{\dot{\gamma}}J_{i})(0)\right)=D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=\tilde{S}^{P}_{\dot{\gamma}(0)}(J_{i}(0)),\quad
i=1,\cdots,k,$ $\displaystyle{\rm
tan}_{\dot{\gamma}(0)}\left((D_{\dot{\gamma}}^{\dot{\gamma}}J_{i})(0)\right)={\rm
tan}_{\dot{\gamma}(0)}(e_{i})=0=\tilde{S}^{P}_{\dot{\gamma}(0)}(J_{i}(0)),\quad
i=k+1,\cdots,n-1.$

Namely, $J_{i}$, $i=1,\cdots,n-1$, are $P$-Jacobi fields. By Lemma 2.10(ii),
$\textsf{nor}_{\gamma}(J_{i})$, $i=1,\cdots,n-1$, are Jacobi fields along
$\gamma$. Observe that

$\displaystyle\textsf{nor}_{\gamma}(J_{i})(0)=e_{i}-\frac{g^{F}_{\dot{\gamma}}(e_{i},\dot{\gamma}(0))}{g^{F}_{\dot{\gamma}}(\dot{\gamma}(0),\dot{\gamma}(0))}\dot{\gamma}(0)=e_{i}=J_{i}(0),\quad
i=1,\cdots,k,$
$\displaystyle\textsf{nor}_{\gamma}(J_{i})(0)=0-\frac{g^{F}_{\dot{\gamma}}(0,\dot{\gamma}(0))}{g^{F}_{\dot{\gamma}}(\dot{\gamma}(0),\dot{\gamma}(0))}\dot{\gamma}(0)=0=J_{i}(0),\quad
i=k+1,\cdots,n-1,$

and that Lemma 2.10(i) implies

$\displaystyle(D_{\dot{\gamma}}^{\dot{\gamma}}\textsf{nor}_{\gamma}(J_{i}))(0)$
$\displaystyle=$
$\displaystyle\textsf{nor}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}))(0)$
$\displaystyle=$ $\displaystyle
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)-\frac{g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0),\dot{\gamma}(0))}{g_{\dot{\gamma}}(\dot{\gamma}(0),\dot{\gamma}(0))}\dot{\gamma}(0)$
$\displaystyle=$ $\displaystyle
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=\tilde{S}^{P}_{\dot{\gamma}(0)}(e_{i}),\quad
i=1,\cdots,k,$
$\displaystyle(D_{\dot{\gamma}}^{\dot{\gamma}}\textsf{nor}_{\gamma}(J_{i}))(0)$
$\displaystyle=$
$\displaystyle\textsf{nor}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}))(0)$
$\displaystyle=$ $\displaystyle
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)-\frac{g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0),\dot{\gamma}(0))}{g_{\dot{\gamma}}(\dot{\gamma}(0),\dot{\gamma}(0))}\dot{\gamma}(0)$
$\displaystyle=$ $\displaystyle
D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=e_{i},\quad i=k+1,\cdots,n-1.$

We obtain $\textsf{nor}_{\gamma}(J_{i})=J_{i}$, $i=1,\cdots,n-1$. That is,
$J_{1},\cdots,J_{n-1}$ are $P$-Jacobi fields along $\gamma$ which are
perpendicular to $\dot{\gamma}$. Suppose $\sum^{n}_{i=1}a_{i}J_{i}=0$ for
reals $a_{1},\cdots,a_{n}$. Then $\sum^{n}_{i=1}a_{i}J_{i}(0)=0$ becomes
$\sum^{k}_{i=1}a_{i}e_{i}=0$ by (2.16). So $a_{1}=\cdots=a_{k}=0$. Moreover

$0=D_{\dot{\gamma}}^{\dot{\gamma}}\left(\sum^{n}_{i=1}a_{i}J_{i}\right)=\sum^{n}_{i=1}a_{i}D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}\quad\hbox{and
hence}\quad\sum^{n}_{i=1}a_{i}D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0)=0,$

which implies $\sum^{n}_{i=k+1}a_{i}e_{i}=0$ by (2.16) and therefore $a_{i}=0$
for $i=k+1,\cdots,n$. Hence $J_{1},\cdots,J_{n}$ form a basis of
$\mathscr{J}^{P}_{\gamma}$. ∎

######  Remark 2.13.

Since $J_{1},\cdots,J_{n}$ are linearly independent, if $\gamma(t_{0})$
($t_{0}\in(0,\tau]$) is not a $P$-focal point along $\gamma$, then for any
$(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{R}^{n}\setminus\\{0\\}$ it holds
that $\sum^{n}_{i=1}\lambda_{i}J_{i}(t_{0})\neq 0$. Therefore
$\\{J_{i}(t_{0})\\}^{n-1}_{i=1}$ and $\\{J_{i}(t_{0})\\}^{n}_{i=1}$ are bases
for $(\dot{\gamma}^{\bot})_{t_{0}}=\\{v\in
T_{\gamma(t_{0})}M\,|\,g_{\dot{\gamma}(t_{0})}(\dot{\gamma}(t_{0}),v)=0\\}$
and $T_{\gamma(t_{0})}M$, respectively.

######  Lemma 2.14.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic. (It must be $C^{4}$.) For an
integer $1\leq\ell\leq 4$ let $V(t)$ ($0\leq t\leq\tau$) be a piecewise
$C^{\ell}$ vector field along $\gamma$ such that $V(0)=0$. Then we have a
piecewise $C^{\ell-1}$-smooth vector field $W$ along $\gamma$ with
$W(0)=D_{\dot{\gamma}}^{\dot{\gamma}}V(0)$ and $W(t)=V(t)/t$ for
$0<t\leq\tau$.

###### Proof.

Starting at any basis $e_{1},\cdots,e_{n}$ for $T_{\gamma(0)}M$ we may
construct a $C^{4}$ parallel orthonormal frame fields $E_{1},\cdots,E_{n}$
along $\gamma$. Write $V(t)=\sum_{i}v^{i}(t)E_{i}(t)$ ($0\leq t\leq\tau$),
where each $v^{i}$ is $C^{\ell}$ and satisfies $v^{i}(0)=0$. Clearly, we have
a small $0<\varepsilon<\tau$ such that $v^{i}$ is $C^{\ell}$ on
$[0,\varepsilon]$. Then $v^{i}(t)=t\int^{1}_{0}(v^{i})^{\prime}(ts)ds$ for all
$t\in[0,\varepsilon]$. Define

$w^{i}(t)=\left\\{\begin{array}[]{ll}\int^{1}_{0}(v^{i})^{\prime}(ts)ds&\quad\hbox{if
$t\in[0,\varepsilon/2]$},\\\ v^{i}(t)/t&\quad\hbox{if
$t\in[\varepsilon/2,\tau]$}.\end{array}\right.$

It is easily seen that $w^{i}$ is piecewise $C^{\ell-1}$ and that
$w^{i}(0)=(v^{i})^{\prime}(0)$ and $w^{i}(t)=v^{i}(t)/t$ for $0<t\leq\tau$.
Set $W(t)=w^{i}(t)E_{i}(t)$. It is piecewise $C^{\ell-1}$, and $W(t)=V(t)/t$
for $0<t\leq\tau$. Clearly,
$D_{\dot{\gamma}}^{\dot{\gamma}}V(t)=\sum_{i}({v}^{i})^{\prime}(t)E_{i}$ for
$t\in[0,\tau]$, and so $D_{\dot{\gamma}}^{\dot{\gamma}}V(0)=W(0)$. ∎

The following result of Riemannian geometry (see [27, III. Lemma 2.9], for
example) carry over to the pseudo-Finsler case immediately. (We shall prove it
for completeness.)

######  Lemma 2.15.

Under the assumptions of Lemma 2.12, $\tilde{\bf I}^{\gamma}_{P,q}$ is
positive definite on $PC^{2}_{P\times q}(\gamma^{\ast}TM)$ provided that $P$
has no focal points along $\gamma$ on $(0,\tau]$.

######  Remark 2.16.

As in (2.12) and (2.12), Corollary 2.3 can only lead to that $\tilde{\bf
I}^{\gamma}_{P,q}$ is positive definite on the subspace $PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{0}$, which is not sufficient for completing the proof of
Proposition 2.9. However, if $P$ is a point, as in (2.13)-(2.14) Lemma 2.15
may be derived from Lemma 2.7.

###### Proof of Lemma 2.15.

Step 1. Let $Y_{1},\cdots,Y_{n}$ be a basis of $\mathscr{J}^{P}_{\gamma}$.
(They are actually $C^{4}$.) We claim that for each $X\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$ there exist piecewise $C^{1}$ functions $f^{i}(t)$
($i=1,\cdots,n$) such that $X=\sum^{n}_{i=1}f^{i}Y_{i}$. Indeed, since $P$ has
no focal points along $\gamma$ on $(0,\tau]$ we deduce

$\\{Y_{i}(t)\\}^{n}_{i=1}$ forms a basis for $T_{\gamma(t)}M$ for
$0<t\leq\tau$. (2.19)

Consider the basis of $\mathscr{J}^{P}_{\gamma}$ constructed in Lemma 2.12,
$J_{1},\cdots,J_{n}$. (They are $C^{4}$.) For each $\alpha=k+1,\cdots,n$,
since $J_{\alpha}(0)=0$, by the proof of Lemma 2.14

$\tilde{J}_{\alpha}(t)=\left\\{\begin{array}[]{ll}D_{\dot{\gamma}}^{\dot{\gamma}}J_{\alpha}(0)&\quad\hbox{if
$t=0$},\\\ J_{\alpha}(t)/t&\quad\hbox{if $t\in(0,\tau]$}\end{array}\right.$

defines a $C^{3}$ vector field along $\gamma$. Recall that
$\\{J_{i}(0)=e_{i}\\}^{k}_{i=1}$ and
$\\{D_{\dot{\gamma}}^{\dot{\gamma}}J_{\alpha}(0)=e_{\alpha}\\}^{n}_{\alpha=k+1}$
form a basis for $T_{\gamma(0)}P$ and a basis for
$(T_{\gamma(0)}P)^{\bot}_{\dot{\gamma}(0)}$, respectively. From this and
(2.19) we obtain that

$\\{J_{j}(t),\;\tilde{J}_{\alpha}(t)\,|\,1\leq j\leq k,\;k+1\leq\alpha\leq
n\\}$ (2.20)

forms a basis of $T_{\gamma(t)}M$ for $0\leq t\leq\tau$. Hence each $X\in
PC^{2}_{P\times q}(\gamma^{\ast}TM)$ may be written as

$X(t)=\sum^{k}_{j=1}h^{j}(t)J_{j}(t)+\sum^{n}_{\alpha=k+1}h^{\alpha}(t)\tilde{J}_{\alpha}(t),$

where $h^{j}(t)$ and $h^{\alpha}(t)$ are piecewise $C^{2}$ functions on
$[0,\tau]$. Since $X(0)\in T_{\gamma(0)}P$, we get $h^{\alpha}(0)=0$ and so
$h^{\alpha}(t)=th^{\alpha}_{1}(t)$ with piecewise $C^{1}$ functions
$h^{\alpha}_{1}$ by the proof of Lemma 2.14. These lead to

$X(t)=\sum^{k}_{j=1}h^{j}(t)J_{j}(t)+\sum^{n}_{\alpha=k+1}h^{\alpha}_{1}(t)J_{\alpha}(t)$

on $[0,\tau]$. Note that $J_{i}=\sum^{n}_{j=1}a_{ij}Y_{j}$ for some matrix
$(a_{ij})\in\mathbb{R}^{n\times n}$. The claim at the beginning follows
immediately.

Step 2. Let us take $Y_{1},\cdots,Y_{n}$ as $J_{1},\cdots,J_{n}$. Note that
each $X\in PC^{2}_{P\times q}(\gamma^{\ast}TM)$ may be written as
$X=\sum^{n}_{i=1}f^{i}J_{i}$ for piecewise $C^{1}$ functions $f^{i}(t)$
($i=1,\cdots,n$). We obtain

$\displaystyle
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}X(t),D_{\dot{\gamma}}^{\dot{\gamma}}X(t))$
$\displaystyle=$
$\displaystyle\sum_{i,j}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(f^{i}(t)J_{i}(t)),D_{\dot{\gamma}}^{\dot{\gamma}}(f^{j}(t)J_{j}(t))$
$\displaystyle=$
$\displaystyle\sum_{i,j}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))$
$\displaystyle+\sum_{i,j}f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),D_{\dot{\gamma}}^{\dot{\gamma}}(J_{j}(t))$
$\displaystyle+\sum_{i,j}[(f^{i}(t))^{\prime}f^{j}(t)+(f^{j}(t))^{\prime}f^{i}(t)]g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))$

because
$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))=g_{\dot{\gamma}}(J_{i}(t),D_{\dot{\gamma}}^{\dot{\gamma}}(J_{j}(t)))$
by Lemma 2.11. Moreover

$\displaystyle
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i},D_{\dot{\gamma}}^{\dot{\gamma}}J_{j})$
$\displaystyle=$
$\displaystyle\frac{d}{dt}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i},J_{j})-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}J_{i},J_{j})$
$\displaystyle=$
$\displaystyle\frac{d}{dt}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i},J_{j})-g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},J_{i})\dot{\gamma},J_{j})$

for $1\leq i,j\leq n$ by (1.14). Then

$\displaystyle
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}X(t),D_{\dot{\gamma}}^{\dot{\gamma}}X(t))$
$\displaystyle=$
$\displaystyle\sum_{i,j}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))$
$\displaystyle-\sum_{i,j}f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},J_{i})\dot{\gamma},J_{j})$
$\displaystyle+\sum_{i,j}\frac{d}{dt}\left[f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))\right].$

and so

$\displaystyle\int^{\tau}_{0}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}X(t),D_{\dot{\gamma}}^{\dot{\gamma}}X(t))dt$
$\displaystyle=$
$\displaystyle\sum_{i,j}\int^{\tau}_{0}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))dt$
$\displaystyle-\sum_{i,j}\int^{\tau}_{0}f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},J_{i})\dot{\gamma},J_{j})dt$
$\displaystyle+\sum_{i,j}\left[f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))\right]^{t=\tau}_{t=0}.$

From this we derive

$\displaystyle\tilde{\bf I}^{\gamma}_{P,q}(X,X)$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},X)\dot{\gamma},X)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}X,D_{\dot{\gamma}}^{\dot{\gamma}}X)\right)dt+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(X(0)),X(0))$
$\displaystyle=$
$\displaystyle\sum_{i,j}\int_{0}^{\tau}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))dt$
$\displaystyle+\sum_{i,j}\left[f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))\right]^{t=\tau}_{t=0}+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(X(0)),X(0)).$

Note that $0=X(\tau)=\sum_{i}f^{i}(\tau)J_{i}(\tau)$ and that $X(0)\in
T_{\gamma(0)}P$ implies $f^{i}(0)=0$ for $i=k+1,\cdots,n$. These and (2.15)
lead to

$\displaystyle\sum_{i,j}\left[f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))\right]^{t=\tau}_{t=0}$
$\displaystyle=$
$\displaystyle-\sum^{k}_{i=1}f^{i}(0)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0),X(0))$
$\displaystyle=$
$\displaystyle-\sum^{k}_{i=1}f^{i}(0)g_{\dot{\gamma}}(\tilde{S}^{P}_{\dot{\gamma}(0)}(e_{i}),X(0))$
$\displaystyle=$ $\displaystyle-
g_{\dot{\gamma}}(\tilde{S}^{P}_{\dot{\gamma}(0)}(X(0)),X(0)).$

Hence we arrive at

$\displaystyle\tilde{\bf
I}^{\gamma}_{P,q}(X,X)=\sum_{i,j}\int_{0}^{\tau}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))dt.$
(2.21)

Since $g_{\dot{\gamma}(t)}$ is positive definite on $[0,\tau]$, it follows
that $\tilde{\bf I}^{\gamma}_{P,q}(X,X)\geq 0$ for each $X\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$. If $\tilde{\bf I}^{\gamma}_{P,q}(X,X)=0$ then for any
$1\leq i\leq n$ we have $(f^{i}(t))^{\prime}\equiv 0$, and hence
$f^{i}\equiv{\rm const}$. Therefore $X$ is a $P$-Jacobi field, i.e., $X\in{\rm
Ker}(\tilde{\bf I}^{\gamma}_{P,q})$. But $\gamma(\tau)$ is not a focal point
of $P$ along $\gamma$. Hence $X=0$, i.e., the desired conclusion is proved. ∎

###### Proof of Proposition 2.9.

Let $0=t_{0}<\cdots<t_{m}=\tau$ be the partition above Proposition 2.9. Then
$\gamma((0,t_{1}])$ contains no $P$-focal points, and for each
$i=1,\cdots,m-1$ and any $t_{i}\leq a<b\leq t_{i+1}$ points $\gamma(a)$ and
$\gamma(b)$ are not mutually conjugate along $\gamma|_{[a,b]}$. It follows
that ${\bf T}\cap PC^{2}(\gamma^{\ast}TM)_{00}=\\{0\\}$, where

${\bf T}=\\{Y\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)\,|\,Y(t_{i})=0,\,i=1,\cdots,m-1\\}.$

(Note that $Y(t_{m})=0$ is clear.) For $X\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$, by Remark 2.13 we have a $C^{2}$ $P$-Jacobi field
$Y_{0}$ along $\gamma|_{[0,t_{1}]}$ such that $Y_{0}(t_{1})=X(t_{1})$, and
$C^{2}$ Jacobi fields $Y_{i}$ along $\gamma|_{[t_{i},t_{i+1}]}$ satisfying
$Y_{i}(t_{i})=X(t_{i})$ and $Y_{i}(t_{i+1})=X(t_{i+1})$ for $i=1,\cdots,m-1$.
Gluing these $Y_{i}$ yields a $Y\in PC^{2}_{P\times q}(\gamma^{\ast}TM)_{00}$
to satisfy the condition $Y(t_{i})=X(t_{i})$ ($i=1,\cdots,m-1$). Define
$Z:=X-Y$. Then $Z(t_{i})=0$ for $i=1,\cdots,t_{m}$ (since
$X(t_{m})=0=Y(t_{m})$), that is, $Z\in{\bf T}$. Hence there is the following
direct sum decomposition of vector spaces

$PC^{2}_{P\times q}(\gamma^{\ast}TM)={\bf T}\oplus PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{00}.$

Claim C. ${\bf T}$ and $PC^{2}_{P\times q}(\gamma^{\ast}TM)_{00}$ are
orthonormal with respect to $\tilde{\bf I}^{\gamma}_{P,q}$.

In fact, for $W\in{\bf T}$ and $V\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)_{00}$, since each $V|_{[t_{i},t_{i+1}]}$ is $C^{2}$ and

$\displaystyle
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,D_{\dot{\gamma}}^{\dot{\gamma}}W)=\frac{d}{dt}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,W)-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}V,W)$

by (1.14), as in (1.20) we deduce

$\displaystyle\tilde{\bf I}^{\gamma}_{P,q}(V,W)$ $\displaystyle=$
$\displaystyle\sum^{m}_{i=0}\int_{t_{i}}^{t_{i+1}}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},W)-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}V,W)\right)dt$
$\displaystyle+\sum^{m}_{i=0}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}V,W)\Big{|}^{t_{i+1}}_{t_{i}}+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0))$
$\displaystyle=$
$\displaystyle\int_{0}^{\tau}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},V)\dot{\gamma},W)-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}V,W)\right)dt$
$\displaystyle-
g_{\dot{\gamma}(0)}(D_{\dot{\gamma}}^{\dot{\gamma}}V(0)-\tilde{S}^{P}_{\dot{\gamma}(0)}(V(0)),W(0))+g_{\dot{\gamma}(1)}(D_{\dot{\gamma}}^{\dot{\gamma}}V(\tau),W(\tau))$
$\displaystyle+\sum^{m-1}_{i=1}g_{\dot{\gamma}(t)}\left(D_{\dot{\gamma}}^{\dot{\gamma}}V(t)\big{|}^{t_{i}+}_{t_{i}-},W(t_{i})\right)=0.$

Claim D. $\tilde{\bf I}^{\gamma}_{P,q}$ is positive definite on ${\bf T}$.

Indeed, let $\gamma_{i}=\gamma|_{[t_{i},t_{i+1}]}$, $i=0,1,\cdots,m-1$. If
$W\in{\bf T}\setminus\\{0\\}$, then there is an $i$ such that
$W|_{[t_{i},t_{i+1}]}\neq 0$. If $i=0$, then $\tilde{\bf
I}^{\gamma_{0}}_{P,\gamma(t_{1})}(W|_{[t_{0},t_{1}]},W|_{[t_{0},t_{1}]})>0$ by
Lemma 2.15. (Note that (2.12) and (2.12) are insufficient for proving this).
If $i>0$ then $\tilde{\bf
I}^{\gamma_{i}}_{\gamma(t_{i}),\gamma(t_{i+1})}(W|_{[t_{i},t_{i+1}]},W|_{[t_{i},t_{i+1}]})>0$
by (2.13)-(2.14). It follows from these that

$\tilde{\bf I}^{\gamma}_{P,q}(W,W)=\tilde{\bf
I}^{\gamma_{0}}_{P,\gamma(t_{1})}(W|_{[t_{0},t_{1}]},W|_{[t_{0},t_{1}]})+\sum^{m-1}_{i=1}\tilde{\bf
I}^{\gamma_{i}}_{\gamma(t_{i}),\gamma(t_{i+1})}(W|_{[t_{i},t_{i+1}]},W|_{[t_{i},t_{i+1}]})>0.$

Therefore we have proved Proposition 2.9. ∎

Step 4(Prove the second part in (iii)). We only need to prove (1.31). The
ideas are the same as those of [25, Theorem 1.2]. Let

$PC^{2}_{P\times Q}(\gamma^{\ast}TM)=\\{\xi\in
PC^{2}(\gamma^{\ast}TM)\,|\,\xi(0)\in T_{\gamma(0)}P,\;\xi(\tau))\in
T_{\gamma(\tau)}Q\\}.$

It is dense in $W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$ and so

${\rm Index}\left({\bf I}^{\gamma}_{P,Q}\right)={\rm Index}\left({\bf
I}^{\gamma}_{P,Q}|_{PC^{2}_{P\times Q}(\gamma^{\ast}TM)}\right).$ (2.22)

Choose a complementary subspace $\mathscr{J}^{P}_{\gamma,0}$ of
$\mathscr{J}^{P}_{\gamma}(\tau)$ in $\mathscr{J}^{P}_{\gamma}$, that is,
$\mathscr{J}^{P}_{\gamma}=\mathscr{J}^{P}_{\gamma,0}\oplus\mathscr{J}^{P}_{\gamma}(\tau)$.
By the assumptions of Theorem 1.1,
$\\{X(\tau)\,|\,X\in\mathscr{J}^{P}_{\gamma}\\}\supseteq
T_{\dot{\gamma}(\tau)}Q$. (This may be satisfied if $\gamma(\tau)$ is not a
$P$-focal point. In fact,
$T_{\gamma(\tau)}Q\subset(\dot{\gamma}^{\bot})_{\tau}=\\{v\in
T_{\gamma(\tau)}M\,|\,g_{\dot{\gamma}(\tau)}(\dot{\gamma}(\tau),v)=0\\}$ and
$\\{J_{i}(\tau)\\}^{n-1}_{i=1}$ spans $(\dot{\gamma}^{\bot})_{\tau}$ by Remark
2.13.) Then for each $V\in PC^{2}_{P\times Q}(\gamma^{\ast}TM)$ we have
$W\in\mathscr{J}^{P}_{\gamma,0}$ such that $W(\tau)=V(\tau)$. Since $Z:=V-W\in
PC^{2}_{P\times q}(\gamma^{\ast}TM)$ and $PC^{2}_{P\times
q}(\gamma^{\ast}TM)\cap\mathscr{J}^{P}_{\gamma,0}=\\{0\\}$ imply

$PC^{2}_{P\times Q}(\gamma^{\ast}TM)=PC^{2}_{P\times
q}(\gamma^{\ast}TM)\oplus\mathscr{J}^{P}_{\gamma,0},$

by (1.20) it is easily seen that ${\bf I}^{\gamma}_{P,Q}(V,W)=0$ for any
$V\in\mathscr{J}^{P}_{\gamma,0}$ and $W\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$. That is, $PC^{2}_{P\times q}(\gamma^{\ast}TM)$ and
$\mathscr{J}^{P}_{\gamma,0}$ are orthonormal with respect to ${\bf
I}^{\gamma}_{P,Q}$. Hence

${\rm Index}\left({\bf I}^{\gamma}_{P,Q}|_{PC^{2}_{P\times
Q}(\gamma^{\ast}TM)}\right)={\rm Index}\left({\bf
I}^{\gamma}_{P,Q}|_{PC^{2}_{P\times q}(\gamma^{\ast}TM)}\right)+{\rm
Index}\left({\bf I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma,0}}\right).$

Clearly, ${\bf I}^{\gamma}_{P,Q}|_{PC^{2}_{P\times
q}(\gamma^{\ast}TM)}=\tilde{\bf I}^{\gamma}_{P,q}$ and

${\bf
I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma,0}}(J_{1},J_{2})=g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{1}(\tau)-\tilde{S}^{Q}_{\dot{\gamma}(\tau)}(J_{1}(\tau)),J_{2}(\tau)).$

Note that $\mathscr{J}^{P}_{\gamma}(\tau)\subset{\rm Ker}\left({\bf
I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma}}\right)$. We have ${\rm
Index}\left({\bf I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma,0}}\right)={\rm
Index}\left({\bf I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma}}\right)$ and so

${\rm Index}\left({\bf I}^{\gamma}_{P,Q}|_{PC^{2}_{P\times
Q}(\gamma^{\ast}TM)}\right)={\rm Index}\left(\tilde{\bf
I}^{\gamma}_{P,q}\right)+{\rm Index}\left({\bf
I}^{\gamma}_{P,Q}|_{\mathscr{J}^{P}_{\gamma}}\right).$

(1.31) follows from (2.22) and this. $\Box$

###### Proof of Proposition B.

(i). In fact, for $X\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$, we obtain
$\textsf{tan}_{\gamma}(X)(0)=0$ and $\textsf{tan}_{\gamma}(X)(\tau)=0$ since
$X(0)\in T_{\gamma(0)}P$, $X(\tau)\in T_{\gamma(\tau)}Q$ and $\gamma$ is
$g_{\dot{\gamma}}$-orthonormal (or perpendicular) to $P$ and $Q$. It follows
from these and (1.18) that

$\displaystyle{\bf
I}^{\gamma}_{P,Q}(\textsf{nor}_{\gamma}(V),\textsf{tan}_{\gamma}(W))$
$\displaystyle=$
$\displaystyle\int_{0}^{\tau}g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\textsf{tan}_{\gamma}(W)dt$
(2.23)
$\displaystyle+\int_{0}^{\tau}g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}\textsf{nor}_{\gamma}(V),D_{\dot{\gamma}}^{\dot{\gamma}}\textsf{tan}_{\gamma}(W))dt.$

By Lemma 2.10(i)
$D_{\dot{\gamma}}^{\dot{\gamma}}\big{(}\textsf{tan}_{\gamma}(W)\big{)}=\textsf{tan}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}W)$
and
$D_{\dot{\gamma}}^{\dot{\gamma}}\big{(}\textsf{nor}_{\gamma}(V)\big{)}=\textsf{nor}_{\gamma}(D_{\dot{\gamma}}^{\dot{\gamma}}V)$.
Hence the second term in the right side of (2.23) vanishes. Moreover, by
(1.10)

$g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\dot{\gamma})=-g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\dot{\gamma})$

and so
$g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\dot{\gamma})=0$,
which implies

$\displaystyle\textsf{tan}_{\gamma}\left(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma}\right)=\frac{g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\dot{\gamma}=0.$

It follows that
$R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma}=\textsf{nor}_{\gamma}\left(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma}\right)$
and hence

$g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},\textsf{nor}_{\gamma}(V))\dot{\gamma},\textsf{tan}_{\gamma}(W)=0.$

(ii). By (i), for all $V,W\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$ it holds
that

$\displaystyle{\bf I}^{\gamma}_{P,Q}(V,W)={\bf
I}^{\gamma,\bot}_{P,Q}(\textsf{nor}_{\gamma}(V),\textsf{nor}_{\gamma}(W))+{\bf
I}^{\gamma}_{P,Q}(\textsf{tan}_{\gamma}(V),\textsf{tan}_{\gamma}(W)).$ (2.24)

If $V\in{\rm Ker}({\bf I}^{\gamma,\bot}_{P,Q})$, then
$V=\textsf{nor}_{\gamma}(V)$ and $\textsf{tan}_{\gamma}(V)=0$. It follows that
$V\in{\rm Ker}({\bf I}^{\gamma}_{P,Q})$ and hence ${\rm Ker}({\bf
I}^{\gamma,\bot}_{P,Q})\subset{\rm Ker}({\bf I}^{\gamma}_{P,Q})$. Conversely,
suppose $V\in{\rm Ker}({\bf I}^{\gamma}_{P,Q})$. (2.24) leads to

$\displaystyle{\bf
I}^{\gamma,\bot}_{P,Q}(\textsf{nor}_{\gamma}(V),\textsf{nor}_{\gamma}(W))+{\bf
I}^{\gamma}_{P,Q}(\textsf{tan}_{\gamma}(V),\textsf{tan}_{\gamma}(W))=0,\quad\forall
W\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM).$ (2.25)

Taking $W\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM)^{\bot}$ so that
$W=\textsf{nor}_{\gamma}(W)$ and $\textsf{tan}_{\gamma}(W)=0$, we derive from
(2.25) that $\textsf{nor}_{\gamma}(V)\in{\rm Ker}({\bf
I}^{\gamma,\bot}_{P,Q})$ and therefore

$\displaystyle{\bf
I}^{\gamma}_{P,Q}(\textsf{tan}_{\gamma}(V),\textsf{tan}_{\gamma}(W))=0,\quad\forall
W\in W^{1,2}_{P\times Q}(\gamma^{\ast}TM).$ (2.26)

Choosing $W=V$ we get

$0={\bf
I}^{\gamma}_{P,Q}(\textsf{tan}_{\gamma}(V),\textsf{tan}_{\gamma}(V))=\int^{\tau}_{0}\left(\frac{d}{dt}\frac{g_{\dot{\gamma}}(V,\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\right)^{2}g_{\dot{\gamma}}(\dot{\gamma}(t),\dot{\gamma}(t))dt=L(\dot{\gamma}(0))\int^{\tau}_{0}\left(\frac{d}{dt}\frac{g_{\dot{\gamma}}(V,\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\right)^{2}dt$

because
$g_{\dot{\gamma}(t)}(\dot{\gamma}(t),\dot{\gamma}(t))=L(\dot{\gamma}(t))\equiv
L(\dot{\gamma}(0))$ by [15, Proposition 2.3]. This implies that $t\mapsto
g_{\dot{\gamma}(t)}(V(t),\dot{\gamma}(t))$ is constant. Note that
$g_{\dot{\gamma}(0)}(V(0),\dot{\gamma}(0))=0$ since $V(0)\in T_{\gamma(0)}P$
and $\dot{\gamma}(0)\in TP^{\bot}$. We obtain that
$g_{\dot{\gamma}(t)}(V(t),\dot{\gamma}(t))\equiv 0$ and so
$V=\textsf{nor}_{\gamma}(V)\in{\rm Ker}({\bf I}^{\gamma,\bot}_{P,Q})$. The
first equality is proved.

In order to prove the second equality, observe that ${\rm Index}({\bf
I}^{\gamma,\bot}_{P,Q})\leq{\rm Index}({\bf I}^{\gamma}_{P,Q})$ by the
definition of Morse indexes. Moreover, we have known that both are finite. Let
$X_{i}$, $i=1,\cdots,m={\rm Index}({\bf I}^{\gamma}_{P,Q})$, be linearly
independent vectors in $W^{1,2}_{P\times Q}(\gamma^{\ast}TM)$ such that ${\bf
I}^{\gamma}_{P,Q}$ is negative definite on the subspace generated by
$X_{1},\cdots,X_{m}$. Then for any
$(c_{1},\cdots,c_{m})\in\mathbb{R}^{m}\setminus\\{0\\}$ and
$Y=\sum^{m}_{i=1}c_{i}X_{i}$ it holds that

$\displaystyle 0>{\bf I}^{\gamma}_{P,Q}(Y,Y)$ $\displaystyle=$
$\displaystyle{\bf
I}^{\gamma}_{P,Q}(\textsf{tan}_{\gamma}(Y),\textsf{tan}_{\gamma}(Y))+{\bf
I}^{\gamma}_{P,Q}(\textsf{nor}_{\gamma}(Y),\textsf{nor}_{\gamma}(Y))$
$\displaystyle=$ $\displaystyle{\bf
I}^{\gamma,\bot}_{P,Q}\left(\sum^{m}_{i=1}c_{i}\textsf{nor}_{\gamma}(X_{i}),\sum^{m}_{i=1}c_{i}\textsf{nor}_{\gamma}(X_{i})\right)+L(\dot{\gamma}(0))\int^{\tau}_{0}\left(\sum^{m}_{i=1}c_{i}\frac{d}{dt}\frac{g_{\dot{\gamma}}(X_{i},\dot{\gamma})}{g_{\dot{\gamma}}(\dot{\gamma},\dot{\gamma})}\right)^{2}dt$
$\displaystyle\geq$ $\displaystyle{\bf
I}^{\gamma,\bot}_{P,Q}\left(\sum^{m}_{i=1}c_{i}\textsf{nor}_{\gamma}(X_{i}),\sum^{m}_{i=1}c_{i}\textsf{nor}_{\gamma}(X_{i})\right).$

This shows that $\sum^{m}_{i=1}c_{i}\textsf{nor}_{\gamma}(X_{i})\neq 0$ and
therefore that
$\textsf{nor}_{\gamma}(X_{1}),\cdots,\textsf{nor}_{\gamma}(X_{m})$ are also
linearly independent. It follows that ${\rm Index}({\bf
I}^{\gamma}_{P,Q})\leq{\rm Index}({\bf I}^{\gamma,\bot}_{P,Q})$. ∎

##  3 Related conclusions and consequences, examples

The following two supplement results of Lemma 2.15 will be helpful in
applications.

######  Proposition 3.1.

Let $\gamma:[0,\tau]\to M$ be an $L$-geodesic which starts from $P$
perpendicularly. Suppose that $g_{\dot{\gamma}(0)}$ is positive definite (and
so each $g_{\dot{\gamma}(t)}$ is positive definite and $L\circ\dot{\gamma}$ is
a positive constant by Remark A below (1.14)).

(i)

If $P$ has no focal point along $\gamma$ on $(0,\tau)$, then $\tilde{\bf
I}^{\gamma}_{P,q}$ is positive semi-definite on $PC^{2}_{P\times
q}(\gamma^{\ast}TM)$.

(ii)

If there exists $s\in(0,\tau)$ such that $\gamma(s)$ is a focal point of $P$
along $\gamma|_{[0,s]}$, then $\tilde{\bf I}^{\gamma}_{P,q}$ is negative
definite on $PC^{2}_{P\times q}(\gamma^{\ast}TM)$. Consequently, there exists
a variation curve $\gamma_{\eta}$ of $\gamma$ in $PC^{2}([0,\tau];M,P,q)$ such
that for a sufficiently small $\eta>0$,
$\mathcal{E}_{P,q}(\gamma_{\eta})<\mathcal{E}_{P,q}(\gamma)$, and so

$\mathcal{L}(\gamma)=\sqrt{2\tau\mathcal{E}_{P,q}(\gamma)}>\sqrt{2\tau\mathcal{E}_{P,q}(\gamma_{\eta})}\geq\mathcal{L}(\gamma_{\eta}),$

where $\mathcal{L}(\alpha)=\int^{\tau}_{0}\sqrt{L(\dot{\alpha}(t))}dt$ for
$\alpha\in PC^{2}([0,\tau];M,P,q)$ close to $\gamma$.

###### Proof.

(i). Let $k=\dim\mathscr{J}^{P}_{\gamma}(\tau)$. Choose a basis of
$\mathscr{J}^{P}_{\gamma}(\tau)$, $Y_{1},\cdots,Y_{k}$. Extend them into a
basis of $\mathscr{J}^{P}_{\gamma}$, $Y_{1},\cdots,Y_{n}$, such that
$Y_{k+1}(\tau),\cdots,Y_{n}(\tau)$ are linearly independent. These $Y_{i}$ are
$C^{4}$ actually. Let us prove that for each $X\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$ there exist piecewise $C^{1}$ functions $f^{i}(t)$
($i=1,\cdots,n$) such that $X=\sum^{n}_{i=1}f^{i}Y_{i}$. By the proof of Step
1 of Lemma 2.15, for any small $\varepsilon>0$ there exist piecewise $C^{1}$
functions $g^{i}(t)$ on $[0,\tau-\varepsilon/2]$ ($i=1,\cdots,n$) such that

$X(t)=\sum^{n}_{i=1}g^{i}(t)Y_{i}(t),\quad\forall t\in[0,\tau-\varepsilon/2].$
(3.1)

(Here the condition $L\circ\dot{\gamma}\neq 0$ is used.) For any two
$Y,Z\in\mathscr{J}^{P}_{\gamma}$, by Lemma 2.11 we have
$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}Y(t),Z(t))=g_{\dot{\gamma}}(Y(t),D_{\dot{\gamma}}^{\dot{\gamma}}Z(t))$
for all $t\in[0,\tau]$ and so

$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}Y_{i}(\tau),Y_{j}(\tau))=g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}Y_{j}(\tau),Y_{i}(\tau))=0,\quad
1\leq i\leq k,\;k+1\leq j\leq n.$ (3.2)

Note that $\gamma$ and $Y_{i}$ are actually well-defined on a larger interval
$[0,\tau+\epsilon)$ for some $\epsilon>0$. If
$\sum^{k}_{i=1}a_{i}D_{\dot{\gamma}}^{\dot{\gamma}}Y_{i}(\tau)=0$ for reals
$a_{1},\cdots,a_{k}$, since $\sum^{k}_{i=1}a_{i}Y_{i}(\tau)=0$ and
$\sum^{k}_{i=1}a_{i}Y_{i}$ is a $P$-Jacobi field it must hold that
$\sum^{k}_{i=1}a_{i}Y_{i}(t)=0$ for all $t\in[0,\tau+\epsilon)$ by the
ordinary differential equation theory, which implies $a_{1}=\cdots=a_{k}=0$.
Hence
$D_{\dot{\gamma}}^{\dot{\gamma}}Y_{1}(\tau),\cdots,D_{\dot{\gamma}}^{\dot{\gamma}}Y_{k}(\tau)$
are linearly independent. This fact and (3.2) imply

$D_{\dot{\gamma}}^{\dot{\gamma}}Y_{1}(\tau),\cdots,D_{\dot{\gamma}}^{\dot{\gamma}}Y_{k}(\tau),Y_{k+1}(\tau),\cdots,Y_{n}(\tau)$

are linearly independent. Then for some small $\varepsilon>0$ and any
$t\in[\tau-\varepsilon,\tau)$ the vectors

$\frac{1}{t-\tau}Y_{i}(t),Y_{j}(t),\quad 1\leq i\leq k,\;k+1\leq j\leq n,$

form a basis for $T_{\gamma(t)}M$. For each $i=1,\cdots,k$, define

$\tilde{Y}_{i}(t)=\left\\{\begin{array}[]{ll}D_{\dot{\gamma}}^{\dot{\gamma}}Y_{i}(\tau)&\quad\hbox{if
$t=\tau$},\\\ Y_{i}(t)/(t-\tau)&\quad\hbox{if
$t\in[\tau-\varepsilon,\tau)$}\end{array}\right.$

These are $C^{3}$ vector fields along $\gamma|_{[\tau-\varepsilon,\tau]}$ and
$\\{\tilde{Y}_{i}(t),\;{Y}_{j}(t)\,|\,1\leq i\leq k,\;k+1\leq j\leq n\\}$
forms a basis of $T_{\gamma(t)}M$ for $t\in[\tau-\varepsilon,\tau]$. Hence we
may write

$X(t)=\sum^{k}_{i=1}\tilde{h}^{i}(t)\tilde{Y}_{i}(t)+\sum^{n}_{j=k+1}h^{j}(t)Y_{j}(t)\quad\forall
t\in[\tau-\varepsilon,\tau]$

where $\tilde{h}^{i}(t)$ and $h^{j}(t)$ are piecewise $C^{2}$ functions on
$[\tau-\varepsilon,\tau]$. Since $X(\tau)=0$ implies $\tilde{h}^{i}(\tau)=0$,
as in the proof of Lemma 2.14 we may get piecewise $C^{1}$ functions $h^{i}$
such that $\tilde{h}^{i}(t)=(t-\tau)h^{i}(t)$ for $i=1,\cdots,k$. These lead
to

$X(t)=\sum^{k}_{i=1}h^{i}(t)Y_{i}(t)+\sum^{n}_{j=k+1}h^{j}(t)Y_{j}(t),\quad\forall
t\in[\tau-\varepsilon,\tau].$

It follows from this and (3.1) that $g^{\ell}=h^{\ell}$ on
$[\tau-\varepsilon,\tau-\varepsilon/2]$ for $\ell=1,\cdots,n$ (because
$Y_{1}(t),\cdots,Y_{n}(t)$ form a basis of $T_{\gamma(t)}M$ for $0<t<\tau$).
Define

$f^{i}(t)=\left\\{\begin{array}[]{ll}g^{i}(t)&\quad\hbox{if
$t\in[0,\tau-\varepsilon]$},\\\ h^{i}(t)&\quad\hbox{if
$t\in[\tau-\varepsilon,\tau]$}\end{array}\right.$

for $i=1,\cdots,n$. They are piecewise $C^{1}$ on $[0,\tau]$ and satisfy
$X(t)=\sum^{n}_{i=1}f^{i}(t)Y_{i}(t)$ for $t\in[0,\tau]$.

Let $J_{1},\cdots,J_{n}$ be the basis of $\mathscr{J}^{P}_{\gamma}$
constructed in Lemma 2.12. Then $J_{i}=\sum^{n}_{j=1}a_{ij}Y_{j}$ for some
matrix $(a_{ij})\in\mathbb{R}^{n\times n}$. Therefore each $X\in
PC^{2}_{P\times q}(\gamma^{\ast}TM)$ may be written as
$X=\sum^{n}_{i=1}f^{i}J_{i}$ for piecewise $C^{1}$ functions $f^{i}(t)$
($i=1,\cdots,n$). Since $g_{\dot{\gamma}(t)}$ is positive definite for each
$t\in[0,\tau]$, the desired conclusion may be derived as in the proof of Step
2 of Lemma 2.15.

(ii). The proof is similar to that of the Riemannian-Finsler case (see $e.g$.
[19, Theorem 1.12.13] and [28, Proposition 4.3]). Since $\gamma(s)$ is a focal
point of $P$ along $\gamma|_{[0,s]}$, there is a nonzero $P$-Jacobi field $J$
along $\gamma|_{[0,s]}$ such that $J(s)=0$. Note that $J$ belongs to ${\rm
Ker}(\tilde{\bf I}^{\gamma_{s}}_{P,\gamma(s)})$ with
$\gamma_{s}=\gamma|_{[0,s]}$ and therefore is $C^{4}$. In particular, (1.29)
implies $g_{\dot{\gamma}(t)}(\dot{\gamma}(t),J(t))=0\;\forall t\in[0,s]$. As
above we may assume that $J$ is actually well-defined on a larger interval
$[0,\tau+\delta)$ for some $\delta>0$. Since the unique $V$ satisfying the
Jacobi equation (1.26) near $s\in[0,\tau+\delta)$ and
$V(s)=0=D_{\dot{\gamma}}^{\dot{\gamma}}V(s)$ is $V=0$, we deduce that
$D_{\dot{\gamma}}^{\dot{\gamma}}J(s)\neq 0$ and so
$g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J,D_{\dot{\gamma}}^{\dot{\gamma}}J)(s)\neq
0$. Define $Z(t)=J(t)$ if $t\in[0,s]$, and $Z(t)=0$ if $t\in[s,\tau]$. Then
$Z\in PC^{2}_{P\times q}(\gamma^{\ast}TM)$ and $\tilde{\bf
I}^{\gamma}_{P,q}(Z,Z)=0$. Take a smooth function $\phi$ on $[0,\tau]$ such
that (a) $\phi(s)=1$; (b) $\phi$ has small support about $s$ (so
$\phi(0)=\phi(\tau)=0$); and (c) $0\leq\phi\leq 1$. Let $Y$ be the parallel
transport of $-D_{\dot{\gamma}}^{\dot{\gamma}}J(s)$ along $\gamma$. It is a
$C^{4}$ parallel vector field along $\gamma$. Set $X_{\epsilon}=Z+\epsilon\phi
Y$ for $\epsilon\in{\mathbb{R}}$. Then $X_{\epsilon}\in PC^{2}_{P\times
q}(\gamma^{\ast}TM)$ and

$\displaystyle\tilde{\bf I}^{\gamma}_{P,q}(X_{\epsilon},X_{\epsilon})$
$\displaystyle=$ $\displaystyle\tilde{\bf
I}^{\gamma}_{P,q}(Z,Z)+2\epsilon\tilde{\bf I}^{\gamma}_{P,q}(Z,\phi
Y)+\epsilon^{2}\tilde{\bf I}^{\gamma}_{P,q}(\phi Y,\phi Y)$ $\displaystyle=$
$\displaystyle
2\epsilon\int_{0}^{s}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},Z)\dot{\gamma},\phi
Y)+g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}Z,D_{\dot{\gamma}}^{\dot{\gamma}}(\phi
Y))\right)dt+\epsilon^{2}\tilde{\bf I}^{\gamma}_{P,q}(\phi Y,\phi Y)$
$\displaystyle=$ $\displaystyle 2\epsilon
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}Z,\phi
Y)(s)+\epsilon^{2}\tilde{\bf I}^{\gamma}_{P,q}(\phi Y,\phi Y)$
$\displaystyle=$ $\displaystyle-2\epsilon
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J,D_{\dot{\gamma}}^{\dot{\gamma}}J)(s)+\epsilon^{2}\tilde{\bf
I}^{\gamma}_{P,q}(\phi Y,\phi Y)$

because the integration in the first term in the second line is equal to

$\displaystyle\int_{0}^{s}\left(g_{\dot{\gamma}}(R_{\dot{\gamma}}(\dot{\gamma},Z)\dot{\gamma},\phi
Y)-g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}D_{\dot{\gamma}}^{\dot{\gamma}}Z,\phi
Y)\right)dt+g_{\dot{\gamma}(s)}(D_{\dot{\gamma}}^{\dot{\gamma}}Z(s),\phi(s)Y(s)).$

It follows that $\tilde{\bf I}^{\gamma}_{P,q}(X_{\epsilon},X_{\epsilon})<0$
for small $\epsilon>0$ such that $\epsilon
g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}J,D_{\dot{\gamma}}^{\dot{\gamma}}J)(s)>0$.

Fix such a $X_{\epsilon}$. Take variation curve $\gamma_{\eta}$ of $\gamma$
generated by $X_{\epsilon}$. Then

$\frac{d}{d\eta}|_{\eta=0}\mathcal{E}_{P,q}(\gamma_{\eta})=0\quad\hbox{and}\quad\frac{d^{2}}{d\eta^{2}}|_{\eta=0}\mathcal{E}_{P,q}(\gamma_{\eta})=\tilde{\bf
I}^{\gamma}_{P,q}(X_{\epsilon},X_{\epsilon})<0.$

Hence for a sufficiently small $\eta>0$ it holds that
$\mathcal{E}_{P,q}(\gamma_{\eta})<\mathcal{E}_{P,q}(\gamma)$. By [12,
Proposition 2.3], $g_{v}(v,v)=L(v)\;\forall v\in A$, and hence
$L(\dot{\gamma}(t))=g_{\dot{\gamma}(t)}(\dot{\gamma}(t),\dot{\gamma}(t))>0$
for all $t\in[0,\tau]$ because $L\circ\dot{\gamma}$ is constant and
$g_{\dot{\gamma}(t)}$ is positive definite for each $t\in[0,\tau]$. It follows
that $g_{v}$ is positive definite in a neighborhood of
$\dot{\gamma}([0,\tau])$ in $A$. Hence for sufficiently small $\eta\geq 0$ we
deduce that
$L(\dot{\gamma}_{\eta}(t))=g_{\dot{\gamma}_{\eta}(t)}(\dot{\gamma}_{\eta}(t),\dot{\gamma}_{\eta}(t))>0$
for all $t\in[0,\tau]$. This leads to

$\mathcal{L}(\gamma_{\eta})=\int^{\tau}_{0}\sqrt{L(\dot{\gamma}_{\eta}(t))}dt\leq\sqrt{\tau}\left(\int^{\tau}_{0}L(\dot{\gamma}_{\eta}(t))dt\right)^{1/2}=\sqrt{2\tau\mathcal{E}_{P,q}(\gamma_{\eta})}<\sqrt{2\tau\mathcal{E}_{P,q}(\gamma)}=\mathcal{L}(\gamma)$

because $L\circ\dot{\gamma}$ is constant. ∎

Let $PC^{2}_{P}(\gamma^{\ast}TM)=\\{\xi\in
PC^{2}(\gamma^{\ast}TM)\,|\,\xi(0)\in T_{\gamma(0)}P\\}$. $\tilde{\bf
I}^{\gamma}_{P,q}$ can be extended to a symmetric bilinear form on
$PC^{2}_{P}(\gamma^{\ast}TM)$, denoted by ${\bf I}^{\gamma}_{P}$. We have the
following pseudo-Finsler version of the so-called index lemma in Riemannian-
Finsler geometry (see $e.g$. [27, III. Lemma 2.10], [25, Lemma 4.2] and [28,
Corollary 4.4]). (The proof the Morse index theorem in [25] used [25, Lemma
4.2].)

######  Proposition 3.2.

Under the assumptions of Proposition 3.1, suppose that $P$ has no focal point
along $\gamma$ on $(0,\tau]$. Then for $X\in PC^{2}_{P}(\gamma^{\ast}TM)$
there exists a unique $P$-Jacobi field $Y\in\mathscr{J}^{P}_{\gamma}$ with
$Y(\tau)=X(\tau)$. Moreover $\tilde{\bf I}^{\gamma}_{P}(Y,Y)\leq\tilde{\bf
I}^{\gamma}_{P}(X,X)$, where equality holds if and only if $Y=X$.

###### Proof.

Since $\gamma(\tau)$ is not a $P$-focal point along $\gamma$ the first
assertion holds. By Step 2 in the proof of Lemma 2.15 we may write
$X=\sum^{n}_{i=1}f^{i}J_{i}$ for piecewise $C^{1}$ functions $f^{i}(t)$
($i=1,\cdots,n$). As the proof above (2.21) we may obtain

$\displaystyle\tilde{\bf I}^{\gamma}_{P}(X,X)$ $\displaystyle=$
$\displaystyle\sum_{i,j}\int_{0}^{\tau}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))dt$
$\displaystyle+\sum_{i,j}\left[f^{i}(t)f^{j}(t)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(t)),J_{j}(t))\right]^{t=\tau}_{t=0}+g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(X(0)),X(0)),$
$\displaystyle
g_{\dot{\gamma}(0)}(\tilde{S}^{P}_{\dot{\gamma}(0)}(X(0)),X(0))=\sum_{i,j}f^{i}(0)f^{j}(0)g_{\dot{\gamma}(0)}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(0),J_{j}(0)).$

It follows that

$\displaystyle\tilde{\bf I}^{\gamma}_{P}(X,X)$ $\displaystyle=$
$\displaystyle\sum_{i,j}\int_{0}^{\tau}g_{\dot{\gamma}}((f^{i}(t))^{\prime}J_{i}(t),(f^{j}(t))^{\prime}J_{j}(t))dt$
(3.3)
$\displaystyle+\sum_{i,j}f^{i}(\tau)f^{j}(\tau)g_{\dot{\gamma}}(D_{\dot{\gamma}}^{\dot{\gamma}}(J_{i}(\tau)),J_{j}(\tau)).$

Note that $\gamma(\tau)$ is not a $P$-focal point along $\gamma$ and that the
$P$-Jacobi fields $Y$ and $\sum_{i}f^{i}(\tau)J_{i}$ along $\gamma$ coincide
at $t=\tau$. We deduce that $Y=\sum_{i}f^{i}(\tau)J_{i}$ and therefore

$\sum_{i,j}f^{i}(\tau)f^{j}(\tau)g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}J_{i}(\tau),J_{j}(\tau))=g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}Y(\tau),Y(\tau))$

Moreover, $g_{\dot{\gamma}(t)}$ is positive definite for each $t\in[0,\tau]$.
Then (3.3) yields

$\tilde{\bf I}^{\gamma}_{P}(X,X)\geq
g_{\dot{\gamma}(\tau)}(D_{\dot{\gamma}}^{\dot{\gamma}}Y(\tau),Y(\tau))=\tilde{\bf
I}^{\gamma}_{P}(Y,Y),$

and equality holds if and only if $(f^{i}(t))^{\prime}J_{i}(t)\equiv
0\;\forall i$. The latter means that $X=Y$. ∎

If $L=F^{2}$ for some $C^{6}$ Finsler metric $F$ on $M$, as a consequence of
Theorem 1.1 and splitting lemmas in [22] we may obtain the following
generalization of [19, Theorem 2.5.10], which strengthens Proposition 3.1.

######  Theorem 3.3.

Let $M$ be an $n$-dimensional, connected $C^{7}$ submanifold of
${\mathbb{R}}^{N}$ of dimension greater than $1$, and let $P$ be a $C^{7}$
submanifold of $M$ of dimension less than $\dim M$. For a $C^{6}$ Finsler
metric $F$ on $M$ let $\gamma:[0,\tau]\to M$ be a (nonzero) constant speed
$F$-geodesic which is perpendicular to $P$, that is, a critical point of the
functional

$\alpha\mapsto\mathcal{E}_{P,q}(\alpha)=\frac{1}{2}\int^{\tau}_{0}[F(\alpha(t),\dot{\alpha}(t))]^{2}dt$

on $W^{1,2}([0,\tau];M,P,q):=\\{\alpha\in H^{1}([0,\tau];M)\,|\,\alpha(0)\in
P,\alpha(\tau)=q\\}$ with $q=\gamma(\tau)$.

(i)

If there are no $P$-focal points on $\gamma((0,\tau])$ then there exists a
neighborhood $\mathscr{U}(\gamma)$ of $\gamma$ in $W^{1,2}([0,\tau];M,P,q)$
such that for all $\alpha\in\mathscr{U}(\gamma)$,
$\mathcal{E}_{P,q}(\alpha)\geq\mathcal{E}_{P,q}(\gamma)$, with
$\mathcal{E}_{P,q}(\alpha)=\mathcal{E}_{P,q}(\gamma)$ only for
$\alpha=\gamma$.

(ii)

Let $m>0$ be the number of $P$-focal points on $\gamma((0,\tau))$, each
counted with its multiplicity. Then there exists a topological embedding

$\varphi:B^{m}\to W^{1,2}([0,\tau];M,P,q)$

of the unit ball $B^{m}=\\{x\in{\mathbb{R}}^{m}\,|\,|x|<1\\}$ such that
$\varphi(0)=\gamma$,
$\mathcal{E}_{P,q}(\varphi(x))\leq\mathcal{E}_{P,q}(\gamma)$ and hence
$L(\varphi(x))\leq L(\gamma)$ for all $x\in B^{m}$, with equality only for
$x=0$. Here $L(\alpha)=\int^{\tau}_{0}F(\alpha(t),\dot{\alpha}(t))dt$.

(iii)

If $\gamma$ is also perpendicular to a submanifold $Q\subset M$ at
$q=\gamma(\tau)$ and $\\{X(\tau)\,|\,X\in\mathscr{J}^{P}_{\gamma}\\}\supseteq
T_{\dot{\gamma}(\tau)}Q$, then for $m>0$ as in (ii) there exists a topological
embedding

$\varphi:B^{m}\to W^{1,2}([0,\tau];M,P,Q)$

such that $\varphi(0)=\gamma$,
$\mathcal{E}_{P,Q}(\varphi(x))\leq\mathcal{E}_{P,Q}(\gamma)$ and hence
$L(\varphi(x))\leq L(\gamma)$, for all $x\in B^{m}$, with equality only for
$x=0$.

###### Proof.

In the present case, we choose a $C^{6}$ Riemannian metric $h$ on $M$ such
that $P$ and $Q$ are totally geodesic near $\gamma(0)$ and $\gamma(\tau)$,
respectively. Then using it we define the norm in (1.15). Let

${\bf B}_{2\rho}(T_{\gamma}\Lambda_{P\times q}(M)):=\\{\xi\in
T_{\gamma}\Lambda_{P\times q}(M)\,|\,\|\xi\|_{1}<2\rho\\}$

for $\rho>0$. If $\rho>0$ is small enough, the map

${\rm EXP}_{\gamma}:{\bf B}_{2\rho}(T_{\gamma}\Lambda_{P\times
q}(M))\to\Lambda_{P\times q}(M)$ (3.4)

given by ${\rm EXP}_{\gamma}(\xi)(t)=\exp_{\gamma(t)}(\xi(t))$, is only a
$C^{2}$ coordinate chart around $\gamma$ on $\Lambda_{P\times q}(M)$, where
$\exp$ is the exponential map defined by the Riemannian metric $h$. The
restriction $\mathcal{E}^{X}_{P,q}$ of $\mathcal{E}_{P,q}$ to the Banach
manifold

$C^{1}([0,\tau];M,P,q):=\\{\alpha\in C^{1}([0,\tau];M)\,|\,\alpha(0)\in
P,\alpha(\tau)=q\\}$

with $q=\gamma(\tau)$ is $C^{2}$ near $\gamma$, and the Hessian
$D^{2}\mathcal{E}^{X}_{P,q}(\gamma)$ (a continuous symmetric bilinear form on
$T_{\gamma}C^{1}([0,\tau];M,P,q)$) can be extended into a continuous symmetric
bilinear form on $W^{1,2}_{P\times
q}(\gamma^{\ast}TM)=T_{\gamma}\Lambda_{P\times q}(M)$ by a localization
argument (cf. [21, 22]), which is exactly equal to ${\bf I}^{\gamma}_{P,q}$
defined below (1) by uniqueness. Our localization arguments ([21, Sect.4])
imply that there exists a bounded linear self-adjoint Fredholm operator $B$ on
the Hilbert space $W^{1,2}_{P\times q}(\gamma^{\ast}TM)$ such that ${\bf
I}^{\gamma}_{P,q}(\xi,\eta)=\langle B\xi,\eta\rangle_{1}$ for all $\xi,\eta\in
W^{1,2}_{P\times q}(\gamma^{\ast}TM)$. (Actually, we showed that $B$ is the
sum of a positive definite operator and a compact linear self-adjoint
operator. This can also be seen from (2.6). ) By [7, Proposition 4.5] or [20,
Proposition B.3], $0$ is at most an isolated point of the spectrum $\sigma(B)$
of $B$. Therefore we can choose $\epsilon>0$ such that
$[-2\epsilon,2\epsilon]\cap\sigma(B)=\\{0\\}$. It follows from this and (the
proof of) [10, Theorem 7.1] that there exists a unique direct sum
decomposition of the space

$T_{\gamma}\Lambda_{P\times q}(M)={\bf H}^{-}({\bf
I}^{\gamma}_{P,q})\oplus{\bf H}^{0}({\bf I}^{\gamma}_{P,q})\oplus{\bf
H}^{+}({\bf I}^{\gamma}_{P,q})$ (3.5)

having the following properties:

(a)

The spaces ${\bf H}^{-}({\bf I}^{\gamma}_{P,q})$, ${\bf H}^{0}({\bf
I}^{\gamma}_{P,q})$ and ${\bf H}^{+}({\bf I}^{\gamma}_{P,q})$ are mutually
orthonormal with respect to both ${\bf I}^{\gamma}_{P,q}$ and
$\langle\cdot,\cdot\rangle_{1}$.

(b)

${\bf I}^{\gamma}_{P,q}$ (or $B$) is negative definite on ${\bf H}^{-}({\bf
I}^{\gamma}_{P,q})$, zero on ${\bf H}^{0}({\bf I}^{\gamma}_{P,q})$, and
positive definite on ${\bf H}^{+}({\bf I}^{\gamma}_{P,q})$.

Let $P^{+}$, $P^{-}$ and $P^{0}$ denote projection operators onto ${\bf
H}^{+}({\bf I}^{\gamma}_{P,q})$, ${\bf H}^{-}({\bf I}^{\gamma}_{P,q})$ and
${\bf H}^{0}({\bf I}^{\gamma}_{P,q})$, respectively. Then they satisfy
$P^{0}P^{-}=P^{0}P^{+}=P^{-}P^{+}=0$, $P^{\ast}B=BP^{\ast},\;\ast=+,-,0$, and
$P^{+}+P^{0}+P^{-}$ is equal to the identity mapping on
$T_{\gamma}\Lambda_{P\times q}(M)$.

By [22, Theorem 1.1] we get $\delta\in(0,2\rho]$ and a unique $C^{2}$-map
$\chi:{\bf B}_{\delta}\bigl{(}{\bf H}^{0}({\bf
I}^{\gamma}_{P,q})\bigr{)}\to{\bf H}^{-}({\bf I}^{\gamma}_{P,q})\oplus{\bf
H}^{+}({\bf I}^{\gamma}_{P,q})$ satisfying $\chi(0)=0$ and $d\chi(0)=0$, and
an origin-preserving homeomorphism $\psi$ from ${\bf
B}_{\delta}(T_{\gamma}\Lambda_{P\times q}(M))$ to an open neighborhood of $0$
in $T_{\gamma}\Lambda_{P\times q}(M)$ such that

${\cal E}_{P,q}\circ{\rm
EXP}_{\gamma}\circ\psi(\xi)=\|P^{+}\xi\|_{1}^{2}-\|P^{-}\xi\|_{1}^{2}+{\cal
E}_{P,q}\circ{\rm EXP}_{\gamma}(P^{0}\xi+\chi(P^{0}\xi))$ (3.6)

for all $\xi\in{\bf B}_{\delta}(T_{\gamma}\Lambda_{P\times q}(M))$.

In case (i), Theorem 1.1 and (1.29) imply, respectively, ${\bf H}^{-}({\bf
I}^{\gamma}_{P,q})=\\{0\\}$ and ${\bf H}^{0}({\bf I}^{\gamma}_{P,q})=\\{0\\}$.
Then (3.6) becomes

${\cal E}_{P,q}\circ{\rm EXP}_{\gamma}\circ\psi(\xi)=\|\xi\|_{1}^{2}+{\cal
E}(\gamma),\quad\forall\xi\in{\bf B}_{\delta}(T_{\gamma}\Lambda_{P\times
q}(M)).$

It follows that $\mathscr{U}(\gamma):={\rm EXP}_{\gamma}\circ\psi\left({\bf
B}_{\delta}(T_{\gamma}\Lambda_{P\times q}(M))\right)$ is a desired
neighborhood of $\gamma$.

In case (ii), by Theorem 1.1 we have $\dim{\bf H}^{-}({\bf
I}^{\gamma}_{P,q})=m$. Let $\xi_{1},\cdots,\xi_{m}$ be a unit orthonormal
basis in ${\bf H}^{-}({\bf I}^{\gamma}_{P,q})$. Define $\varphi:B^{m}\to
W^{1,2}([0,\tau];M,P,q)$ by

$\varphi(x_{1},\cdots,x_{m})={\rm
EXP}_{\gamma}\circ\psi\left(\delta\sum^{m}_{i=1}x_{i}\xi_{i}\right).$

This is a topological embedding satisfying $\varphi(0)=\gamma$. By (3.6) we
have

${\cal
E}_{P,q}\big{(}\varphi(x_{1},\cdots,x_{m})\big{)}=-\delta^{2}\sum^{m}_{i=1}x_{i}^{2}+{\cal
E}_{P,q}({\gamma}),\quad\forall(x_{1},\cdots,x_{m})\in B^{m}.$

Note that $L(\varphi(x))^{2}\leq 2\tau\mathcal{E}_{P,q}(\varphi(x))\leq
2\tau\mathcal{E}_{P,q}(\gamma)=L(\gamma)^{2}$. The expected assertions follow
from these immediately.

In order to get (iii), by shrinking $\rho>0$ (if necessary) we have a $C^{2}$
coordinate chart around $\gamma$ on $\Lambda_{P\times Q}(M)$,

${\rm EXP}_{\gamma}:{\bf B}_{2\rho}(T_{\gamma}\Lambda_{P\times
Q}(M))\to\Lambda_{P\times Q}(M)$

given by ${\rm EXP}_{\gamma}(\xi)(t)=\exp_{\gamma(t)}(\xi(t))$. Then by [22,
Theorem 1.1] we get $\delta\in(0,2\rho]$ and a unique $C^{2}$-map $\chi:{\bf
B}_{\delta}\bigl{(}{\bf H}^{0}({\bf I}^{\gamma}_{P,Q})\bigr{)}\to{\bf
H}^{-}({\bf I}^{\gamma}_{P,Q})\oplus{\bf H}^{+}({\bf I}^{\gamma}_{P,Q})$
satisfying $\chi(0)=0$ and $\chi(0)=0$, and an origin-preserving homeomorphism
$\psi$ from ${\bf B}_{\delta}(T_{\gamma_{0}}\Lambda_{P\times Q}(M))$ to an
open neighborhood of $0$ in $T_{\gamma}\Lambda_{P\times Q}(M)$ such that

${\cal E}_{P,Q}\circ{\rm
EXP}_{\gamma}\circ\psi(\xi)=\|P^{+}\xi\|_{1}^{2}-\|P^{-}\xi\|_{1}^{2}+{\cal
E}_{P,Q}\circ{\rm EXP}_{\gamma}(P^{0}\xi+\chi(P^{0}\xi))$

for all $\xi\in{\bf B}_{\delta}(T_{\gamma}\Lambda_{P\times Q}(M))$. Since
$\dim{\bf H}^{-}({\bf I}^{\gamma}_{P,Q})\geq m$ by (1.31), repeating the proof
of (ii) we may obtain the assertions. (Clearly, $m$ may be replaced by $m+{\rm
Index}(\mathcal{A}_{\gamma})$. ) ∎

Finally, we discuss the exponential map of a $C^{6}$ conic Finsler manifold
$(M,L)$ with domain $A\subset TM\setminus 0_{TM}$. For $p\in M$, let
$\mathcal{D}_{p}$ be the set of vectors $v$ in $A\cap T_{p}M$ such that the
unique maximal geodesic $\gamma_{v}:[0,b_{v})\to M$ satisfying
$\gamma_{v}(0)=p$ and $\dot{\gamma}_{v}(0)=v$, where $b_{v}>1$. In the present
case, by the proof of [15, Prop.2.15] we see that $\mathcal{D}_{p}$ is an open
subset of $A\mathop{\cap}T_{p}M$ with property that $v\in\mathcal{D}_{p}$
implies $tv\in\mathcal{D}_{p}$ for all $0<t\leq 1$. Define the exponential map
of $(M,L)$ at $p$ by

$\exp^{L}_{p}:\mathcal{D}_{p}\to M,\quad
v\mapsto\exp^{L}_{p}(v):=\gamma_{v}(1).$

It is $C^{3}$. When $A\cap T_{p}M=T_{p}M\setminus\\{0\\}$ the map
$\exp^{L}_{p}$ can be extended to a $C^{1}$ map in an open neighborhood of
$0_{p}$ by putting $\exp^{L}_{p}(0_{p})=p$ ([15, Prop.2.15]). Moreover, for
$v\in\mathcal{D}_{p}$ and $w\in T_{v}(T_{p}M)$ it holds that
$D\exp^{L}_{p}(v)[w]=J(1)$, where $J$ is the unique Jacobi field along
$\gamma_{v}$ such that $J(0)=0$ and $J^{\prime}(0)=w$ ([15, Prop.3.15]).

Let $P\subset M$ be a $C^{7}$ submanifold of dimension $k<n$ as before. Since
$g_{\lambda v}=g_{v}$ for any $v\in A$ and $\lambda>0$, by (1.11) we see that
$TP^{\bot}$ has the cone property, i.e., $v\in TP^{\bot}$ implies $\lambda
v\in TP^{\bot}$ for all $\lambda>0$. Let $\mathcal{D}=\cup_{p\in
M}\mathcal{D}_{p}$. It is an open subset in $A$ and $\mathcal{D}\cap
TP^{\bot}$ is also an $n$-dimensional submanifold of $TM$. For each $v\in
TP^{\bot}$, since $tv\in\mathcal{D}\cap TP^{\bot}$ for small $t>0$, we have
$\pi(\mathcal{D}\cap TP^{\bot})=\pi(TP^{\bot})=P_{0}$. Therefore the map
$\pi:\mathcal{D}\cap TP^{\perp}\rightarrow P_{0}$ is a submersion. Let

$(\mathcal{D}\cap TP^{\bot})^{\ast}=\left\\{v\in\mathcal{D}\cap
TP^{\bot}\;\big{|}\;\hbox{$g_{v}|_{T_{\pi(v)}P\times T_{\pi(v)}P}$ is
nondegenerate}\right\\}.$

This is an open subset of $\mathcal{D}\cap TP^{\bot}$, and
$v\in(\mathcal{D}\cap TP^{\bot})^{\ast}$ implies $tv\in(\mathcal{D}\cap
TP^{\bot})^{\ast}$ for all $0<t\leq 1$. Denote by $\exp^{LN}$ the restriction
of $\exp^{L}$ to $(\mathcal{D}\cap TP^{\bot})^{\ast}$.

For $v\in(\mathcal{D}\cap TP^{\bot})^{\ast}$ the unique maximal geodesic
$\gamma_{v}$ is orthogonal to $P$ at $\gamma_{v}(0)=\pi(v)$. Let $p=\pi(v)\in
P_{0}$, $u\in T_{v}(\mathcal{D}\cap TP^{\bot})^{\ast}$ and let
$\alpha:(-\epsilon,\epsilon)\to(\mathcal{D}\cap TP^{\bot})^{\ast}$ be a
$C^{4}$ curve such that $\alpha(0)=v$ and $\alpha^{\prime}(0)=u$. Put
$c=\pi\circ\alpha$. Then $\alpha(s)\in\mathcal{D}_{c(s)}\cap T_{c(s)}P^{\bot}$
for $s\in(-\epsilon,\epsilon)$. Define

$\Lambda(t,s)=\exp^{LN}(t\alpha(s))=\gamma_{\alpha(s)}(t),\quad\forall(t,s)\in[0,1]\times(-\epsilon,\epsilon).$

Then $\Lambda(t,0)=\gamma_{v}(t)$, $\Lambda(0,s)=c(s)\in P_{0}$ and
$\partial_{t}\Lambda(0,s)=\alpha(s)$ for all $s$.

######  Proposition 3.4.

$t\mapsto J(t)=\partial_{s}\Lambda(t,0)$ is a $P$-Jacobi field along
$\gamma_{v}$, and $D\exp^{LN}(v)[u]=J(1)$.

###### Proof.

By [15, Prop. 3.13], $J$ is a Jacobi field along $\gamma_{v}$.
$J(1)=\frac{d}{ds}\Lambda(1,s)|_{s=0}=D\exp^{LN}(v)[u]$. By the proof above
[28, (4.4)] we have ${\rm
tan}^{P}_{\dot{\gamma}_{v}(0)}\big{(}D_{\dot{\gamma}_{v}}^{\dot{\gamma}_{v}}J(0))=\tilde{S}^{P}_{\dot{\gamma}_{v}(0)}(J(0))$,
namely, $J$ is a $P$-Jacobi field. ∎

For $0<\tau\leq 1$ let
$\Lambda_{\tau}(t,s)=\exp^{LN}(t\tau\alpha(s))=\Lambda(\tau t,s)$. Denote by
$\tau\alpha$ the curve in $(\mathcal{D}\cap TP^{\bot})^{\ast}$ given by
$s\mapsto\tau\alpha(s)$. Then $\tau\alpha(0)=\tau v$,
$(\tau\alpha)^{\prime}(0)=\tau u$ and

$D\exp^{LN}(\tau v)[\tau u]=\frac{\partial}{\partial
s}\Lambda_{\tau}(1,s)|_{s=0}=\frac{\partial}{\partial s}\Lambda(\tau
1,s)|_{s=0}=J(\tau).$

By the conclusion (i) of Theorem 1.1, if $s>0$ is small enough,
$\gamma_{v}(s)$ is not a $P$-focal point along $\gamma_{v}|_{[0,s]}$, and
therefore $D\exp^{LN}(\tau v)$ is an isomorphism because $\dim(\mathcal{D}\cap
TP^{\bot})^{\ast}=\dim M=n$.

Note that for $v\in(\mathcal{D}\cap TP^{\bot})^{\ast}$ there exists $a_{v}>1$
such that $a_{v}v\notin(\mathcal{D}\cap TP^{\bot})^{\ast}$ and that
$ta_{v}v\in(\mathcal{D}\cap TP^{\bot})^{\ast}$ for all $0<t<1$. By the above
arguments there exists $0<d_{v}\leq a_{v}$ such that $d_{v}v$ depends only on
the ray $[v]:=\\{tv\,|\,t>0\\}$ and that $D\exp^{LN}(tv)$ is an isomorphism
for each $0<t<d_{v}$. Set $(\mathcal{D}\cap
TP^{\bot})^{\ast\ast}:=\\{tv\,|\,v\in(\mathcal{D}\cap
TP^{\bot})^{\ast},\;0<t<d_{v}\\}$. We have

######  Proposition 3.5.

For each $v\in(\mathcal{D}\cap TP^{\bot})^{\ast\ast}$, the restriction of
$\exp^{LN}$ to a small neighborhood of $v$ in $\mathcal{D}\cap TP^{\bot}$ is a
local $C^{3}$ diffeomorphism.

Specially, if $P$ is a point $p$, by the conclusion (ii) of Theorem 1.1 we may
deduce

######  Proposition 3.6.

For each $v\in\mathcal{D}_{p}$, there exists $d_{v}>0$ such that
$td_{v}v\in\mathcal{D}_{p}$ for all $t\in(0,1]$, $d_{v}v$ depends only on the
ray $[v]:=\\{tv\,|\,t>0\\}$ and that the restriction of $\exp^{L}$ to a small
neighborhood of $tv$ in $\mathcal{D}_{p}$ is $C^{3}$ diffeomorphism.

If $L=F^{2}$ for some conic Finsler metric $F$ on $M$ there is a better
result, see [14, Proposition 3.21].

Using the theory developed in this paper it is possible to give
generalizations of related results in Riemannian-Finsler geometry. They will
be studied elsewhere.

Examples. There exist many examples of (conic) pseudo-Finsler metrics and
conic Finsler metrics (cf. [5] and [14, 17]). A popular conic Finsler metric
on $M$ is the Kropina metric $K$ given by $K(v)=-\frac{h(v,v)}{2\omega(v)}$,
where $h$ is a Riemannian metric on $M$ and $\omega$ is a nonvanishing
$1$-form on $M$. $K$ is defined on $TM\setminus{\rm Ker}(\omega)$, and usually
it is restricted to $A=\\{v\in TM\,|\,\omega(v)<0\\}$, on which we have
$K(v)>0$. The pair $(M,K)$ is often called a Kropina space, and is related to
the Zermelo’s navigation problem which consists in finding the paths between
two points $p$ and $q$ that minimize the travel time of a ship or an airship
moving in a wind in a Riemannian landscape $(M,h)$. Recently, Cheng et al.
gave a variational characterization of chains in CR geometry as geodesics of a
certain Kropina metric ([6]). Caponio et al. studied the existence of
connecting and closed geodesics in a manifold endowed with a Kropina metric
([3]).

######  Theorem 3.7 ([3, Theorem 4.4]).

Let $(M,K)$ be above and let $p$, $q$ be two different points in $M$ such that
$\Omega_{p,q}(A)$ the set of the piecewise smooth, $A$-admissible curves
between $p$ and $q$ is nonempty. Assume that $(M,h)$ is complete and there
exists a point $\bar{x}$ and a positive constant $C_{\bar{x}}$ such that
$\|\omega\|_{x}\leq C_{\bar{x}}(d_{h}(x,\bar{x})+1)$. Then for each connected
component $\mathscr{C}$ of $W^{1,2}([0,1];M,p,q)$ there exists a geodesic of
the Kropina space from $p$ to $q$, which is a minimizer of the Kropina length
functional on $\mathscr{C}$, provided that there exists an $A$-admissible
curve in $\mathscr{C}$. Moreover, if $\mathscr{C}$ corresponds to a non-
trivial element of the fundamental group of $M$, a geodesic loop in
$\mathscr{C}$ exists when $p$ and $q$ coincide.

Let $\gamma:[0,1]\to M$ be the geodesic of the Kropina space $(M,K)$ from $p$
to $q$ given by Theorem 3.7. By Theorem 1.1 and (2.2), $p$ has only finitely
many conjugate points along $\gamma$ and

${\rm Index}(\tilde{\bf I}^{\gamma}_{p,q})={\rm Index}({\bf
I}^{\gamma}_{p,q})=\sum\limits_{t_{0}\in(0,\tau)}\mu^{p}(t_{0}).$

We conclude that there is no $s\in(0,1)$ such that $\gamma(s)$ is a conjugate
point of $p$ along $\gamma|_{[0,s]}$ and therefore ${\rm Index}(\tilde{\bf
I}^{\gamma}_{p,q})={\rm Index}({\bf I}^{\gamma}_{p,q})=0$. Otherwise,
Proposition 3.1 leads to a contradiction because $\gamma$ is a minimizer of
the Kropina length functional on $\mathscr{C}$.

Similarly, the closed geodesic obtained by [3, Theorem 5.1] contains no
conjugate points and has index zero.

## Acknowledgments

The author is deeply grateful to the anonymous referee for some interesting
questions, numerous comments and improved suggestions.

## REFERENCES

  * [1] D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry. Springer, Berlin (2000).
  * [2] G. Buttazzo, M. Giaquinta and S. Hildebrandt, _One-dimensional Variational Problems: An Introduction_. Clarendon Press Oxford 1998.
  * [3] E. Caponio, F. Giannoni, A. Masiello, S. Suhr, Connecting and closed geodesics of a Kropina metric. Adv. Nonlinear Stud. 21(2021), no. 3, 683-695. MR4294172
  * [4] E. Caponio, M. Á. Javaloyes, A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes. Math.Ann., 351(2011), no.2, 365-392.
  * [5] E. Caponio, M. Á. Javaloyes, M. Sánchez, Wind Finslerian structures: From Zermelo’s navigation to the causality of spacetimes, preprint (2014), https://arxiv. org/abs/1407.5494.
  * [6] J. H. Cheng, T. Marugame, V.S. Matveev, Chains in CR geometry as geodesics of a Kropina metric. Adv. Math. 350(2019), 973-999. MR3948689
  * [7] J. B. Conway, _A Course in Functional Analysis_ , Springer, New York, 1990. (MR1070713)
  * [8] P. Hartman, A. Wintner, On disconjugate differential systems. Canadian J. Math., 8(1956), 72-81. MR0074595.
  * [9] R.A. Hermann, Differential geometry and the calculus of variations. Mathematics in Science and Engineering, Vol. 49 Academic Press, New York-London 1968 x+440 pp.
  * [10] M.R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math., 1(1951), 525-581.
  * [11] M. Huber, M. Á. Javaloyes, The flag curvature of a submanifold of a Randers-Minkowski space in terms of Zermelo data. Results Math., 77(2022), no. 3, Paper No. 124, 33 pp.
  * [12] M. Á. Javaloyes, Chern connection of a pseudo-Finsler metric as a family of affine connections. Publ. Math. Debrecen, 84(2014), pp. 29-43.
  * [13] M. Á. Javaloyes, Corrigendum to “Chern connection of a pseudo-Finsler metric as a family of affine connections”. Publ. Math. Debrecen, 84(2014), pp. 481-487.
  * [14] M. Á. Javaloyes, M. Sánchez, On the definition and examples of Finsler metrics. Ann. Sc. Norm. Super. Pisa Cl. Sci., (5)13(2014), no. 3, 813-858.
  * [15] M. Á. Javaloyes, B. L. Soares, Geodesics and Jacobi fields of pseudo-Finsler manifolds. Publ. Math. Debrecen, 87(2015), pp. 57-78.
  * [16] M. Á. Javaloyes, Curvature computations in Finsler geometry using a distinguished class of anisotropic connections. Mediterr. J. Math. 17(2020), no. 4, Paper No. 123, 21 pp.
  * [17] M. Á. Javaloyes, M. Sánchez, On the definition and examples of cones and Finsler spacetimes, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas volume 114, Article number: 30(2020).
  * [18] M. Á. Javaloyes, B. L. Soares, Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds. Classical Quantum Gravity, 38(2021), no. 2, Paper No. 025002, 16 pp.
  * [19] W. Klingenberg, _Riemannian Geometry_ , de Gruyter, New York, 1995.
  * [20] G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces I, Discrete and Continuous Dynamical Systems, 33(2013), no.7, pp. 2939-2990. arXiv:1102.2062.
  * [21] G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds, Nonlinear Analysis, 113(2015), 230-282. arXiv:1212.2078 v3, Aug 22, 2013.
  * [22] G. Lu,  Splitting lemmas for the Finsler energy functional on the space of $H^{1}$-curves,  _Proc. London Math. Soc._ , 113(2016), no.3,24-76.
  * [23] J. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N.J., 1973. MR0163331.
  * [24] P. Piccione and D. Tausk, A note on the Morse Index Theorem for Geodesics between Submanifolds in Semi-Riemannian Geometry, Journal of Mathematical Physics, 40(12):6682-6688, 1999.
  * [25] I. R. Peter, On the Morse index theorem where the ends are submanifolds in Finsler geometry, Houston J. Math., 32(2006), no. 4, 995-1009.
  * [26] Hans-Bert, Rademacher, Nonreversible Finsler metrics of positive flag curvature. A sampler of Riemann-Finsler geometry, 261-302, Math. Sci. Res. Inst. Publ., 50, Cambridge Univ. Press, Cambridge, 2004.
  * [27] Takashi Sakai, Riemannian geometry. Translated from the 1992 Japanese original by the author. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996.
  * [28] Bing-Ye Wu, Volume comparison theorem for tubular neighborhoods of submanifolds in Finsler geometry and its applications, Ann. Polon. Math., 112(2014), no. 3, 267-286.

School of Mathematical Sciences, Beijing Normal University
---
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, The People’s Republic of China
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