Transnormal Functions and Focal Varieties on Finsler Manifolds ${}^{*}\,$

††${}^{*}\,$ Project supported by AHNSF (No.2108085MA11).
${}^{\dagger}\,$<EMAIL_ADDRESS>
Yali Chen1${}^{\dagger}\,$, Qun He1

1 School of Mathematical Sciences, Tongji University, Shanghai, 200092, China.

Abstract: In this paper, we study transnormal functions and their level sets
and focal varieties on complete Finsler manifolds. We prove that the focal
varieties of a $C^{2}$ transnormal function are smooth submanifolds and each
regular level set is a tube over either of the focal varieties.

Key words: transnormal function; focal variety; tube; submanifold; Morse
index.

Mathematics Subject Classification (2010): 53C60, 53C42, 34D23.

##  1 Introduction

In Riemannian geometry, the study on isoparametric hypersurfaces has a long
history. In the 1930s, Levi-Civita was the first to give the definition of
isoparametric functions and isoparametric hypersurfaces[1]. B. Segre gave a
complete classification of isoparametric hypersurfaces in an Euclidean
space[2]. E. Cartan began to study the isoparametric hypersurfaces in real
space forms with constant sectional curvature $c$ systematically[3]. Then many
mathematicians started working on isoparametric functions and the
classification of isoparametric hypersurfaces in a space form[4, 5, 6]. The
basic work of isoparametric functions and isoparametric hypersurfaces on
general Riemannian manifolds was established by Q.M.Wang who proved

######  Theorem A.

[7] Let $M$ be a connected, complete, smooth Riemannian manifold and $f$ a
transnormal function on $M$. Then
(1)The focal varieties of $f$ are smooth submanifolds of $M$,
(2)Each regular level set of $f$ is a tube over either of the focal varieties.

Then, Z.Z. Tang and J.Q. Ge made a further study of isoparametric functions
and isoparametric hypersurfaces on general Riemannian manifolds[8, 9] and gave
the relationship between focal sets and focal varieties of transnormal
functions.

In Finsler geometry, [10] introduced the concept of isoparametric functions
and transnormal functions. Let $(N,F,d\mu)$ be an $n$-dimensional Finsler
manifold with volume form $d\mu$. A $C^{1}$ non-constant function $f$ on
$(N,F,d\mu)$ is called isoparametric if it is smooth on $N_{f}$ and there are
functions $a(t)$ and $b(t)$ such that

$\left\\{\begin{aligned} &F^{2}(\nabla f)=a(f),\\\ &\Delta
f=b(f),\end{aligned}\right.$ (1.1)

where $\nabla f$ denotes the gradient of $f$, which is defined by means of the
Legendre transformation, and $\Delta f$ is a nonlinear Finsler-Laplacian of
$f$. A function $f$ satisfying only the first equation of (1.1) is called
transnormal.

Similar to Riemannian geometry, the study of Finsler isoparametric functions
and isoparametric hypersurfaces started from Finsler space forms. The flag
curvature of a Finsler manifold is a natural generalization of the sectional
curvature and we call a (forward) complete and simply connected Finsler
manifold with constant flag curvature a (forward) Finsler space form. There
are many kinds of special (forward) Finsler space forms, such as Minkowski
spaces(with zero flag curvature), Funk spaces(with negative constant flag
curvature) and Randers spheres(with positive constant flag curvature). [10,
11] classified isoparametric hypersurfaces in Minkowski spaces and Funk
spaces. [13, 12] got a complete classification of $d\mu_{BH}-$isoparametric
hypersurfaces in a Randers space form and a Funk-type space. [14] studied
isoparametric hypersurfaces in Finsler space forms by considering focal
points, tubes and parallel hypersurfaces of submanifolds. [15] gave the global
expression of geodesics, isoparametric functions and isoparametric families on
a Randers sphere. In addition to Finsler space forms, there are few results on
isoparametric functions and isoparametric hypersurfaces in other cases. M. Xu
discussed the correspondences of local isoparametric functions and geodesics
by homothetic navigation[16]. Then he used the same technique to study
isoparametric hypersurfaces induced by homothetic navigation in Lorentz
Finsler geometry[17].

In order to give a further study on isoparametric functions and isoparametric
hypersurfaces in Finsler geometry, in this papar, we study transnormal
functions and focal varieties in Finsler manifolds. Unlike the Riemannian
case, besides the regularity of isoparametric functions is worse, the Finsler
metrics are not necessarily reversible, that is, the geodesics are not
necessarily reversible, which cause more difficulties for the research.

For a connected Finsler manifold $(N,F)$, denote the focal varieties of $f$ by
$V=\\{p\in N\big{|}\nabla f(p)=0\\}$ and $M_{\pm}$ the set where $f$ attains
its global maximum value or global minimum value (if either of them exists).
As a generalization of Theorem A, set $J=f(N)$, we have

######  Theorem 1.1.

Let $(N,F)$ be a complete, connected Finsler manifold and $f$ a $C^{2}$
transnormal function on $N$. If $a(t)\neq 0$ for $t\in\textmd{int}J$ and
$a^{\prime}(t)\neq 0$ for $t\in\partial J$, then we have
(1)each focal variety of $f$ is a smooth submanifold of $N$.
(2)each regular level set of $f$ is a tube over either of the focal varieties
with respect to $\overleftarrow{F}$.

######  Remark 1.2.

From the condition in Theorem 1.1, we obviously have $V=M_{\pm}$. Compare with
Theorem A, there are more conditions on $f$ and $a(t)$ in Theorem 1.1 which
hold in Riemannian case naturally. There exist functions which satisfy Theorem
1.1(See Example 6.1). Furthermore, from Lemma 3.3 and Lemma 3.4, we know that
if $a(t)\in C^{2}(\textmd{int}J)$, then $a(t)\neq 0$ for $t\in\textmd{intJ}$
and if $a(t)\in C^{2}(J)$, then $a^{\prime}(t)\neq 0$ for $t\in\partial J$.
There also exist functions which satisfy $a(t)\in C^{2}(J)$ and $f\in
C^{2}(N)$(See Example 6.2).

##  2 Preliminaries

In this section, we will give some definitions and lemmas that will be used in
the proof of our main results.

###  2.1 Finsler manifolds

Let $N$ be an $n$-dimensional manifold and let $TN=\cup_{x\in N}T_{x}N$ be the
tangent bundle of $N$, where $T_{x}N$ is the tangent space at $x\in N$. From
now on, we use the convention of index ranges $1\leq i,j\leq n$, $1\leq
a,b\leq n-1$.

A Finsler metric is a Riemannian metric without quadratic restriction.
Precisely, a function $F(x,y)$ on $TN$ is called a Finsler metric on a
manifold $N$ with local coordinates $(x,y)$, where $x=(x^{i})$ and
$y=y^{i}\frac{\partial}{\partial x^{i}}$ , if it has the following properties:

(i) Regularity: $F(x,y)$ is $C^{\infty}$ on $TN\backslash\\{0\\}$;

(ii) Positive homogeneity: $F(ty)=tF(y),\ \forall t>0,y\in T_{x}N$;

(iii) Strong convexity: The $n\times n$ matrix
$(\frac{\partial^{2}F^{2}}{\partial y^{i}\partial y^{j}}(x,y))(y\neq 0)$ is
positive definite.

The fundamental form $g$ of $(N,F)$ is given by

$g=g_{ij}(x,y)dx^{i}\otimes
dx^{j},~{}~{}~{}~{}~{}~{}~{}g_{ij}(x,y)=\frac{1}{2}[F^{2}]_{y^{i}y^{j}}.$

The projective sphere bundle of $(N,F)$ is defined by

$SN=\\{y\in TN\setminus\\{0\\}\big{|}F(y)=1\\}.$

For a given $x\in N$, the fiber $S_{x}N$ is called the indicatrix of $(N,F)$
at $x$.

The projection $\pi:TN\rightarrow N$ gives rise to the pull-back bundle
$\pi^{\ast}TN$. There exists the unique _Chern connection_ $\nabla$ with
$\nabla\frac{\partial}{\partial x^{i}}=\omega_{i}^{j}\frac{\partial}{\partial
x^{j}}=\Gamma^{i}_{jk}dx^{k}\otimes\frac{\partial}{\partial x^{j}}$ on the
pull-back bundle $\pi^{\ast}TN$. For $X=X^{i}\frac{\partial}{\partial
x^{i}}\in\Gamma(TN)$, the covariant derivative of $X$ along
$v=v^{i}\frac{\partial}{\partial x^{i}}\in T_{x}N$ with respect to a reference
vector $w\in T_{x}N\setminus\\{0\\}$ is defined by

$\nabla^{w}_{v}X(x)=\\{v^{j}\frac{\partial X^{i}}{\partial
x^{j}}(x)+\Gamma^{i}_{jk}(w)v^{j}X^{k}(x)\\}\frac{\partial}{\partial x^{i}}.$

The equation of geodesics can be expressed by
$\nabla^{\dot{\gamma}}_{\dot{\gamma}}\dot{\gamma}\equiv 0$. If a vector field
$J(t)$ along $\gamma(t)$ satisfies

$\displaystyle\nabla_{\dot{\gamma}}^{\dot{\gamma}}\nabla_{\dot{\gamma}}^{\dot{\gamma}}J+\textbf{R}_{\dot{\gamma}}(J)=0,$
(2.1)

then $J(t)$ is called the Jacobi field along $\gamma(t)$, where
$\textbf{R}_{y}$ is the flag curvature tensor which can be expressed by
_geodesic coefficients_

$\displaystyle
G^{i}=\frac{1}{4}g^{il}\left\\{[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}\right\\}$

and

$R^{i}_{k}(y)=2\frac{\partial G^{i}}{\partial x^{k}}-y^{j}\frac{\partial
G^{i}}{\partial x^{j}\partial y^{k}}+2G^{j}\frac{\partial G^{i}}{\partial
y^{j}\partial y^{k}}-\frac{\partial G^{i}}{\partial y^{j}}\frac{\partial
G^{j}}{\partial y^{k}}.$

Let ${\mathcal{L}}:TN\rightarrow T^{\ast}N$ denote the Legendre
transformation, satisfying ${\mathcal{L}}(\lambda y)=\lambda{\mathcal{L}}(y)$
for all $\lambda>0,~{}y\in TN$. Moreover, we know from [18] that

$\displaystyle\mathcal{L}^{-1}(\xi)=F^{*}(\xi)[F^{*}]_{\xi^{i}}(\xi)\frac{\partial}{\partial
x^{i}},\ \ \forall\xi\in T^{*}N\setminus\\{0\\},\ \ \mathcal{L}^{-1}(0)=0,$

where $F^{*}$ is the dual metric of $F$. For a smooth function
$f:N\rightarrow\mathbb{R}$, the gradient vector of $f$ at $x$ is defined as
$\nabla f(x)={\mathcal{L}}^{-1}(df(x))\in T_{x}N$. Hessian of $f$ is defined
by

$\displaystyle\textrm{Hes}f(X,Y):=g_{\nabla
f}(\nabla^{2}f(X),Y)=X(Yf)-(\nabla^{\nabla f}_{X}Y)f,$ (2.2)

where $X,Y\in T_{x}N$, $\nabla^{2}f(x)=\nabla^{\nabla f}_{X}\nabla f\in
T_{x}^{*}N\otimes T_{x}N$ for $x\in N_{f}=\\{x\in N|df(x)\neq 0\\}$. From
[19], $\textrm{Hes}f(X,Y)=\textrm{Hes}f(Y,X)$.

###  2.2 Anisotropic submanifolds of a Finsler manifold

Let $(N,F)$ be an $n$-dimensional Finsler manifold and $\phi:M\to(N,F)$ be an
$m$-dimensional immersion. For simplicity, we will denote $d\phi(x)$ by $x$
and $d\phi X$ by $X$. Let

$\mathcal{V}(M)=\\{(x,\xi)~{}|~{}x\in M,\xi\in T_{x}^{*}N,\xi(X)=0,\forall
X\in T_{x}M\\},$

which is called the normal bundle of $M$ [20]. Set
$\mathcal{N}M={\mathcal{L}}^{-1}(\mathcal{V}(M))\subset TN$. In general,
$\mathcal{N}M$ is not a vector bundle. For any locally defined normal vector
$\textbf{n}\in\mathcal{N}_{x}M$ which satisfies
$g_{\textbf{n}}(\textbf{n},X)=0$ for any $X\in T_{x}M$, we can define a
Riemannian metric $\hat{g}:=\phi^{*}g_{\textbf{n}}$. We call $(M,\hat{g})$ an
anisotropic submanifold of $(N,F)$. For any $X\in T_{x}M$, the _shape
operator_ ${A}_{\mathbf{\textbf{n}}}:T_{x}M\rightarrow T_{x}M$ is defined by

${A}_{\mathbf{\textbf{n}}}X=-\left(\nabla^{\mathbf{\textbf{n}}}_{X}\mathbf{\textbf{n}}\right)^{\top}_{g_{\mathbf{\textbf{n}}}}.$
(2.3)

Furthermore, we denote the unit normal bundle by

$\mathcal{V}^{0}(M)=\\{\nu\in\mathcal{V}(M)|F^{*}(\nu)=1\\}.$

Set $\mathcal{N}^{0}M={\mathcal{L}}^{-1}(\mathcal{V}^{0}(M))$,
$\textbf{n}\in\mathcal{N}^{0}_{x}M$ denotes the unit normal vector of $M$.

If $m=n-1$, there exists a global normal vector field n. Then $(M,\hat{g})$ is
an oriented Riemannian manifold which is called an oriented anisotropic
hypersurface. In this case,

${A}_{\mathbf{\textbf{n}}}X=-\nabla^{\mathbf{\textbf{n}}}_{X}\mathbf{\textbf{n}}.$

###  2.3 Focal sets and tubes

Define the normal exponential map $E:\mathcal{N}M\rightarrow N$ satisfying
$E(x,\textbf{n})=\exp_{x}\textbf{n}$, which is $C^{\infty}$ on
$TN\setminus\\{0\\}$. The focal points of $M$ are the critical values of the
normal exponential map $E$. The focal point $p\in N$ has multiplicity $m$
means that $E_{*}$ at $(x,\textbf{n})$ has nullity $m>0$.

For the immersion $\phi:M\rightarrow(\overrightarrow{N},F)$ with codimension
$n-m\geq 1$, if $n-m>1$, we define $\phi_{s}M(s>0)$ by the map
$\phi_{s}:\mathcal{N}^{0}M\rightarrow N$ satisfying
$\phi_{s}(x,\textbf{n})=E(x,s\textbf{n})$. If $(x,s\textbf{n})$ is not a
critical point of $E$, then $\phi_{s}$ is an $(n-1)$-dimensional immersion in
$(N,F)$. From [14], in the neighborhood $U$ of $x\in\phi(M)$, there exists a
local tube over $U$ belonging to $\phi_{s}M$ for all sufficiently small $s>0$.
We call $\phi_{s}M$ a tube over $M$.

If $M$ is a hypersurface, then $\mathcal{N}^{0}M$ is a double covering of $M$.
In this case, for a given unit normal field n of $M$, define
$\phi_{s}:M\rightarrow(\overrightarrow{N},F)$ by

$\displaystyle\phi_{s}(x)=E(x,s\textbf{n}(x)).$ (2.4)

$\phi_{s}$ is a local immersion for sufficiently small $s$. If there is no
focal point of $M$ on $\phi_{s}M$, $\phi_{s}M$ is a parallel hypersurface over
$M$ as $s>0$ and $M$ is a parallel hypersurface over $\phi_{s}M$ as $s<0$ with
respect to n. For simplicity, we call that $\phi_{s}M$ is a parallel
hypersurface over $M$.

###  2.4 The reverse metric

Let $(N,F)$ be an oriented smooth Finsler manifold. The reverse metric of $F$
is defined by $\overleftarrow{F}(x,y)=F(x,-y)$, where $y\in T_{x}N$, $x\in N$.
Then $\overleftarrow{F}^{*}(\xi)=F^{*}(-\xi)$, where $\xi\in T^{*}_{x}N$.
Obviously, by direct computation,

$\displaystyle\overleftarrow{g}_{ij}(y)=g_{ij}(-y),~{}~{}\overleftarrow{\Gamma}^{i}_{jk}(y)=\Gamma^{i}_{jk}(-y),~{}~{}\overleftarrow{N}^{i}_{j}(y)=-N^{i}_{j}(-y).$

Let $f:N\to\mathbb{R}$ be a $C^{1}$ non-constant function on $(N,F)$. Then
from the definition of gradient vector of $f$, we have

$\overleftarrow{\nabla}(-f)=-\nabla f.$

Then we know if $f$ is a transnormal function with respect to $F$ if and only
if $-f$ is a transnormal function with respect to $\overleftarrow{F}$ which
satisfies

$\overleftarrow{F}(\overleftarrow{\nabla}(-f))=a(f).$

Furthermore, for the level set $M_{t}$, $\textbf{n}=\frac{\nabla f}{F(\nabla
f)}$ and
$\overleftarrow{\textbf{n}}=\frac{\overleftarrow{\nabla}(-f)}{\overleftarrow{F}(\overleftarrow{\nabla}(-f))}$
are the unit normal vector fields of $M$.

##  3 Some properties of transnormal functions

In this section, let $f$ be a transnormal function on a connected and forward
complete Finsler manifold $(N,F)$, which is $C^{1}$ on $N$ and smooth on
$N_{f}$. Recall that a geodesic segment $\gamma(s)$ is called an $f-$segment
if $f(\gamma(s))$ is an increasing function of $s$, where $s$ is an arc-length
parameter.

######  Lemma 3.1.

[10] We have the following results on $\textmd{int}J$.
(1)There is a function $\rho$ defined on $\textmd{int}J$ such that
$\nabla\rho=\frac{\nabla f}{F(\nabla f)}$.
(2)The integral curves of the gradient vector field $\nabla f$ in
$\textmd{int}J$ are all geodesics($f$-segments).
(3)The regular level hypersurfaces $M_{t}=f^{-1}(t)$ are parallel along the
direction of $\nabla f$.
(4)$d(x,M_{t_{2}})=d(M_{t_{1}},y)=\int^{t_{2}}_{t_{1}}\frac{1}{\sqrt{a(t)}}dt$
for any $[t_{1},t_{2}]\in\textmd{int}J$, $x\in M_{t_{1}}$, $y\in M_{t_{2}}$
and the $f$-segments are the shortest curves among all curves connecting
$M_{t_{1}}$ and $M_{t_{2}}$.

######  Lemma 3.2.

Suppose that $\beta$ is the only critical value of $f$ in
$[\alpha,\beta]\subseteq J$, then

$d(M_{\alpha},M_{\beta})=\int^{\beta}_{\alpha}\frac{df}{\sqrt{a(f)}}$ (3.1)

and the improper integral in (3.1) converges.

###### Proof.

Let $\\{p\\}$ be any point in $M_{\beta}$. Since $\\{f\geqslant\beta\\}$ in
$N$ is closed and $N$ is connected, then there are points
$\\{p_{i}\\}\in\\{f<\beta\\}$ in $N$ such that
$\lim\limits_{i\rightarrow\infty}d(p,p_{i})=0$. Set $f(p_{i})=c_{i}$.

Centered at $p$, there exists a normal geodesic $\gamma(s)=\exp_{p}sX$ through
$p_{i}$ which satisfies $\gamma(0)=p$, $\gamma^{\prime}(0)=X$, where $X\in
S_{p}N$. Let $U(p)$ be a compact neighborhood covering the geodesic sphere
determined by $\gamma(s)$ with radius less than $\max d(p,p_{i})$ and
$\lambda_{0}=\max\limits_{U(p)}\frac{\overleftarrow{F}(y)}{F(y)}$. Then we
have

$\displaystyle d(p_{i},p)\leq
L(\overleftarrow{\gamma}(s))=\int^{s_{i}}_{0}|F(-\gamma^{\prime}_{i}(s))|ds=\int^{s_{i}}_{0}|\overleftarrow{F}(\gamma^{\prime}_{i}(s))|ds\leq\lambda_{0}\int^{s_{i}}_{0}|F(\gamma^{\prime}_{i}(s))|ds.$
(3.2)

Hence,

$\displaystyle d(p_{i},p)\leq\lambda_{0}s_{i}=\lambda_{0}d(p,p_{i}).$ (3.3)

Namely,

$\displaystyle\lim_{i\rightarrow\infty}d(p_{i},p)=0.$ (3.4)

Let $\sigma_{i}(t)$ be any piecewise $C^{1}$ curve which goes from
$M_{\alpha}$ to $p$ through $p_{i}$. For every $p_{i}\in(\alpha,\beta)$,
$\sigma_{1_{i}}$ is a curve from $M_{\alpha}$ to $M_{p_{i}}$ and
$\sigma_{2_{i}}$ is a curve from $M_{p_{i}}$ to $p$ such that
$\sigma_{i}=\sigma_{1_{i}}\cup\sigma_{2_{i}}$. Furthermore, $\inf
L(\sigma_{i})=\inf L(\sigma_{1_{i}})+\inf L(\sigma_{2_{i}})$. Combine (3.4),
we obtain

$\displaystyle
d(M_{\alpha},M_{\beta})=\lim_{i\rightarrow\infty}d(M_{\alpha},M_{p_{i}}).$

This completes the proof. ∎

######  Lemma 3.3.

$f$ has no critical value(i.e. $a(t)\neq 0$) in $\textmd{int}J$ when $a(t)\in
C^{2}(\textmd{int}J)$.

###### Proof.

Suppose $\beta$ the only critical value of $f$ in $\textmd{int}J$, then we
have $a(\beta)=0$ and $a^{\prime}(\beta)=0$. If
$t\in[\beta-\varepsilon,\beta]\subseteq\textmd{int}J$, from Taylor’s theorem,

$\displaystyle
a(t)=a(\beta)+a^{\prime}(\beta)(t-\beta)+\frac{1}{2}a^{\prime\prime}(\xi)(t-\beta)^{2}=\frac{1}{2}a^{\prime\prime}(\xi)(t-\beta)^{2},\
\ \ \ \xi\in(t,\beta).$ (3.5)

Since $a(t)\in C^{2}([\beta-\varepsilon,\beta])$, there exists a constant $K$
such that $|a^{\prime\prime}(\xi)|\leq 2K$. From (1.1) and (3.5), $0\leq
a(t)\leq K(t-\beta)^{2}$, then the integral in (3.1) would diverge. There
exists a contradiction with Lemma 3.2. ∎

Combine Lemma 3.2 and $a(t)\neq 0$ on $\textmd{int}J$,
$s(t)=\int_{t_{0}}^{t}\frac{1}{\sqrt{a(x)}}dx$ is convergent, where
$t_{0},t\in J$. Hence, from (2.4), set $r$ the distance from $M_{c}$ to
$M_{\pm}$, the smooth map $\phi_{r}:M_{c}\rightarrow N$ is

$\displaystyle\phi_{r}(x)=\exp_{x}(r\textbf{n}).$ (3.6)

It’s easy to show that the focal points of $M_{c}$ are also the critical
values of $\phi_{r}$. Hence, $\phi_{r}$ is called the focal map of $M_{c}$.
Suppose $\max J=\beta<\infty$ if the maximal value of $J$ exists,
$M_{+}=f^{-1}(\beta)=\phi_{r}M_{c}$. Note that fix a $c$ in $\textmd{int}J$,
the hypersurface $M_{c}$ may not be connected. Define
$\textmd{Foc}_{t}(f)=\\{\phi_{s(t)}(x)\in N\big{|}d\phi_{s(t)}X=0,X\in
T_{x}M_{c}\setminus\\{0\\}\\}$ the focal set of the level set
$M_{t}=\phi_{s(t)}M=f^{-1}(t)$. Since $\phi_{s(t)}:M\rightarrow M_{t}$ is a
diffeomorphism if $s<r$, we easily get $\textmd{Foc}(f)\subseteq M_{\pm}$. For
two different level hypersurfaces $M_{t_{1}}$ and $M_{t_{2}}$, from Lemma 3.2,
we have $\textmd{Foc}_{t_{1}}(f)=\textmd{Foc}_{t_{2}}(f)$. Hence, we can
simply write $\textmd{Foc}_{t}(f)$ as $\textmd{Foc}(f)$.

######  Lemma 3.4.

If $a(t)\in C^{2}(J)$, then $a^{\prime}\neq 0$ on $\partial J$. Moreover,
$a^{\prime}<0$(resp.$a^{\prime}>0$) at the maximal(resp.minimal) point of $J$
if it exists.

###### Proof.

Suppose $\beta$ the only critical value of $f$ in $[\beta-\varepsilon,\beta]$
for small $\varepsilon$. From Taylor’s theorem,

$\displaystyle
a(t)=a(\beta)+a^{\prime}(\beta)(t-\beta)+\frac{1}{2}a^{\prime\prime}(\xi)(t-\beta)^{2}=a^{\prime}(\beta)(t-\beta)+\frac{1}{2}a^{\prime\prime}(\xi)(t-\beta)^{2},$

where $t\in(\beta-\varepsilon,\beta)$. Firstly, we can prove
$a^{\prime}(\beta)\neq 0$. If $a^{\prime}(\beta)=0$, similar to the analysis
in Lemma 3.3, there exists a contradiction with Lemma 3.2. Moreover, we claim
$a^{\prime}(\beta)<0$. If $a^{\prime}(\beta)>0$, we have $a(t)<0$ due to the
continuity of $a(t)$. This is in contradiction with $a(t)\geq 0$. ∎

From Lemma 3.1, Lemma 3.2 and the first variation formula for arc length, we
have

######  Lemma 3.5.

(1) Given a point $x$ in $f^{-1}(\textmd{int}J)$, there is a unique maximal
$f$-segment through $x$.
(2) Every maximal $f$-segment intersects each regular level set of $f$ exactly
once and orthogonally.
(3) Let $\gamma(s)$ be an $f$-segment, then for $s_{1}<s_{2}$,
$d(M_{f(\gamma(s_{1}))},M_{f(\gamma(s_{2}))})=s_{2}-s_{1}$.
(4) The $f$-segment $\gamma(s)$ is also perpendicular to
$M_{+}$(resp.$M_{-}$).

###### Proof.

(1), (2), (3) are direct corollaries of Lemma 3.1, Lemma 3.2. Here, we focus
on the proof of (4).

Set $\sigma(t)$ any curve on $M_{+}$ through $p$ satisfying $\sigma(0)=p$.
$\gamma(s)$ is a geodesic which intersects $M_{c}$ and $M_{+}$ at
$x=\gamma(0)$ and $p=\gamma(r)$, respectively. Let $\widetilde{\sigma}(t)$ be
any curve on $M_{c}$ through $x$ satisfying $\widetilde{\sigma}(0)=x$.
Obviously,
$\langle\widetilde{\sigma}^{\prime}(0),\gamma^{\prime}(0)\rangle=0$. We only
need to prove $\langle\sigma^{\prime}(0),\gamma^{\prime}(r)\rangle$=0.

Set $\Phi(s,t)$ any smooth variation which satisfies $\Phi(s,0)=\gamma(s)$,
$\Phi(0,t)=\widetilde{\sigma}(t)$ and $\Phi(r,t)=\sigma(t)$. Set
$V(s)=\frac{\partial\Phi}{\partial t}\big{|}_{t=0}$ and
$T(s)=\frac{\partial\Phi}{\partial s}\big{|}_{t=0}$ which satisfy

$V(0)=\widetilde{\sigma}^{\prime}(0),\ \ \ \ V(r)=\sigma^{\prime}(0),\ \ \ \
T(0)=\gamma^{\prime}(0),\ \ \ \ T(r)=\gamma^{\prime}(r),\ \ \ \ \langle
T(0),V(0)\rangle=0.$

From the first variation formula for arc length, we know

$\displaystyle 0=L^{\prime}(0)=\langle
T(s),V(s)\rangle\big{|}^{s=r}_{s=0}-\int^{r}_{0}g_{T(s)}(\nabla^{T(s)}_{T(s)}T(s),V(s))ds=\langle
V(r),T(r)\rangle=\langle\sigma^{\prime}(0),\gamma^{\prime}(r)\rangle.$

This completes the proof of (4). ∎

##  4 Focal varieties and tubes

###  4.1 Hessian

From now on, suppose $f\in C^{2}(N)$. Hessian on $N_{f}$ is defined by (2.2).
Now, we need to give the definition of Hessian on focal varieties which
satisfies $\nabla f=0$. In this case, $g_{\nabla f}$ is not defined. For $p\in
N$, $U_{p}(x^{i})$ is a local coordinate system at $p$. For the convenience of
writing, we use $f_{ij}$ and $f_{k}$ instead of $\frac{\partial^{2}f}{\partial
x^{i}\partial x^{j}}$ and $\frac{\partial f}{\partial x^{k}}$, respectively.

For any $p\in M_{+}$, $X_{p},Y_{p}\in T_{p}M_{+}$, extend $X_{p},Y_{p}$ to
$U_{p}$. Set $q\in U_{p}\setminus M_{+}$. Define

$\displaystyle\textmd{Hes}f(X_{p},Y_{p})$ $\displaystyle=\lim_{q\rightarrow
p}\textmd{Hes}f(X_{q},Y_{q})$ $\displaystyle=\lim_{q\rightarrow
p}X^{i}_{q}Y^{j}_{q}f_{ij}(q)-f_{k}(q)\Gamma^{k}_{ij}(\nabla
f)X^{i}_{q}Y^{j}_{q}.$

Since $\lim\limits_{q\rightarrow p}f_{k}(q)=0$ and $\Gamma^{k}_{ij}(\nabla f)$
is homogenous of degree $0$ which is bounded in $U_{p}\setminus M_{+}$, we
have

$\displaystyle\textmd{Hes}f(X_{p},Y_{p})=\lim_{q\rightarrow
p}X^{i}_{q}Y^{j}_{q}f_{ij}(q)=X^{i}_{p}Y^{j}_{p}f_{ij}(p).$ (4.1)

Obviously, $\textmd{Hes}f(X_{p},Y_{p})$ is self-adjoint. For the convenience
of writing, we will use $H$ instead of Hes in the following. Then, for
$\xi=\mathcal{L}(X)$, we define

$\displaystyle H^{*}(\xi)(Y)=H(X,Y).$ (4.2)

Using $H$ and $H^{*}$, for any $p\in N$, denote

$E_{0}(p)=\\{X\in T_{p}N\big{|}~{}H(X)=0\\},$ $E^{*}_{0}(p)=\\{\xi\in
T^{*}_{p}N\big{|}~{}H^{*}(\xi)=0\,\\},$ $E_{\mu}^{*}(p)=\\{\xi\in
T^{*}_{p}N\big{|}~{}H^{*}(\xi)=\mu\xi\\},$ $E^{\bot}_{0}(p)=\\{\xi\in
T^{*}_{p}N\big{|}~{}\xi(X)=0,X\in E_{0}(p)\\}$

and $\mathcal{U}E$ as the unit vectors in a vector space $E$ with respect to
$F$(or $F^{*}$).

######  Remark 4.1.

Since $H^{*}$ is not necessarily a linear operator, $E^{*}_{\mu}$ is not
necessarily a vector space.

Set $\gamma_{x}(s)=\exp_{x}s\textbf{n}$ a normal geodesic in $N$, where
$x=\gamma_{x}(0)\in M_{c}$ and $\textbf{n}=\gamma^{\prime}_{x}(0)$ is the unit
normal vector of $M_{c}$. Define the map $\sigma^{*}:M_{c}\rightarrow T^{*}N$
which satisfies $\sigma^{*}(x)=\mathcal{L}(\gamma^{\prime}_{x}(r))$. For $p\in
M_{+}$, set $A_{p}=\phi^{-1}_{r}(p)\subseteq M_{c}$. The relationship between
$\sigma^{*}A_{p}$, $\mathcal{U}E^{*}_{\lambda}(p)$ and $E^{\bot}_{0}(p)$ can
be obtained.

######  Lemma 4.2.

(1) $\sigma^{*}A_{p}\subseteq\mathcal{U}E^{*}_{\lambda}(p)$, where
$\lambda=\frac{1}{2}a^{\prime}(\beta)$. (2) $E^{*}_{\lambda}(p)\subseteq
E^{\bot}_{0}(p)$.

###### Proof.

(1) For any $\tau\in\sigma^{*}A_{p}$, there exist $x\in A_{p}$ and
$\textbf{n}\in\mathcal{N}_{x}^{0}M_{c}$ such that
$\gamma_{x}(s)=\exp_{x}s\textbf{n}$ satisfying $p=\gamma_{x}(r)$ and
$\eta=\mathcal{L}^{-1}(\tau)=\gamma_{x}^{\prime}(r)$. Let $\tilde{U}(p)$ be a
neighborhood of $p$ in $N$ and $T(s)=\gamma_{x}^{\prime}(s)$. Extend $T$ to
the vector field $\widetilde{T}$ in $\widetilde{U}(p)\subseteq N$ such that
$\widetilde{T}=\frac{\nabla f}{\sqrt{a}}$ in $\widetilde{U}(p)\setminus M_{+}$
and $\widetilde{T}\big{|}_{\gamma(s)}=T$. For any $Y\in TN$,
$q\in\widetilde{U}(p)\setminus M_{+}$, from (4.2),

$\displaystyle H^{*}(\tau)(Y)$
$\displaystyle=H(\eta,Y)=\lim\limits_{q\rightarrow
p}g_{\widetilde{T}}(\nabla_{\widetilde{T}}^{\widetilde{T}}(\sqrt{a}\widetilde{T}),Y)$
$\displaystyle=\lim\limits_{q\rightarrow
p}\widetilde{T}(\sqrt{a})g_{\widetilde{T}}(\widetilde{T},Y)+\lim\limits_{q\rightarrow
p}\sqrt{a}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{\widetilde{T}}\widetilde{T},Y)$
$\displaystyle=\lim\limits_{q\rightarrow
p}\frac{1}{2\sqrt{a}}a^{\prime}g_{\widetilde{T}}(\widetilde{T},\nabla
f)g_{\widetilde{T}}(\widetilde{T},Y)=\frac{1}{2}a^{\prime}(\beta)\tau(Y).$

Hence, $H^{*}(\tau)=\lambda\tau$, where
$\lambda=\frac{1}{2}a^{\prime}(\beta)$.

(2) For any $\xi\in E^{*}_{\lambda}(p)$, we have $H^{*}(\xi)=\lambda\xi$,
where $\lambda\neq 0$. For any $X\in E_{0}$, we have

$\displaystyle 0=H^{*}(\xi)(X)=H(\mathcal{L}^{-1}(\xi),X)=\lambda\xi(X).$

Hence, $\xi(X)=0$. Then we have $\xi\in E^{\bot}_{0}(p)$. Namely,
$E^{*}_{\lambda}(p)\subseteq E^{\bot}_{0}(p)$. ∎

###  4.2 Tubes of smooth submanifolds

Let $U$ be an open subset in $M_{c}$ such that rank$(\phi_{r}\big{|}_{U})=k$
and $W=\phi_{r}(U)$ a smooth $k$-dimentional submanifold of $N$. Set $x\in U$,
$p=\phi_{r}(x)$. In this section, we denote $A_{p}=\phi^{-1}_{r}(p)\cap U$.
From Lemma 3.5, we obviously know that $\sigma^{*}$ maps $A_{p}$ to
$\mathcal{V}^{0}_{p}W$. Consider the relationship between $T_{p}W$,
$E_{0}(p)$, $\mathcal{V}^{0}_{p}W$ and $\mathcal{U}E^{\bot}_{0}(p)$, we have

######  Lemma 4.3.

(1) $T_{p}W\subseteq E_{0}(p)$, (2)
$\mathcal{V}^{0}_{p}W=\mathcal{U}E^{\bot}_{0}(p)$.

###### Proof.

(1) Set $\forall X\in T_{p}W$. Extend $X$ to $\widetilde{U}(p)$ such that
$g_{\widetilde{T}}(\widetilde{T},X)=0$, where $\widetilde{T}$ and
$\widetilde{U}(p)$ are defined in the proof of (1) in Lemma 4.2. For any $Y\in
TN$ and $q\in\widetilde{U}(p)\setminus M_{+}$, from (4.1),

$\displaystyle H(X,Y)(p)$ $\displaystyle=\lim_{q\rightarrow
p}H(X,Y)(q)=\lim_{q\rightarrow
p}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{X}(\sqrt{a}\widetilde{T}),Y)$
$\displaystyle=\lim_{q\rightarrow
p}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}(\sqrt{a}\widetilde{T}),X)$
$\displaystyle=\lim_{q\rightarrow
p}Y(\sqrt{a})g_{\widetilde{T}}(\widetilde{T},X)+\lim_{q\rightarrow
p}\sqrt{a}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}\widetilde{T},X)$
$\displaystyle=\lim_{q\rightarrow
p}\sqrt{a}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}\widetilde{T},X)$

Since

$\displaystyle
g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}\widetilde{T},X)+g_{\widetilde{T}}(\widetilde{T},\nabla^{\widetilde{T}}_{Y}X)+2C_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}\widetilde{T},\widetilde{T},X)=Yg_{\widetilde{T}}(\widetilde{T},X),$
(4.3)

we have

$\lim_{q\rightarrow
p}g_{\widetilde{T}}(\nabla^{\widetilde{T}}_{Y}\widetilde{T},X)=-\lim_{q\rightarrow
p}g_{\widetilde{T}}(\widetilde{T},\nabla_{Y}^{\widetilde{T}}X)=-g_{\eta}(\eta,\nabla_{Y}^{\eta}X).$

In addition, $\lim\limits_{q\rightarrow p}\sqrt{a}=0$. Hence, we have
$H(X,Y)(p)=0$ for any $Y\in TN$. Namely, $T_{p}W\subseteq E_{0}(p)$.

(2) From Lemma 3.5, $\sigma^{*}(A_{p})\subseteq\mathcal{V}^{0}_{p}W$. Combine
Lemma 4.2, $\sigma^{*}(A_{p})\subseteq\mathcal{U}E^{\bot}_{0}(p)$. Namely,
$\sigma^{*}(A_{p})\subseteq\mathcal{V}^{0}_{p}W\cap\mathcal{U}E^{\bot}_{0}(p)$.
Since $\sigma^{*}$ is injective and $A_{p}=\phi^{-1}_{r}(p)\cap U$ is
$(n-1-k)$-dimensional, $\dim\sigma^{*}(A_{p})=n-1-k$. In addition,
$\mathcal{V}_{p}W$ and $E^{\bot}_{0}(p)$ are two $(n-k)$-dimensional linear
spaces. Hence, $\mathcal{V}^{0}_{p}W=\mathcal{U}E^{\bot}_{0}(p)$. ∎

######  Remark 4.4.

(2) in Lemma 4.3 means $E_{0}(p)=T_{p}W$.

We can also express $\sigma^{*}(x)=\mathcal{L}(\gamma^{\prime}_{x}(r))$ as
$\sigma^{*}=\mathcal{L}\circ\Phi_{r}\circ\textbf{n}$, where $\textbf{n}(x)$
embeds $M_{c}$ into the unit tangent bundle $SN$ and $x\in M_{c}$. Define
$\Phi_{s}:SN\rightarrow SN$ by
$\Phi_{s}(x,\textbf{n})=(\exp_{x}s\textbf{n},\frac{d}{ds}\exp_{x}s\textbf{n})$,
then we obtain

######  Lemma 4.5.

$\Phi_{r}:SN\rightarrow SN$ is a diffeomorphism induced by the geodesic flow
of $N$ for the fixed length $r$.

###### Proof.

Since
$\Phi_{r}(x,\gamma^{\prime}_{x}(0))=(\gamma_{x}(r),\gamma^{\prime}_{x}(r))$,
it is obvious that $\Phi_{r}$ is injective and smooth. For
$(\widetilde{x},\widetilde{\textbf{n}})\in SN$, set
$\overleftarrow{\gamma}_{\widetilde{x}}(s)=\exp_{\widetilde{x}}s(-\widetilde{\textbf{n}})$,
where $\overleftarrow{\gamma}_{\widetilde{x}}(0)=\widetilde{x}$,
$\overleftarrow{\gamma}^{\prime}_{\widetilde{x}}(0)=-\widetilde{\textbf{n}}$.
Then for $x=\overleftarrow{\gamma}_{\widetilde{x}}(r)$ and
$\textbf{n}=-\frac{d}{ds}\overleftarrow{\gamma}_{\widetilde{x}}(s)\big{|}_{s=r}$,
we have $\Phi_{r}(x,\textbf{n})=(\tilde{x},\tilde{\textbf{n}})$. Therefore,
$\Phi_{r}$ is a surjection. This completes the proof. ∎

From Lemma 4.2, Lemma 4.3 and Lemma 4.5,

######  Lemma 4.6.

$\phi_{r}^{-1}W$ is a tube of radius $r$ over $W$ with respect to
$\overleftarrow{F}$.

###### Proof.

Since $\mathcal{N}^{0}(M_{c})$ is closed in $SN$, $\Phi_{r}$ and
$\mathcal{L}:SN\rightarrow S^{*}N$ are both diffeomorphism, $\sigma^{*}M_{c}$
is closed in $S^{*}N$. Moreover, $\sigma^{*}M_{c}\cap\mathcal{V}^{0}_{p}W$ is
closed in $\mathcal{V}^{0}_{p}W$. For any $x\in M_{c}$, set $U_{x}$ an open
neighborhood in $N$. Meanwhile, there exists an open neighborhood $V_{x}$ of
$\textbf{n}(U_{x})$ in $SN$ such that

$\sigma^{*}(U_{x}\cap
M_{c})\cap\mathcal{V}^{0}_{p}W=\mathcal{L}\circ\Phi_{r}(V_{x})\cap\mathcal{V}^{0}_{p}W.$

Since $\mathcal{L}\circ\Phi_{r}(V_{x})$ is open in $S^{*}N$,
$\sigma^{*}(U_{x}\cap M_{c})\cap\mathcal{V}^{0}_{p}W$ is open in
$\mathcal{V}^{0}_{p}W$. Hence, $\sigma^{*}M_{c}\cap\mathcal{V}^{0}_{p}W$ is
open in $\mathcal{V}^{0}_{p}W$. From above analysis,
$\mathcal{V}^{0}W=\sigma^{*}\phi_{r}^{-1}W$.

For any $(p,\eta)\in\mathcal{N}^{0}W$, set
$\overleftarrow{\phi}_{s}(p,-\eta)=\overleftarrow{E}(p,-s\eta)=\overleftarrow{\exp}_{p}s(-\eta)$.
In addition, $x=\overleftarrow{\exp}_{p}r(-\eta)\in\phi_{r}^{-1}W\subseteq
M_{c}$ is not a critical point of $\overleftarrow{E}$. Hence,
$\phi_{r}^{-1}W=\overleftarrow{\phi}_{r}\overleftarrow{\mathcal{N}}^{0}W$ is
the tube of radius $r$ over $W$ with respect to $\overleftarrow{F}$. ∎

Combine Lemma 4.2 and Lemma 4.6, we have

######  Corollary 4.7.

$E^{\bot}_{0}(p)=E^{*}_{\lambda}(p)$, where $p\in W$.

###### Proof.

On the one hand, for any
$\tau\in\mathcal{U}E^{\bot}_{0}(p)=\mathcal{V}^{0}_{p}W$, from Lemma 4.6, we
know $\tau\in\sigma^{*}\phi_{r}^{-1}p$. Similar to the proof of (1) in Lemma
4.2, $\sigma^{*}\phi_{r}^{-1}p\subseteq\mathcal{U}E^{*}_{\lambda}(p)$. Namely,
$\mathcal{U}E^{\bot}_{0}(p)\subseteq\mathcal{U}E^{*}_{\lambda}(p)$. On the
other hand, for any $\tau\in\mathcal{U}E_{\lambda}^{*}(p)$, from (2) in Lemma
4.2, $\mathcal{U}E^{*}_{\lambda}(p)\subseteq\mathcal{U}E^{\bot}_{0}(p)$.
Hence, $\mathcal{U}E^{\bot}_{0}(p)=\mathcal{U}E^{*}_{\lambda}(p)$. This
completes the proof. ∎

From Lemma 4.6, we have

######  Lemma 4.8.

For any $p\in W$, there is a neighborhood $W_{1}$ of $p$ in $W$ such that
$U_{\varepsilon}(W_{1})\cap M_{+}=W_{1}$ for a sufficiently small
$\varepsilon>0$, where $U_{\varepsilon}(W_{1})$ is the tubular neighborhood of
$W_{1}$ of radius $\varepsilon$ with respect to $F$.

###### Proof.

For any $\eta\in\mathcal{N}^{0}_{p}W$,
$\tau=\mathcal{L}(\eta)\in\mathcal{V}^{0}_{p}W$, there exists a unique
geodesic $\gamma_{\eta}(s)$ determined by $p=\gamma_{\eta}(0)$ and
$\eta=\gamma_{\eta}^{\prime}(0)$ for $F$. Denote the function
$f_{\eta}(s)=f(\gamma_{\eta}(s))$, we have $f_{\eta}(0)=\beta$,
$f^{\prime}_{\eta}(0)=df|_{p}(\eta)=0$. Combine Lemma 4.3 and Corollary 4.7,
$\mathcal{U}E^{\bot}_{0}(W)=\mathcal{U}E^{*}_{\lambda}(W)=\mathcal{V}^{0}W$,
we have

$\displaystyle
f^{\prime\prime}_{\eta}(0)=H(\eta,\eta)=H^{*}(\tau)(\eta)=\lambda\tau(\eta)=\lambda.$

From Lemma 3.4 and (1) of Lemma 4.2, $\lambda<0$. Hence,
$f^{\prime\prime}_{\eta}(0)<0$. Lifting $f$ to $SN$,
$f^{\prime\prime}_{\eta}(s)$ is continuous in $\eta$ and $s$. Hence, there
exists a neighborhood $W_{1}$ of $p$ in $W$ and a sufficiently small
$\varepsilon$ such that $f^{\prime\prime}_{\eta}(s)<\frac{\lambda}{2}<0$ for
$s\in(0,\varepsilon)$ and $\eta=\mathcal{L}^{-1}(\tau)$, where
$\tau\in\mathcal{U}E^{\bot}_{0}(W_{1})$. Hence,

$\displaystyle
f_{\eta}(s)=f_{\eta}(0)+f^{\prime}_{\eta}(0)s+\frac{1}{2}f^{\prime\prime}_{\eta}(\xi)s^{2}<\beta,\
\ \ \ s\in(0,\varepsilon).$

Namely, $f$ is strictly less than $\beta$ if restricted to
$U_{\varepsilon}(W_{1})\setminus W_{1}$. ∎

###  4.3 Maximal rank problem

Denote $m=\min\limits_{p\in M_{+}}\\{\dim E_{0}(p)\\}$, $M_{+}^{m}=\\{p\in
M_{+}\big{|}\dim E_{0}(p)=m\\}=\\{p\in
M_{+}\big{|}\max\textmd{rank}H_{p}=n-m\\}$,
$M^{m}_{c}=\phi^{-1}_{r}(M^{m}_{+})$, $\mathring{M}^{m}_{c}$ the subset of
$M^{m}_{c}$ where $\phi_{r}$ has maximal rank $m$ and
$\mathring{M}^{m}_{+}=\phi_{r}\mathring{M}^{m}_{c}$, we have

######  Lemma 4.9.

$\mathring{M}^{m}_{c}$ is open and dense in $M^{m}_{c}$.

###### Proof.

For any $x\in\mathring{M}^{m}_{c}$, there exists
$U(x)\subseteq\mathring{M}^{m}_{c}\subseteq M^{m}_{c}$ in which
rank$\phi_{r}\big{|}_{U(x)}\geq m$. Consider the condition $\phi_{r}$ has
maximal rank $m$, we have $\phi_{r}$ has constant rank $m$ on $U(x)$. Hence,
$\mathring{M}^{m}_{c}$ is open in $M^{m}_{c}$.

Suppose
$\widetilde{U}=\textmd{int}(M^{m}_{c}-\mathring{M}^{m}_{c})\neq\varnothing$,
$\phi_{r}$ has maximal rank $k$ in $\widetilde{U}$ and $k<m$. In analogy to
the analysis of maximal rank above, there exists $U\subseteq\widetilde{U}$
such that rank$\phi_{r}\big{|}_{U}=k$ and $\phi_{r}U=W$ is a $k$-dimensional
submanifold. From Lemma 4.3, $\dim T_{p}W=\dim E_{0}(p)$. There exists a
contradiction. ∎

######  Remark 4.10.

From Lemma 4.9 and the constant rank theorem, for any
$x\in\mathring{M}^{m}_{c}$, there exists a neighborhood $U(x)$ in
$\mathring{M}^{m}_{c}$ such that $W(p)=\phi_{r}U(x)$ is a submanifold of $N$,
where $p=\phi_{r}x\in\mathring{M}^{m}_{+}$.

Due to $H$ has maximal rank $n-m$ in $M^{m}_{+}$, we easily get $M^{m}_{+}$ is
open in $M_{+}$. Furthermore, we have

######  Lemma 4.11.

$\mathring{M}^{m}_{+}$ is dense in $M^{m}_{+}$.

###### Proof.

Set $U(p)\subseteq M^{m}_{+}=\phi_{r}M^{m}_{c}$ a neighborhood of $\forall
p\in M^{m}_{+}$, we only need to prove
$U(p)\cap\mathring{M}^{m}_{+}\neq\varnothing$.

$U(p)\cap\mathring{M}^{m}_{+}=U(p)\cap\phi_{r}\mathring{M}^{m}_{c}\supseteq\phi_{r}(\phi_{r}^{-1}U(p)\cap\mathring{M}^{m}_{c}).$

From Lemma 4.9, $\mathring{M}^{m}_{c}$ is dense in $M^{m}_{c}$,
$\phi_{r}^{-1}U(p)\cap\mathring{M}^{m}_{c}\neq\varnothing$. Hence,
$U(p)\cap\mathring{M}^{m}_{+}\neq\varnothing$. ∎

Let $(x^{1},x^{2},\dots,x^{n})$ be a local coordinate system at $p$, where
$p\in N$. Due to the definition of $H^{*}$ in (4.2), we give the expression of
$H^{*}$ using local coordinate,

$\displaystyle H^{*}(\xi)=g^{*ik}(\xi)f_{kj}\xi_{i}dx^{j},$ (4.4)

where $\xi\in T_{p}^{*}N$.

######  Lemma 4.12.

Let $(\xi_{1},\xi_{2},\dots,\xi_{n})$ be a local coordinate system of
$T_{p}^{*}N$. Then for the linear map
${(dH^{*}_{p})}_{\xi}:T_{\xi}T_{p}^{*}N\rightarrow T_{\xi}T_{p}^{*}N$, we have
the following results.
(1)For any $\xi\in T^{*}_{p}N$,

$\displaystyle{(dH^{*}_{p})}_{\xi}=f_{kj}g^{*ik}(p,\xi)\frac{\partial}{\partial\xi_{j}}\otimes
d\xi_{i},$ (4.5)

(2)For any $p\in\mathring{M}^{m}_{+}$, $\xi\in E^{\bot}_{0}(p)$,

${(dH^{*}_{p})}_{\xi}^{2}=\lambda{(dH^{*}_{p})}_{\xi},\ \ \ \ \ \
{(dH^{*}_{p})}_{\xi}=(n-m)\lambda.$

###### Proof.

From (4.4), we immediately get the expression of $dH^{*}$ in (1) by directly
calculation.

In order to prove (2), for any $p\in\mathring{M}^{m}_{+}$ and $\xi_{0}\in
E^{\bot}_{0}(p)=E_{\lambda}^{*}(p)$, let $\xi(t)$ be a curve in
$E^{\bot}_{0}(p)$ satisfying $\xi(0)=\xi_{0}$. From Corollary 4.7 and Remark
4.10, $H^{*}(\xi(t))=\lambda\xi(t)$. Set $X=\xi^{\prime}(0)\in
T_{\xi_{0}}E^{\bot}_{0}(p)$, we have $(dH^{*}_{p})_{\xi_{0}}X=\lambda X$.
Define

$\displaystyle E_{0}^{*}(p,\xi)=\\{X^{i}g_{ij}^{*}(p,\xi)\big{|}X\in
E_{0}\\}.$

From (4.5), $dH^{*}(E_{0}^{*}(p,\xi))=0$, where
$\textmd{rank}E_{0}^{*}(\xi)=\textmd{rank}E_{0}=m$.

$T_{\xi}T^{*}_{p}N=E_{0}^{*}(p,\xi)\oplus T_{\xi}E^{\bot}_{0}(p)$

and the following algebraic equations hold,

$\displaystyle{(dH^{*}_{p})}_{\xi}^{2}=\lambda{(dH^{*}_{p})}_{\xi},\ \ \ \ \ \
{(dH^{*}_{p})}_{\xi}=(n-m)\lambda.$ (4.6)

This completes the proof. ∎

######  Lemma 4.13.

$M^{m}_{+}$ is open and closed in $M_{+}$, that is, $M^{m}_{+}$ is a union of
some connected components of $M_{+}$.

###### Proof.

(4.6) also holds on the closure of $\mathring{M}^{m}_{+}$ in $M_{+}$ by the
continuity of $dH^{*}$. Combine Lemma 4.11, we have
$\overline{\mathring{M}^{m}_{+}}=\overline{M^{m}_{+}}$. Hence, (4.6) holds on
$\overline{M^{m}_{+}}$ either. Namely, $dH^{*}$ has maximal rank $n-m$ on
$\overline{M^{m}_{+}}$. Due to the definition of $M^{m}_{+}$, $M^{m}_{+}$ is
closed in $M_{+}$. ∎

From Lemma 4.13, we also have $M^{m}_{c}$ is open and closed in $M_{c}$, that
is, $M^{m}_{c}$ is a connected component of $M_{c}$. Replacing $M_{+}$ by
$M_{+}\backslash M^{m}_{+}$ and iterating the whole process above by limited
steps, we have

######  Corollary 4.14.

(1) $\dim E_{0}(p)$ is constant on each connecting component of $M_{+}$,
(2) $\mathring{M}_{+}=\mathop{\cup}\limits_{m\in N^{+}}\mathring{M}^{m}_{+}$
is dense in $M_{+}$.
(3) $M_{+}$ is a disjoint union of $M^{m}_{+}$, where $m\in N^{+}$.

###  4.4 Tubes of focal varieties

######  Lemma 4.15.

For any $p\in M_{+}^{m}$, we have
$\phi_{r}^{-1}(p)=\\{\overleftarrow{\exp}_{p}(-r\eta)\big{|}\eta=\mathcal{L}^{-1}\tau,\tau\in\mathcal{U}E^{\bot}_{0}(p)\\}$,
which is a $(k-1)$-dimensional closed submanifold in the geodesic sphere
$S^{n-1}_{\overleftarrow{F}}(p;r)$, where $k=n-m=\dim E^{\bot}_{0}(p)$.

###### Proof.

If $p\in\mathring{M}^{m}_{+}$, from Lemma 4.6 and Remark 4.10, obviously,
$\phi_{r}^{-1}(p)=\\{\overleftarrow{\exp}_{p}(-r\eta)\big{|}\eta=\mathcal{L}^{-1}\tau,\tau\in\mathcal{U}E^{\bot}_{0}(p)\\}$.
Suppose that $p\in M_{+}^{m}\setminus\mathring{M}^{m}_{+}$ and
$\tau\in\mathcal{U}E^{\bot}_{0}(p)$. Let $\overleftarrow{\gamma}(s)$ be a
geodesic which satisfies $\overleftarrow{\gamma}(0)=p$ and
$\overleftarrow{\gamma}^{\prime}(0)=-\eta=-\mathcal{L}^{-1}(\tau)$. Since
$\mathring{M}_{+}^{m}$ is dense in $M_{+}^{m}$ and $E^{\bot}_{0}$ is a smooth
subbundle of $T^{*}N$ on $M_{+}^{m}$, there exist
$\\{p_{i}\\}\in\mathring{M}_{+}^{m}$ and
$\tau_{i}\in\mathcal{U}E^{\bot}_{0}(p_{i})$ such that
$\lim\limits_{i\rightarrow+\infty}p_{i}=p$ and
$\lim\limits_{i\rightarrow+\infty}\tau_{i}=\tau$.
$\overleftarrow{\gamma}_{i}(s)$ is the geodesics determined by
$(p_{i},-\eta_{i})$ which satisfies $f(\overleftarrow{\gamma_{i}}(r))=c$,
where $\eta_{i}=\mathcal{L}^{-1}(\tau_{i})$. Due to continuous dependence of
geodesics on initial conditions,
$\lim\limits_{i\rightarrow+\infty}\overleftarrow{\gamma}_{i}(s)=\overleftarrow{\gamma}(s)$
for $s\in[0,r]$, $f(\overleftarrow{\gamma}(r))=c$. This implies
$x=\overleftarrow{\gamma}(r)\in M_{c}$. Hence,
$\\{\overleftarrow{\exp}_{p}(-r\eta)\big{|}\eta=\mathcal{L}^{-1}\tau,\tau\in\mathcal{U}E^{\bot}_{0}(p)\\}\subseteq\phi_{r}^{-1}p$.

Furthermore, there exists a geodesic $\gamma(s)$ determined by
$x\in\phi_{r}^{-1}p$ and $\textbf{n}\in\mathcal{N}^{0}_{x}(M_{c})$ which
satisfies $\gamma(r)=p$ and $\gamma^{\prime}(r)=\eta=\mathcal{L}^{-1}(\tau)$.
Similar to the proof in (1) of Lemma 4.2, for any $X\in E_{0}(p)$,

$0=H^{*}(\tau)(X)=H(\eta,X)=\frac{1}{2}a^{\prime}(\beta)\tau(X).$

Namely, we have $\tau\in\mathcal{U}E_{0}^{\bot}(p)$.

Since $\mathcal{L}:SN\rightarrow S^{*}N$ is a diffeomorphism and
$\mathcal{U}E_{0}^{\bot}(p)=E_{0}^{\bot}(p)\cap S^{*}_{p}N$, where
$E_{0}^{\bot}(p)$ is a $k$-dimensional subspace in $T^{*}_{p}N$,
$\\{\overleftarrow{\exp}_{p}(-r\eta)\big{|}\eta=\mathcal{L}^{-1}\tau,\tau\in\mathcal{U}E^{\bot}_{0}(p)\\}$
is a closed submanifold in the geodesic sphere
$S^{n-1}_{\overleftarrow{F}}(p;r)$. ∎

From Lemma 4.15, the focal map $\phi_{r}:M_{c}\rightarrow M_{+}$ is
subjective. Moreover, we have

######  Lemma 4.16.

For any $x\in M_{c}$, there exist a neighborhood $U(x)$ and a sufficiently
small $h\in(0,d(M_{-},M_{+}))$ such that $\gamma_{z}(s)\big{|}_{[0,r+h]}$
passes $M_{+}$(resp.$M_{-}$) once for any $z\in U(x)$.

###### Proof.

For a unique geodesic $\gamma_{x}(s)$ determined by $x=\gamma_{x}(0)$ and
$\gamma^{\prime}_{x}(0)$, set $p=\gamma_{x}(r)\in M_{+}$ and
$\eta=\mathcal{L}^{-1}(\tau)=\gamma_{x}^{\prime}(r)$, where
$\tau\in\mathcal{U}E_{0}^{\perp}(p)$. Denote the function
$f_{x}(s)=f(\gamma_{x}(s))$, similar to the calculation in Lemma 4.8, using
Taylor expansion for $f_{x}(s)$ at $s=r$,

$f_{x}(s)=\beta+\frac{1}{2}f_{x}^{\prime\prime}(\xi)s^{2},\ \ \ \
s\in(r,r+d(M_{-},M_{+})),$

From Lemma 4.2 and Lemma 4.15, similar to the proof in Corollary 4.7,
$E^{\bot}_{0}(p)=E^{*}_{\lambda}(p)$ for any $p\in M_{+}$. Hence,

$f^{\prime\prime}_{x}(r)=H(\eta,\eta)=H^{*}(\tau)(\eta)=\lambda\tau(\eta)=\lambda<0.$

Then there exists a sufficiently small $h(x)$ such that
$f^{\prime\prime}_{x}(s)<0$ for any $s\in(r,r+h(x))$. Namely, if $\forall
s\in(r,r+h(x))$, $f_{x}(s)<\beta$. Since $f_{x}(s)$ is continuous in $x$, for
any $x\in M_{c}$, there must exist a neighborhood $U(x)$ of $x$ in $M_{c}$ and
a sufficiently small $h\in(0,d(M_{-},M_{+}))$ such that
$\gamma_{z}(s)\big{|}_{[0,r+h]}$ intersect $M_{+}$ once for $z\in U(x)$. This
completes the proof. ∎

###  4.5 Morse Index Theorem

Let $(N,F)$ be a Finsler manifold, $P$ a submanifold of $N$ and
$\gamma:[0,a]\rightarrow N$ an unit geodesic with $x=\gamma(0)\in P$ and
$\textbf{n}(x)=\dot{\gamma}(0)\in\mathcal{N}_{x}^{0}P$. Recall that the Morse
index form $I$ along the normal geodesic $\gamma:[0,a]\rightarrow N$ is the
symmetric bilinear form

$\displaystyle
I(X,Y)=\int_{0}^{a}[g_{\dot{\gamma}}(\nabla^{\dot{\gamma}}_{\dot{\gamma}}X,\nabla_{\dot{\gamma}}^{\dot{\gamma}}Y)-g_{\dot{\gamma}}(\textbf{R}_{\dot{\gamma}}(X),Y)]ds,$
(4.7)

where $X$, $Y$ are piecewise $C^{\infty}$ vector fields along $\gamma$.

$\mathcal{X}^{P}_{x}$ is the subspace of all piecewise smooth vector fields
$X$ along $\gamma$ such that $X(0)\in T_{x}P\setminus\\{0\\}$, which satisfies
$X$ is orthogonal to $\dot{\gamma}$ along $\gamma$ and $X(a)=0$. Denote
$I^{P}:\mathcal{X}^{P}_{x}\times\mathcal{X}^{P}_{x}\rightarrow\mathbb{R}$,

$\displaystyle I^{P}(X,Y)=I(X,Y)-g_{\textbf{n}}(A_{\textbf{n}}X(0),Y(0)),$
(4.8)

where $A_{\textbf{n}}$ is the shape operator defined in (2.3) with respect to
n. Define the Morse index $I^{P}$ of $\gamma$ as follows,

$\textmd{ind}~{}I^{P}=\max\\{\dim(\mathcal{B})~{}|~{}\mathcal{B}~{}\textmd{is}~{}\textmd{a}~{}\textmd{subspace}~{}\textmd{of}~{}\mathcal{X}^{P}_{x}~{}\textmd{with}~{}I^{P}|_{\mathcal{B}}<0\\}.$

A $P$-Jacobi field $J$ is a Jacobi field which satisfies in addition

$\displaystyle J(0)\in T_{x}P,\ \ \ \
\nabla_{\textbf{n}}^{\textbf{n}}J(0)+A_{\textbf{n}}J(0)\in\mathcal{N}_{x}^{0}P.$
(4.9)

A point $\gamma(s_{0})$, $s_{0}\in[0,a]$ is called a $P$-focal point along
$\gamma$ if there exists a non-null $P$-Jacobi field $J$ along $\gamma$ with
$J(s_{0})=0$. The multiplicity $\mu^{P}(s_{0})$ of a $P$-focal point
$\gamma(s_{0})$ is the dimension of the vector space of all $P$-Jacobi fields
along $\gamma$ that vanish in $s_{0}$.

Recall the Morse Index Theorem in Finsler geometry,

######  Lemma 4.17.

[21] Let $(N,F)$ be a Finsler manifold, $P$ a submanifold of $N$ and
$\gamma:[0,a]\rightarrow N$ a geodesic with $\gamma(0)\in P$ and
$\dot{\gamma}(0)\in\mathcal{N}^{0}_{x}P$. Then

$\displaystyle\textmd{ind}~{}I^{P}=\sum\limits_{s_{0}\in(0,a)}\mu^{P}(s_{0})<\infty.$
(4.10)

######  Remark 4.18.

In [21], the Morse index form $I$, $P$-Jacobi field $J$ and the the shape
operator $A$ are defined in the horizontal subspaces of
$T(TN\setminus\\{0\\})$. By a direct calculation, we know that they can be
express on $N$ by (4.7), (4.9) and (2.3).

Similar to the proof of Lemma 4.6 in $II$ of [22], we get

######  Lemma 4.19.

Let $J(s)$ be a vector field along a normal geodesic $\gamma:[0,a]\rightarrow
N$ with $\gamma(0)\in P$ and $\dot{\gamma}(0)\in\mathcal{N}_{x}^{0}P$. Then
$J(s)$ is a $P$-Jacobi field if and only if there exists a $C^{\infty}$
variation of $\gamma$ given by

$\Psi:(-\varepsilon,\varepsilon)\times[0,a]\rightarrow N,\ \ \ \
\Psi(0,s)=\gamma(s),$

where for each $t\in(-\varepsilon,\varepsilon)$ such that the variation curves
$\Psi_{t}(s)=\Psi(t,s)$ are all geodesic perpendicular to $P$ at $s=0$ and
$J(s)$ is given by $\frac{\partial\Psi}{\partial t}\big{|}_{t=0}$.

In fact, the $C^{\infty}$ variation in Lemma 4.19 can be given as
$\Psi(t,s)=E(\sigma(t),s\textbf{n}(\sigma(t)))$, where $\sigma(t)$ is a curve
in $P$ and $\textbf{n}(\sigma(t))\in\mathcal{N}^{0}_{\sigma(t)}P$. Moreover,
from Lemma 3.2 and the proof of Lemma 3.3 in [14], we immediately have

######  Lemma 4.20.

For $s_{0}\in(0,a)$, $p=\gamma(s_{0})$ is a focal point of $P$ if and only if
there exists a $P$-Jacobi field along $\gamma(s)$ such that $J(s_{0})=0$.

Hence, the focal point defined by the normal exponential map $E$ is equivalent
to the $P$-focal point defined by the $P$-Jacobi field.

##  5 Proof of Theorem 1.1

######  Lemma 5.1.

The rank of the focal map is constant on each connected component.

###### Proof.

Let $\gamma_{x}(s)$ be an $f$-segment through $x\in M_{c}$ which satisfies
$\gamma_{x}(0)=x$, $\gamma^{\prime}(0)=\textbf{n}(x)$, where
$\textbf{n}(x)\in\mathcal{N}^{0}_{x}M_{c}$. For any point $z$ which is
sufficiently close to $x$, it is obvious that
$\textmd{rank}\phi_{r}(z)\geqslant\textmd{rank}\phi_{r}(x)$. In order to prove
rank$\phi_{r}$ is constant, we only need to prove
rank$\phi_{r}(z)\leq$rank$\phi_{r}(x)$.

From Lemma 4.16, there exist a neighborhood $U(x)$ of $x$ and a sufficiently
small $h\in(0,d(M_{-},M_{+}))$ such that for $z\in U(x)$, the geodesic segment
$\Gamma_{z}(s)=\gamma_{z}(s)\big{|}_{[0,r+h]}$ passes $M_{+}$(resp. $M_{-}$)
at $\gamma_{z}(r)$ only. If $\gamma_{x}(r)$ is not a focal point of $M_{c}$
along $\Gamma_{x}(s)$, then rank$\phi_{r}(x)=n-1$ and
$n-1=\textmd{rank}\phi_{r}(z)\geq\textmd{rank}\phi_{r}(x)=n-1$. That is,
rank$\phi_{r}$ is constant locally. If $\gamma_{x}(r)$ is a focal point of
$M_{c}$ along $\Gamma_{x}(s)$, then $\gamma_{x}(r)$ is the unique one. Denote
$m(\Gamma_{x}(s))$ as the multiplicity of $\gamma_{x}(r)$, then
$m(\Gamma_{x}(s))=n-1-\textmd{rank}_{x}\phi_{r}$. From Lemma 4.17 and Lemma
4.20 ,
$\textmd{ind}I^{M_{c}}_{x}=m(\Gamma_{x}(s))=n-1-\textmd{rank}_{x}\phi_{r}$.

Set $\\{e_{i}\\}=\\{e_{a},\textbf{n}(x)\\}$ an unit orthonormal frame at $x\in
M_{c}$ with repect to $g_{\textbf{n}(x)}$. Firstly, translate parallel
$\\{e_{i}\\}$ along $\Gamma_{x}(s)$, the unit orthonormal frame field
$\\{E_{i}\\}=\\{E_{a},T\\}$ can be given, where $T=\Gamma^{\prime}_{x}$.
Secondly, translate parallel $\\{e_{i}\\}$ from $x$ to $z$ along the geodesic
on $M_{c}$ with respect to the Riemannian metric $g_{\textbf{n}}$, we can get
the unit orthonormal frame
$\\{\overline{e}_{i}\\}=\\{\overline{e}_{a},\textbf{n}(z)\\}$ at $z$ with
repect to $g_{\textbf{n}(z)}$. Thirdly, translate parallel
$\\{\overline{e}_{i}\\}$ along $\Gamma_{z}(s)$, the unit orthonormal frame
field $\\{\overline{E}_{i}\\}=\\{\overline{E}_{a},\overline{T}\\}$ is
obtained, where $\overline{T}=\Gamma^{\prime}_{z}$. Set
$X(s)=b^{a}(s)E_{a}(s)$ for any $X\in\mathcal{B}_{x}$, where $b^{a}(r+h)=0$.
Set $\overline{X}(s)=b^{a}(s)\overline{E}_{a}(s)\in\mathcal{X}^{M_{c}}_{z}$,

$\displaystyle
I^{M{c}}(\overline{X},\overline{X})=\int^{r+h}_{0}\big{(}\dot{b}_{a}\dot{b}^{a}-b^{a}b^{b}g_{\overline{T}}(R_{\overline{T}}(\overline{E}_{a}),\overline{E}_{b})\big{)}ds+b^{a}(0)b^{b}(0)g_{\textbf{n}(z)}(\nabla^{\textbf{n}(z)}_{\overline{e}_{a}}\textbf{n}(z),\overline{e}_{b}).$
(5.1)

Since $\gamma_{z}(s)$, $\overline{E}_{a}$ and $\overline{T}$ are determined by
$z\in M_{c}$, from (5.1), $I(\overline{X},\overline{X})$ only depends on $z$
continuously. Due to $I(X,X)<0$, there exists $\overline{U}(x)\subseteq U(x)$
such that $I(\overline{X},\overline{X})<0$ for all $z\in\overline{U}(x)$.
Namely, $\overline{X}\in\mathcal{B}_{z}$ if $z\in\overline{U}(x)$. Hence,
$\textmd{dim}\mathcal{B}_{z}\geq\textmd{dim}\mathcal{B}_{x}$. Therefore,

$m(\Gamma_{z})=\textmd{ind}I^{M_{c}}_{z}=\max\textmd{dim}\mathcal{B}_{z}\geq\max\textmd{dim}\mathcal{B}_{x}=\textmd{ind}I^{M_{c}}_{x}=m(\Gamma_{x}).$

Due to the expression of $m(\Gamma_{x})$,
rank$\phi_{r}(z)\leq$rank$\phi_{r}(x)$. Hence, The rank of the focal map is
locally constant.

Furthermore, if rank$\phi_{r}\big{|}_{U(x)}=m$, then we have
$\phi_{r}\big{|}_{\overline{U(x)}}=m$. Hence, the set of $\phi_{r}$ with
constant rank is open and closed. Namely, it is a connected component of
$M_{c}$. This completes the proof. ∎

Combine Lemma 4.8 and Lemma 5.1, $M^{m}_{+}$ is a submanifold of $(N,F)$.
According to the analysis of Lemma 4.6 about tubes, $\phi_{r}^{-1}M^{m}_{+}$
is a tube of radius $r$ with respect to $\overleftarrow{F}$. We completes the
proof of Theorem 1.1 while the focal variety is $M_{+}$. When the focal
variety is $M_{-}$, we just need to change $F$ to $\overleftarrow{F}$ in all
proofs above.

From Theorem 1.1, similar to the proof in [9], we have

######  Corollary 5.2.

Each connected component of $M_{\pm}$ has codimension not less than 2 if and
only if $M_{\pm}=\textmd{Foc}(f)$.

##  6 Example

Recall $(\alpha,\beta)$-metric in Finsler geometry which can be expressed by
$F=\alpha\phi(s)$, where $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian
metric, $\beta=b_{i}(x)y^{i}$ is a 1-form satisfying
$b:=\|\beta\|_{\alpha}<b_{0}$ $s=\frac{\beta}{\alpha}$ for any $x\in N$ and
$\phi:(-b_{0},b_{0})\rightarrow(0,+\infty)$ is a smooth positive function,
which satisfies

$\displaystyle\phi(s)>0,\ \ \ \
(\phi(s)-s\phi^{\prime}(s))+(b^{2}-s^{2})\phi^{\prime\prime}(s)>0,$ (6.1)

where $s$ and $b$ are arbitrary numbers with $|s|\leq b<b_{0}$.

Let $\alpha$ be a standard Euclidean sphere metric, which its dual metric
denotes by $\alpha^{*}$. Set $\varphi(s):[-1,1]\rightarrow(0,+\infty)$ a
positive function which is expressed by $\varphi(s)=1+\varepsilon s^{k}$,
where $0<\varepsilon<\frac{1}{2k-1}$ and $k\in N_{+}$. By a direct
computation, we can verify that (6.1) holds. Denote
$x=(\overline{x}_{1},\overline{x}_{2})\in\mathbb{R}^{n+1}$, where
$\overline{x}_{1}=(x_{1},x_{2},\dots,x_{p+1})$,
$\overline{x}_{2}=(x_{p+2},x_{p+3}\dots,x_{p+q+2})$, $p,q\in N^{+}$ and
$p+q=n-1$. For $x\in\mathbb{S}^{n}$, define $V(x)$ as the projection of
$\widetilde{x}=(\overline{x}_{1},-\overline{x}_{2})$ onto $\mathbb{S}^{n}$,
then $V(x)=\tilde{x}-\langle\tilde{x},x\rangle x$. Set
$\beta^{*}(\xi)=\xi(V)$. Then we equip an $(\alpha,\beta)$-metric $F$ on
$\mathbb{S}^{n}$, which its dual metric is expressed by
$F^{*}=\alpha^{*}\varphi(s)$, where $s=\frac{\beta^{*}}{\alpha^{*}}$.

Set $\Phi(x)=|\overline{x}_{1}|^{2}-|\overline{x}_{2}|^{2}$ and
$f=\Phi\big{|}_{\mathbb{S}^{n}}$, we have

$\big{(}\alpha^{*}(df)\big{)}^{2}=|\nabla^{\alpha}f|^{2}=\langle\nabla^{E}\Phi-2\Phi
x,\nabla^{E}\Phi-2\Phi x\rangle=4(1-f^{2}),$
$\beta^{*}(df)=\langle\nabla^{\alpha}f,V\rangle_{\alpha}=\langle\nabla^{E}\Phi-2\Phi
x,\tilde{x}-\langle\tilde{x},x\rangle x\rangle=2(1-f^{2}).$

Hence,

$F(\nabla
f)=F^{*}(df)=\alpha^{*}(df)\varphi(\frac{\beta^{*}(df)}{\alpha^{*}(df)})=2\sqrt{1-f^{2}}(1+\varepsilon(1-f^{2})^{\frac{k}{2}}).$

Namely, $f$ is a $C^{\infty}$ transnormal function on $(\mathbb{S}^{n},F)$. In
this case, $J=[-1,1]$.

######  Example 6.1.

If $k=1$,

$a(t)=4(1-f^{2})+8\varepsilon(1-f^{2})^{\frac{3}{2}}+4\varepsilon^{2}(1-f^{2})^{2}\in
C^{1}(J).$

Hence, $a(t)\neq 0$ for $t\in(-1,1)$ and $a^{\prime}(\pm 1)=\mp 8\neq 0$. In
this case, $f$ and $a(t)$ satisfy the conditions in Theorem 1.1.

######  Example 6.2.

If $k=3$,

$a(t)=4(1-f^{2})+8\varepsilon(1-f^{2})^{\frac{5}{2}}+4\varepsilon^{2}(1-f^{2})^{4}\in
C^{2}(J),$

If $k=2l$, $l\in N_{+}$,

$a(t)=4(1-f^{2})+8\varepsilon(1-f^{2})^{l+1}+4\varepsilon^{2}(1-f^{2})^{2l+1}\in
C^{\infty}(J).$

Hence, there exists a smooth function $f$ on $N=\mathbb{S}^{n}$ which
satisfies $a\in C^{2}(J)$.

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Yali Chen
School of Mathematical Sciences, Tongji University, Shanghai, 200092, China
E-mail<EMAIL_ADDRESS>

Qun He
School of Mathematical Sciences, Tongji University, Shanghai, 200092, China
E-mail<EMAIL_ADDRESS>