# Isoparametric hypersurfaces induced by navigation in Lorentz Finsler
geometry

Ming Xu, Ju Tan and Na Xu  Ming Xu
School of Mathematical Sciences, Capital Normal University, Beijing 100048,
P.R. China<EMAIL_ADDRESS>Ju Tan, corresponding author
School of Mathematics and Physics, Anhui University of Technology, Maanshan
243032, P.R. China<EMAIL_ADDRESS>Na Xu
School of Mathematics and Physics, Anhui University of Technology, Maanshan
243032, P.R. China<EMAIL_ADDRESS>
###### Abstract.

Using a navigation process with the datum $(F,V)$, in which $F$ is a Finsler
metric and the smooth tangent vector field $V$ satisfies $F(-V(x))>1$
everywhere, a Lorentz Finsler metric $\tilde{F}$ can be induced. Isoparametric
functions and isoparametric hypersurfaces with or without involving a smooth
measure can be defined for $\tilde{F}$. When the vector field $V$ in the
navigation datum is homothetic, we prove the local correspondences between
isoparametric functions and isoparametric hypersurfaces before and after this
navigation process. Using these correspondences, we provide some examples of
isoparametric functions and isoparametric hypersurfaces on a Funk space of
Lorentz Randers type.

Mathematics Subject Classification (2010): 52B40, 53C42, 53C60

Key words: Finsler metric, homothetic vector field, isoparametric function,
isoparametric hypersurface, Lorentz Finsler metric, Zermelo navigation

##  1\. Introduction

A smooth function $f$ on a Riemannian manifold $(M,g)$ is called isoparametric
if the pointwise norm of its gradient field $|\nabla f|$ and its Laplacian
$\Delta f$ only depends on the values of $f$. The level set of a regular or
critical value of $f$ is called an isoparametric hypersurface or a focal
subset respectively [4]. These notions have been extensively studied and have
many applications in geometry and topology [8][21][26]. They were generalized
to pseudo-Riemannian geometry and to high codimension during the 1980’s
[9][25]. The classification of isoparametric hypersurfaces in unit spheres was
one of most celebrated geometric problem [30]. This problem has a history of
eighty years [2], and was completely solved by Q.S. Chi in 2020 [3]. See [28]
for more references.

In recent years, isoparametric theory has also been studied in Finsler
geometry. Using the nonlinear analogs of gradient vector field and Laplacian
operators, Q. He et al defined two types of isoparametric functions on a
Finsler manifold $(M,F)$, i.e., the isoparametric functions involving a smooth
measure $\mathrm{d}\mu$ [13] (usually $\mathrm{d}\mu$ is taken to be the B.H.
measure $\mathrm{d}\mu^{F}_{\mathrm{Bh}}$ of $F$), and those without involving
any smooth measure [12][27]. In this paper, they are specified as
$\mathrm{d}\mu$-isoparametric functions and isoparametric functions
respectively. By a systematical approach in Finsler submanifold geometry, Q.
He and her coworkers proved Cartan type formulae[12][13] and classified
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric hypersurfaces in Minkowski
spaces [13], Funk spaces [14], and Randers spheres with $K\equiv 1$ [10].
Notice that these Finsler space forms have constant S-curvature [24][29], so
the isoparametric property coincides with the
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric property [27].

From another view point, the classifications for isoparametric hypersurfaces
in Funk spaces and Randers space forms can be explained by the correspondence
between isoparametric hypersurfaces before and after a homothetic navigation
[29]. Let $(F,V)$ be the datum in which $V$ is a homothetic field for the
Finsler metric $F$, and $\tilde{F}$ the Finsler metric induced by the
corresponding navigation process where $F(-V(x))<1$. Locally any hypersurface
is $\mathrm{d}\mu^{\tilde{F}}_{\mathrm{BH}}$-isoparametric for the metric
$\tilde{F}$ iff it is $\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric for the
metric $F$ (see Theorem 1.5 in [29]). The correspondence between isoparametric
functions is a little complicated, but we may simplify it by only considering
those isoparametric functions normalized at some $x_{0}\in M$ (see Theorem 7.5
in [29]). See [6][7][11][19][29] for more geometric correspondences for
homothetic navigation in Finsler geometry.

This correspondence between isoparametric hypersurfaces totally ignored the
domain where $F(-V(x))>1$. In this domain, the navigation process with the
datum $(F,V)$ induces a (conic) Lorentz Finsler metric
$\tilde{F}:\mathcal{A}\rightarrow\mathbb{R}_{>0}$, in which
$\mathcal{A}=\coprod\mathcal{A}_{x}$ for all $x\in M$ with $F(-V(x))>1$ and
$\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$ consists of all
$\tilde{y}=y+F(x,y)V(x)$ satisfying $\langle y,V(x)\rangle_{y}^{F}<-F(x,y)$
(see Lemma 2.2). Moreover, $\tilde{F}$ satisfies Condition (C), i.e., each
nonempty $\mathcal{A}_{x}$ is convex in $T_{x}M$. Condition (C) is useful for
us to uniquely define the gradient field $\nabla^{\tilde{F}}f$ of some regular
real function $f$. This $\tilde{F}$ was found and studied by E. Caponio et al
[5]. The special case, wind Riemannian $\tilde{F}$ induced by Riemannian $F$,
was studied by M.A. Javaloyes and M. Sánchez [15][16]. The geodesic and flag
curvature correspondences for a homothetic navigation in pseudo-Finsler
geometry was explored by M.A. Javaloyes and H. Vitório [17].

The $\mathrm{d}\mu$-isoparametric (and isoparametric) functions can be
generalized to Lorentz Finsler geometry, using similar gradient field
$\nabla^{\tilde{F}}\tilde{f}$ and Laplacian $\Delta^{\tilde{F}}\tilde{f}$ (and
$\Delta^{\tilde{F}}_{\mathrm{d}\mu}\tilde{f}$ respectively). See Section 3.2
for the precise definitions. The crucial issue here is that the gradient field
$\nabla^{\tilde{F}}\tilde{f}$ for a general real regular function $f$ may not
exist, and moreover it may not be unique. But for the situation we discuss
below $\nabla^{\tilde{F}}\tilde{f}$ can be uniquely determined.

The main purpose of this paper is to prove the following main theorem, which
generalizes the correspondence between isoparametric hypersurfaces before and
after a homothetic navigation with Finsler $F$ and Lorentz Finsler
$\tilde{F}$, and provides some examples.

######  Theorem 1.1.

Let $V$ be a homothetic vector field with dilation $c$ on the Finsler manifold
$(M,F)$, satisfying $F(-V(x))>1$ everywhere, and $\tilde{F}$ the conic Lorentz
Finsler metric induced by the navigation process with the datum $(F,V)$.
Denote $\Psi_{t}$ the flow of homothetic translations generated by $V$,
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$ the B.H. measure of $F$, and
$\alpha_{c}(t)=t$ when $c=0$ or $\alpha_{c}(t)=\tfrac{e^{2ct}-1}{2c}$
otherwise. Suppose $f$ is a normalized
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric (or isoparametric) function
for $F$ around $x_{0}$, satisfying
$\langle\nabla^{F}f(x),V(x)\rangle^{F}_{\nabla^{F}f(x)}<-1$, then the function
$\tilde{f}$ determined by $\tilde{f}^{-1}(t)=\Psi_{t}(f^{-1}(\alpha_{c}(t)))$
around $x_{0}$ is a normalized $\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric
(or isoparametric) function for $\tilde{F}$. In particular,
$f^{-1}(0)=\tilde{f}^{-1}(0)$ is a
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric (or isoparametric
respectively) hypersurface around $x_{0}$ for both $F$ and $\tilde{F}$.

To prove Theorem 1.1, we use the geodesic correspondence for homothetic
navigation to prove $\tilde{f}$ is a normalized transnormal function for
$\tilde{F}$ (see Lemma 4.2). A comparison between
$\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f$ (see Lemma 4.3) and
$\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}$ indicates
$\tilde{f}$ is $\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric iff $f$ is
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric.

To find the relation between $\Delta^{F}f$ and $\Delta^{\tilde{F}}\tilde{f}$,
and prove $\tilde{f}$ is isoparametric for $\tilde{F}$ when $f$ is for $F$, we
need a few more ingredients. For the Finsler metric $F$ and the transnormal
function $f$, $\Delta^{F}f$ only differs from
$\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f$ by the S-curvature
$S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,\nabla^{F}f(x))$ (see Lemma 14.1.2
in [23] or Lemma 3.1 below). For the Lorentz Finsler metric $\tilde{F}$ and
the transmoral function $\tilde{f}$, we can prove a similar equality (see
Lemma 3.4). The S-curvatures for $F$ and $\tilde{F}$ are related by Lemma 2.6,
which is an analog of Theorem 1.3 in [29]. Then the comparison between
$\Delta^{F}f$ and $\Delta^{\tilde{F}}\tilde{f}$ can be easily summarized from
above observations (see Lemma 4.4).

######  Remark 1.2.

We believe that almost the same argument can also prove the backward
correspondence, i.e., when $\tilde{f}$ in Theorem 1.1 is a
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric (or isoparametric) function
for $\tilde{F}$ normalized at $x_{0}$, then $f$ is a
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric (or isoparametric for $F$
respectively) function normalized at $x_{0}$. Isoparametric functions,
isoparametric hypersurfaces, and their correspondence for homothetic
navigation can be further generalized to pseudo-Finsler geometry with
arbitrary signature type, as long as the existence and choice for the gradient
field can be settled.

Theorem 1.1 can provide many examples of isoparametric functions and
isoparametric hypersurfaces in Lorentz Finsler geometry. Take a Lorentz Funk
metric $\tilde{F}$ for example. When $\tilde{F}$ is a Lorentz Randers metric
defined on the complement $M$ of the closed unit ball in $\mathbb{R}^{n}$, it
is induced by a homothetic navigation with the datum $(F,V)$, in which $F$ is
the standard Euclidean metric and $V(x)=-x$. By [22] and Theorem 1.1,
hyperplanes, spheres and spherical cylinders provide isoparametric
hypersurfaces for $\tilde{F}$ around any $x_{0}$ with $|x_{0}|>1$. See Section
4.2 for the precise description of these examples and some explicit
isoparametric functions. When $\tilde{F}$ is not Randers, it is induced by the
homothetic navigation with the datum $(F,V)$, in which $F$ is a Minkowski
metric and $V(x)=-x$ is the same. Then we may use Theorem 1.1 and [13] to
similarly find isoparametric hypersurfaces for $\tilde{F}$.

This paper is organized as following. In Section 2, we recall some preliminary
knowledge on Finsler metric, Lorentz Finsler metric, navigation process and
geometric correspondences for homothetic navigation. In Section 3, we recall
the definitions of isoparametric functions and isoparametric hypersurfaces for
Finsler metrics (with or without a smooth measure involved), and generalize
these notions to Lorentz Finsler metrics. In Section 4, we prove Theorem 1.1
and provide some examples of isoparametric functions and isoparametric
hypersurfaces in Funk space of Lorentz Randers type.

##  2\. Preliminaries

###  2.1. Finsler metric and Lorentz Finsler metric

A Finsler metric on $M$ is a continuous function $F:TM\rightarrow\mathbb{R}$
satisfying:

  1. (1)

the positiveness and smoothness, i.e., $F$ is positive and smooth when
restricted to $TM\backslash 0$;

  2. (2)

the positive 1-homogeneity, i.e., $F(x,\lambda y)=\lambda F(x,y)$ for any
$x\in M$, $y\in T_{x}M\backslash\\{0\\}$ and $\lambda\geq 0$;

  3. (3)

the strong convexity, i.e., for any $y\in T_{x}M\backslash\\{0\\}$, the
fundamental tensor

$\langle u,v\rangle_{y}^{F}=\frac{1}{2}\frac{\partial^{2}}{\partial s\partial
t}|_{s=t=0}F^{2}(y+su+tv)$

is an inner product on $T_{x}M$.

Using standard local coordinates $(x^{i},y^{i})$, i.e., $x=(x^{i})\in M$ and
$y=y^{i}\partial_{x^{i}}\in T_{x}M$, the fundamental tensor can also be
presented as $\langle u,v\rangle^{F}_{y}=g_{ij}u^{i}v^{j}$, where
$u=u^{i}\partial_{x^{i}}$, $v=v^{j}\partial_{x^{j}}$ and
$(g_{ij})=(\frac{1}{2}[F^{2}]_{y^{i}y^{j}})$ is the Hessian metric [1].

More generally, a (conic) pseudo-Finsler metric on $M^{n}$ is defined on a
conic open subset $\mathcal{A}$ in $TM\backslash 0$. The conic property of
$\mathcal{A}\subset TM\backslash 0$ is referred to
$\mathcal{A}_{x}=\mathbb{R}_{>0}\mathcal{A}_{x}$ for each
$\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$. We call the function
$F:\mathcal{A}\rightarrow\mathbb{R}_{>0}$ a pseudo-Finsler metric on $M$ if
the positiveness, smoothness and positive 1-homogeneity are satisfied, and the
fundamental tensor $\langle\cdot,\cdot\rangle_{y}^{F}$ for any
$y\in\mathcal{A}$ is non-degenerate [17]. See [18] for the original of pseudo-
Finsler metrics in physics.

In this paper, we will study a pseudo-Finsler metric
$F:\mathcal{A}\rightarrow\mathbb{R}_{>0}$ satisfying the following properties:

  1. (1)

the signature type of $\langle\cdot,\cdot\rangle_{y}^{F}$ is $(1,n-1)$
everywhere;

  2. (2)

it satisfies the following Condition (C): for every $x\in M$,
$\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$ is convex.

We will simply call this $F$ a Lorentz Finsler metric satisfying Condition
(C).

Many geometric notions in Finsler geometry [1] can be naturally generalized to
pseudo-Finsler geometry.

The geodesic spray $G$ is a smooth tangent vector field on $\mathcal{A}$. With
respect to standard local coordinates $(x^{i},y^{i})$, the geodesic spray has
the presentation

$G=y^{i}\partial_{x^{i}}-2G^{i}\partial_{y^{i}},\mbox{ in which
}G^{i}=\frac{1}{4}g^{il}([F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}).$

The smooth curve $c(t)$ is called a geodesic (with positive constant speed) if
its lifting $(c(t),\dot{c}(t))$ is an integral curve of the geodesic spray
$G$. Using standard local coordinates, the geodesic $c(t)=(c^{i}(t))$ is
determined by the ODE system

$\ddot{c}^{i}(t)+2G^{i}(c(t),\dot{c}(t))=0,\quad\forall i.$

Let $\mathrm{d}\mu$ be a smooth measure with local coordinates presentation
$\mathrm{d}\mu=\sigma(x)\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$, in which
$\sigma(x)$ is a positive smooth function. Then the S-curvature
$S^{F}_{\mathrm{d}\mu}:\mathcal{A}\rightarrow\mathbb{R}$ for $F$ and
$\mathrm{d}\mu$ is the derivative of the distortion function

$\tau^{F}_{\mathrm{d}\mu}(x,y)=\ln\left(\frac{\sqrt{|\det(g_{ij}(x,y))|}}{\sigma(x)}\right)$

in the direction of the geodesic spray $G$.

######  Remark 2.1.

Notice that in Finsler geometry, there is a canonical smooth measure called
the B.H. measure, i.e.,
$\mathrm{d}\mu^{F}_{\mathrm{BH}}=\sigma(x)\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$
with

$\sigma(x)=\frac{\mathrm{Vol}(\\{(y^{i})|(\sum_{i=1}^{n}(y^{i})^{2})^{1/2}\leq
1\\})}{\mathrm{Vol}(\\{(y^{i})|F(x,y^{i}\partial_{x^{i}})\leq 1\\})},$

where $\mathrm{Vol}(\cdot)$ is the standard measure in the Euclidean space
$\mathbb{R}^{n}$. Generally speaking, B.H. measure is not well defined in
pseudo-Finsler geometry.

###  2.2. Finsler and Lorentz Finsler metrics induced by navigation

Consider a Finsler metric $F$ on $M$ and a smooth tangent field $V$. Then the
Zermelo navigation process formally defines a new metric $\tilde{F}$ by the
equality

$F(x,y)=\tilde{F}(x,y+F(x,y)V(x)),\quad\forall x\in M,y\in
T_{x}M\backslash\\{0\\}.$ (2.1)

Geometrically, we parallelly shift the indicatrix $S_{x}^{F}M=\\{y\in
T_{x}M\mbox{ with }F(x,y)=1\\}$ of $F$ by the vector $V(x)$ in $T_{x}M$, then
we get another indicatrix which uniquely determines the metric $\tilde{F}$.

To be precise, the new metric $\tilde{F}$ is still a Finsler metric where
$F(-V(x))<1$ is satisfied. For example, the navigation presentation for the
Finsler metric $\tilde{F}$ of Randers type is

$\tilde{F}(x,y)=\frac{\sqrt{(1-\langle V(x),V(x)\rangle)\langle
y,y\rangle+\langle y,V(x)\rangle^{2}}-\langle V(x),y\rangle}{1-\langle
V(x),V(x)\rangle}.$

Here $\langle\cdot,\cdot\rangle=F(\cdot)^{2}$ is a Riemannian metric and $V$
is a tangent field satisfying $\langle V(x),V(x)\rangle<1$ everywhere.

The relation between the fundamental tensors
$\langle\cdot,\cdot\rangle_{y}^{F}$ and
$\langle\cdot,\cdot\rangle^{\tilde{F}}_{\tilde{y}}$ with
$F(x,y)=\tilde{F}(x,\tilde{y})=1$ can be revealed by taking the mixed second
derivative of (2.1) with respect to the smooth parameters $s$ and $t$ in
$y=y(s,t)\in S^{F}_{x}M$ with $\frac{\partial}{\partial s}y(0,0)=u$ and
$\frac{\partial}{\partial t}y(0,0)=v$ in
$T_{y}(S^{F}_{x}M)=T_{\tilde{y}}(S^{\tilde{F}}_{x}M)$, and evaluating it at
$s=t=0$, i.e.,

$\langle
u,v\rangle^{F}_{y}=\frac{1}{1-\langle\tilde{y},V(x)\rangle^{\tilde{F}}_{\tilde{y}}}\langle
u,v\rangle^{\tilde{F}}_{\tilde{y}}.$ (2.2)

Using the inverse navigation process, $F$ can be induced by the datum
$(\tilde{F},-V)$, so we also have

$\langle u,v\rangle^{\tilde{F}}_{\tilde{y}}=\frac{1}{1+\langle
y,V(x)\rangle^{F}_{y}}\langle u,v\rangle^{F}_{y}.$ (2.3)

Notice that above calculation and formulae for fundamental tensors are also
valid when $\tilde{F}$ is conic Finsler or conic Lorentz Finsler.

If we have $F(-V(x))\equiv 1$, the induced $\tilde{F}$ is a conic Finsler
metric metric. For example, the navigation presentation for the Kropina metric
$\tilde{F}$ is

$F(x,y)=\frac{\langle y,y\rangle}{2\langle y,V(x)\rangle},\quad\forall x\in
M,y\in T_{x}M\mbox{ with }\langle y,V(x)\rangle>0,$

where $\langle\cdot,\cdot\rangle=F(\cdot)^{2}$ is a Riemannian metric and the
tangent field $V$ satisfies $F(V(x))\equiv 1$ everywhere.

The case that $F(-V(x))>1$ is the main concern in this paper. Figure 2 in [17]
visualizes the two conic metrics induced by the navigation process. In later
discussion, we just ignore the conic Finsler one and concentrate in the
Lorentz Finsler one. Rigorous description for this Lorentz Finsler metric
$\tilde{F}$ satisfying Condition (C), induced by the navigation process, is
given by the following lemma.

######  Lemma 2.2.

Let $F$ be a Finsler metric and $V$ a smooth tangent field satisfying
$F(-V(x))>1$ everywhere on $M$. Then (2.1), i.e.,
$F(x,y)=\tilde{F}(x,y+F(x,y)V(x))$ for $y\in T_{x}M$ satisfying $\langle
y,V(x)\rangle^{F}_{y}<-F(x,y)$, defines a Lorentz Finsler metric
$\tilde{F}:\mathcal{A}\rightarrow\mathbb{R}_{>0}$, such that each
$\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$ is the nonempty conic convex open
subset in $T_{x}M$ consisting of all $\tilde{y}=y+F(x,y)V(x)\in T_{x}M$ with
$\langle y,V(x)\rangle^{F}_{y}<-F(x,y)$.

Lemma 2.2 is a consequence of Proposition 2.2 in [17]. Here we present a self
contained proof for it.

Proof. It is obvious that each $\mathcal{A}_{x}$ is conic. We will prove

$\mathcal{A}_{x}=\\{t(y^{\prime}+V(x))|\forall t>0,y^{\prime}\in T_{x}M\mbox{
with }F(x,y^{\prime})<1\\}.$ (2.4)

Consider any ray contained in the right side of (2.4), i.e.,
$\mathbb{R}_{>0}(y^{\prime}+V(x))$ with $F(x,y^{\prime})<1$. Denote
$y(t)=y^{\prime}-t(V(x)+y^{\prime})$ and $\varphi(t)=F(y(t))$. The function
$\varphi(t)$ is continuous and non-negative, and it is smooth when $y(t)\neq
0$. Direct calculation shows

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\varphi(t)=\frac{\langle
y(t),-V(x)-y^{\prime}\rangle^{F}_{y(t)}}{F(y(t))}$ (2.5)

and by Cauchy inequality,

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\varphi(t)=\frac{\langle-V(x)-y^{\prime},-V(x)-y^{\prime}\rangle_{y(t)}^{F}\langle
y(t),y(t)\rangle_{y(t)}^{F}-(\langle-V(x)-y^{\prime},y(t)\rangle_{y(t)}^{F})^{2}}{F(y(t))^{3}}\geq
0,$ (2.6)

so $\varphi(t)$ is convex. Since $\varphi(0)=F(y^{\prime})<1$ and
$\varphi(1)=F(-V(x))>1$, there exists $y=y(t_{0})$ for some $t_{0}\in(0,1)$
with $\varphi(t_{0})=F(y)=1>f(0)$. By the convexity of $f(t)$ and (2.5),

$\frac{\mathrm{d}}{\mathrm{d}t}\varphi(t_{0})=\frac{\langle
y,-V(x)-y^{\prime}\rangle_{y}^{F}}{F(y)}=\frac{\langle
y,-V(x)-y\rangle_{y}^{F}}{(1-t_{0})F(y)}=\frac{-\langle
y,V\rangle_{y}^{F}-1}{(1-t_{0})F(y)}>0.$

So $\langle y,V(x)\rangle_{y}^{F}<-F(y)=-1$ and

$\mathbb{R}_{>0}\tilde{y}=\mathbb{R}_{>0}(y+V(x))=\mathbb{R}_{>0}((1-t_{0})(y^{\prime}+V(x)))=\mathbb{R}_{>0}(y^{\prime}+V(x))$

is contained in $\mathcal{A}_{x}$.

On the other hand, consider any ray $\mathbb{R}_{>0}\tilde{y}$ in
$\mathcal{A}_{x}$ where $\tilde{y}=y+V(x)$ with
$g_{y}^{F}(y,V(x))<-F(x,y)=-1$. Similar calculation as (2.5) shows
$\varphi(t)=F(y-t(y+V(x)))$ has positive derivative at $t=0$, so for
$y^{\prime}=y-t^{\prime}(y+V(x))$ with negative $t^{\prime}$ sufficiently
close to 0, $F(y^{\prime})<1$ and
$\mathbb{R}_{>0}(y+V(x))=\mathbb{R}_{>0}(y^{\prime}+V(x))$ is contained in the
right side of (2.4).

To summarize, above argument proves the presentation (2.4) for
$\mathcal{A}_{x}$. By this presentation, we see $\mathcal{A}_{x}$ is a cone
generated by the nonempty convex open subset $\\{y+V(x),\forall y\in
T_{x}M\mbox{ with }F(y)<1\\}$. So $\mathcal{A}_{x}$ is a nonempty convex conic
open subset in $T_{x}M$. On the other hand, the presentation (2.4) depends
continuously on $x\in M$, so $\mathcal{A}=\coprod_{x\in M}\mathcal{A}_{x}$ is
an open subset in $TM\backslash 0$, i.e., a Lorentz Finsler metric may be
defined on it.

Secondly, we proves (2.1) defines a positive smooth function $\tilde{F}$ on
$\mathcal{A}$. Denote the open subset $\mathcal{A}^{\prime}=\coprod_{x\in
M}\mathcal{A}^{\prime}_{x}$ in $TM\backslash 0$ in which
$\mathcal{A}^{\prime}_{x}$ is the subset of all $y\in T_{x}M\backslash 0$
satisfying $\langle y,V(x)\rangle^{F}_{y}<-F(x,y)$, and
$\phi(x,y)=(x,y+F(x,y)V(x))$ the surjective smooth map from
$\mathcal{A}^{\prime}$ to $\mathcal{A}$. Now we prove $\phi$ is a
diffeomorphism. Direct calculation shows its tangent map is isomorphic
everywhere. To see it is injective, we consider the possibility of
$y_{1}+F(x,y_{1})V(x)=y_{2}+F(x,y_{2})V(x)=v$ for any
$y_{1},y_{2}\in\mathcal{A}^{\prime}_{x}$. If $F(x,y_{1})=F(x,y_{2})$, we get
immediately $y_{1}=y_{2}$. Otherwise, we may assume $F(x,y_{1})<F(x,y_{2})$.
By similar calculation as (2.5) and (2.6), $\psi(t)=F(x,v-tV(x))$ is a
continuous convex non-negative function with

$\tfrac{\mathrm{d}}{\mathrm{d}t}\psi(F(x,y_{1}))=\frac{\langle
y_{1},-V(x)\rangle^{F}_{y_{1}}}{F(x,y_{1})}>1\quad\mbox{and}\quad\tfrac{\mathrm{d}}{\mathrm{d}t}\psi(F(x,y_{2}))=\frac{\langle
y_{2},-V(x)\rangle^{F}_{y_{2}}}{F(x,y_{2})}>1,$

and $\psi(t)$ is smooth when $\psi(t)\neq 0$. So the restriction of $\psi(t)$
to the interval $[F(x,y_{1}),F(x,y_{2})]$ is positive and smooth with
$\tfrac{\mathrm{d}}{\mathrm{d}t}\psi(t)>1$ everywhere. This is a contradiction
to Lagrange’s Mean Value Theorem because $\psi(F(x,y_{i}))=F(x,y_{i})$ for
$i=1,2$. To summarize, $\phi$ is a diffeomorphism from $\mathcal{A}^{\prime}$
to $\mathcal{A}$, and then (2.1) implies $\tilde{F}=F\circ\phi^{-1}$ is a
positive smooth function on $\mathcal{A}$. The positive 1-homogeneity of
$\tilde{F}$ is obvious.

Finally, we prove the fundamental tensor of $\tilde{F}$ has the signature type
$(1,n-1)$. At each $\tilde{y}=y+F(x,y)V(x)$ with
$\tilde{F}(x,\tilde{y})=F(x,y)=1$ and $\langle y,V(x)\rangle^{F}_{y}<-1$, the
fundamental tensor $\langle\cdot,\cdot\rangle^{\tilde{F}}_{\tilde{y}}$ is
obviously positive definite in the direction $\mathbb{R}\tilde{y}$. Apply
$\langle y,V(x)\rangle^{F}_{y}<-F(x,y)=-1$ to (2.3), we see
$\langle\cdot,\cdot\rangle^{\tilde{F}}_{\tilde{y}}$ is negative definite when
restricted to $T_{\tilde{y}}S^{\tilde{F}}_{x}M=T_{y}S^{F}_{x}M$, i.e., the
$\langle\cdot,\cdot\rangle^{\tilde{F}}_{\tilde{y}}$-orthogonal complement of
$\mathbb{R}\tilde{y}$. So $\langle\cdot,\cdot\rangle^{\tilde{F}}_{\tilde{y}}$
has the signature type $(1,n-1)$.

To summarize, $\tilde{F}:\mathcal{A}\rightarrow\mathbb{R}_{>0}$ is a Lorentz
Finsler metric, and it satisfies Condition (C).

###  2.3. Homothetic navigation and geometric correspondences

A smooth tangent vector field $V$ on the Finsler manifold $(M,F)$ is called
homothetic if it generates a one-parameter local subgroup of local homothetic
translations $\Psi_{t}$ satisfying

$(\Psi_{t}^{*}F)(x,y)=F(\Psi_{t}(x),(\Psi_{t})_{*}(y))=e^{-2ct}F(x,y),$ (2.7)

for each $x\in M$, $y\in T_{x}M$ and $t\in\mathbb{R}$, whenever $\Psi_{t}(x)$
is defined. The constant $c$ in (2.7) is called the dilation of $V$. We call
the homothetic field $V$ a Killing field when its dilation $c=0$. Using the
standard local coordinates $(x^{i},y^{i})$ on $TM$, the homothetic property
for $V$ can be equivalently described by $V^{\mathrm{c}}F=-2cF$, in which
$V^{\mathrm{c}}=V^{i}\partial_{x^{i}}+y^{j}\frac{\partial}{\partial
x^{j}}V^{i}\partial_{y^{i}}$ is the complete lifting of
$V=V^{i}(x)\partial_{x^{i}}$. This observation implies the following useful
fact (see Lemma 3.1 in [29]).

######  Lemma 2.3.

The restriction of the homothetic field $V$ with dilation $c$ to a unit speed
geodesic $\gamma(t)$ on the Finsler manifold $(M,F)$ satisfies

$\langle V(\gamma(t)),\dot{\gamma}(t)\rangle^{F}_{\dot{\gamma}(t)}\equiv
c_{0}-2ct,$ (2.8)

in which the constant $c_{0}$ may be taken as $c_{0}=\langle
V(\gamma(0)),\dot{\gamma}(0)\rangle^{F}_{\dot{\gamma}(0)}$ when $\gamma(t)$ is
defined for $t$ around $0$.

The equality (2.7) for $\Psi_{t}$ generated by $V$ immediately implies

$\langle(\Psi_{t})_{*}u,(\Psi_{t})_{*}v\rangle_{(\Psi_{t})_{*}y}^{F}=e^{-4ct}\langle
u,v\rangle_{y}^{F},\quad\forall u,v\in T_{x}M,$ (2.9)

from which we see the tangent map $(\Psi_{t})_{*}:TM\rightarrow TM$ preserves
the distortion function $\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,y)$
(defining the S-curvature $S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$ for $F$
and its B.H. measure) on $TM\backslash 0$. Meanwhile, $(\Psi_{t})_{*}$
preserves the geodesic spray $G$ because $\Psi_{t}$ maps any constant speed
geodesic to a constant speed geodesic, so $(\Psi_{t})_{*}$ preserves the
S-curvature $S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$, i.e.,
$S^{F}(x,y)=S^{F}(\Psi_{t}(x),(\Psi_{t})_{*}y)$, $\forall(x,y)\in TM\backslash
0$ and $t\in\mathbb{R}$ where $\Psi_{t}$ is defined. This observation is valid
not only for those $\Psi_{t}$ generated by a homothetic vector field but also
for all local homothetic translations, which can be summarized as following.

######  Lemma 2.4.

Any local homothetic translation on a Finsler manifold $(M,F)$ preserves the
S-curvature $S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$.

We say $\tilde{F}$ is induced by a homothetic (or Killing) navigation if the
tangent vector field $V$ in the navigation datum $(F,V)$ is a homothetic (or
Killing respectively) field. The geometric correspondences between $F$ and
$\tilde{F}$ have been extensively explored in Finsler and pseudo-Finsler
geometry [17][29]. In this paper, we will use the correspondences for the
geodesics and for S-curvatures before and after a homothetic navigation when
$F$ is Finsler and $\tilde{F}$ is Lorentz Finsler.

To unify the discussion for Killing navigation and non-Killing homothetic
navigation, we denote $\alpha_{0}(t)=t$ and
$\alpha_{c}(t)=\frac{e^{2ct}-1}{2c}$ for $c\in\mathbb{R}\backslash\\{0\\}$,
i.e., $\alpha_{c}(t)$ satisfies
$\frac{\mathrm{d}}{\mathrm{d}t}\alpha_{c}(t)=e^{2ct}$ and $\alpha_{c}(0)=0$.
The variable $t$ here and later can always be assumed to be sufficiently close
to 0, because our discussion is local.

######  Lemma 2.5.

Let $\tilde{F}$ be the Lorentz Finsler metric induced by the navigation
process with the datum $(F,V)$, in which $V$ is a homothetic field for $F$
with dilation $c$, satisfying $F(-V(x))>1$ everywhere. Denote $\Psi_{t}$ the
flow of local diffeomorphisms generated by $V$. Suppose $\gamma(t)$ is a unit
speed geodesic on $(M,F)$ satisfying
$c_{0}=\langle\dot{\gamma}(0),V(\gamma(0))\rangle^{F}_{\dot{\gamma}(0)}<-1$.
Then $\tilde{\gamma}(t)=\Psi_{t}(\gamma(\alpha_{c}(t)))$ (for $t$ sufficiently
close to 0) is a unit speed geodesic on $(M,\tilde{F})$. Further more, for any
tangent vectors $v_{1},v_{2}\in T_{\gamma(\alpha_{c}(t))}M$ with

$\langle
v_{1},\dot{\gamma}(\alpha_{c}(t))\rangle^{F}_{\dot{\gamma}(\alpha_{c}(t))}=\langle
v_{2},\dot{\gamma}(\alpha_{c}(t))\rangle^{F}_{\dot{\gamma}(\alpha_{c}(t))}=0,$

$(\Psi_{t})_{*}(v_{1})$ and $(\Psi_{t})_{*}(v_{2})$ in
$T_{\tilde{\gamma}(t)}M$ satisfy

$\langle(\Psi_{t})_{*}(v_{1}),\dot{\tilde{\gamma}}(t)\rangle^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)}=\langle(\Psi_{t})_{*}(v_{2}),\dot{\tilde{\gamma}}(t)\rangle^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)}=0,$

and

$\langle(\Psi_{t})_{*}(v_{1}),(\Psi_{t})_{*}(v_{2}),\rangle^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)}=\frac{1}{c_{0}+1}e^{-2ct}\langle
v_{1},v_{2}\rangle^{F}_{\dot{\gamma}(\alpha_{c}(t))}.$ (2.10)

Proof. The first statement in Lemma 2.5 is a special case of Theorem 1.2 in
[17]. For the convenience in later discussion, we denote
$y(t)=(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))$ and
present

$\dot{\tilde{\gamma}}(t)=y(t)+V(\tilde{\gamma}(t)).$ (2.11)

By the homothetic property of $\Psi_{t}$,

$y(t)=(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))=e^{2ct}(\Psi_{t})_{*}(\dot{\gamma}(\alpha_{c}(t)))$

is a $F$-unit vector. Using (2.8) and (2.9), we get

$\displaystyle\langle y(t),V(\tilde{\gamma}(t))\rangle^{F}_{y(t)}$
$\displaystyle=$
$\displaystyle\langle(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t))),(\Psi_{t})_{*}(V({\gamma}(\alpha_{c}(t)))\rangle^{F}_{(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))}$
(2.12) $\displaystyle=$ $\displaystyle
e^{-4ct}\langle\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)),V(\gamma(\alpha_{c}(t)))\rangle^{F}_{\dot{\gamma}(\alpha_{c}(t))}$
$\displaystyle=$ $\displaystyle
e^{-2ct}\langle\dot{\gamma}(\alpha_{c}(t)),V(\gamma(\alpha_{c}(t)))\rangle^{F}_{\dot{\gamma}(\alpha_{c}(t))}$
$\displaystyle=$ $\displaystyle e^{-2ct}(c_{0}-2c\alpha_{c}(t))$
$\displaystyle=$ $\displaystyle e^{-2ct}(c_{0}+1)-1,$

in which $c_{0}=\langle\dot{\gamma}(0),V(\gamma(0))\rangle<-1$. So

$\langle
y(t),V(\tilde{\gamma}(t))\rangle^{F}_{y(t)}<-F(\tilde{\gamma}(t),y(t))=-1,$

and $\dot{\tilde{\gamma}}(t)=y(t)+V(\tilde{\gamma}(t))$ is a $\tilde{F}$-unit
vector in $\mathcal{A}_{\tilde{\gamma}(t)}$.

The other statements in Lemma 2.5 then follows (2.3) and (2.12) by almost the
same argument as for Lemma 5.2 in [29].

######  Lemma 2.6.

Let $\tilde{F}$ be the Lorentz Finsler metric on $M^{n}$ induced by the
homothetic navigation process with the datum $(F,V)$, in which the homothetic
tangent vector field $V$ has the dilation constant $c$ and satisfies
$F(-V(x))>1$ everywhere. Then with respect to the B.H. measure
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$ of $F$, the S-curvatures
$S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$ for $F$ and
$S^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$ for $\tilde{F}$, are related
by the following equality,

$S^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,\tilde{y})=S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,y)+(n+1)cF(x,y),$
(2.13)

for any $y\in T_{x}M\backslash\\{0\\}$ satisfying $\langle
y,V(x)\rangle_{y}^{F}<-F(y)$ and $\tilde{y}=y+F(x,y)V(x)$.

Proof. This proof is very similar to that of Theorem 1.3 in [29]. We only need
to prove (2.13) when $F(x,y)=\tilde{F}(x,\tilde{y})=1$. Let $\gamma(t)$ be the
unit speed geodesic on $(M,F)$ satisfying $\gamma(0)=x$ and
$\dot{\gamma}(0)=y$, and denote
$\tilde{\gamma}(t)=\Psi_{t}(\gamma(\alpha_{c}(t)))$ in which $\Psi_{t}$ is the
flow of local diffeomorphisms generated by $V$.

We choose the smooth vector fields $e_{i}(t)$, $1\leq i\leq n$, along
$\gamma(t)$, such that $e_{1}(t)=\dot{\gamma}(t)$ and they are orthonormal
w.r.t. $\langle\cdot,\cdot\rangle^{F}_{\dot{\gamma}(t)}$ at each $\gamma(t)$.
Then we have the induced smooth vector fields along $\tilde{\gamma}(t)$:

$\displaystyle\bar{e}_{1}(t)$ $\displaystyle=$
$\displaystyle(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))=e^{2ct}(\Psi_{t})_{*}(e_{1}(\alpha_{c}(t))),$
$\displaystyle\bar{e}_{i}(t)$ $\displaystyle=$
$\displaystyle(\Psi_{t})_{*}(e_{i}(\alpha_{c}(t))),\quad\forall 1<i\leq n;$
$\displaystyle\tilde{e}_{1}(t)$ $\displaystyle=$
$\displaystyle\dot{\tilde{\gamma}}(t)=\bar{e}_{1}(t)+V(\tilde{\gamma}(t)),$
$\displaystyle\tilde{e}_{i}(t)$ $\displaystyle=$
$\displaystyle\bar{e}_{i}(t),\quad\forall 1<i\leq n.$

Then at each $\tilde{\gamma}(t)$, $\\{\bar{e}_{i}(t),\forall 1\leq i\leq n\\}$
and $\\{\tilde{e}_{i}(t),\forall 1\leq i\leq n\\}$ are two bases for
$T_{\tilde{\gamma}(t)}M$. Denote

$\displaystyle\mathrm{vol}(t)$ $\displaystyle=$
$\displaystyle\mathrm{Vol}(\\{(y^{i})|F(\gamma(t),y^{i}e_{i}(t))\leq 1\\}),$
$\displaystyle\overline{\mathrm{vol}}(t)$ $\displaystyle=$
$\displaystyle\mathrm{Vol}(\\{(y^{i})|F(\tilde{\gamma}(t),y^{i}\bar{e}_{i}(t))\leq
1\\}),$ $\displaystyle\widetilde{\mathrm{vol}}(t)$ $\displaystyle=$
$\displaystyle\mathrm{Vol}(\\{(y^{i})|\tilde{F}(\tilde{\gamma}(t),y^{i}\tilde{e}_{i}(t))\leq
1\\}),$

in which $\mathrm{Vol}(\cdot)$ is the standard measure in the Euclidean space
$\mathbb{R}^{n}$.

The distortion function $\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$ can be
calculated at each $(\gamma(t),\dot{\gamma}(t))$ using $\\{e_{i}(t),\forall
1\leq i\leq n\\}$, i.e.,

$\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\gamma(t),\dot{\gamma}(t))=\ln\sqrt{\det(\langle
e_{i}(t),e_{j}(t)\rangle^{F}_{\dot{\gamma}(t)})}+\ln\mathrm{vol}(t)+C_{0},$
(2.14)

in which $C_{0}$ is some universal constant depending on $n$. Similarly the
distortion function $\tau^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}$ can
be presented as

$\tau^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\gamma(t),\dot{\gamma}(t))=\ln\sqrt{|\det(\langle\tilde{e}_{i}(t),\tilde{e}_{j}(t)\rangle^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)})|}+\ln\widetilde{\mathrm{vol}}(t)+C_{0}.$
(2.15)

Using Lemma 2.5, the relation between the two determinants in (2.14) and
(2.15) can be given as

$|\det(\langle\tilde{e}_{i}(t),\tilde{e}_{j}(t)\rangle^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)})|=\frac{1}{|1+c_{0}|^{n-1}}e^{-2c(n-1)t}\det(\langle
e_{i}(\alpha_{c}(t)),e_{j}(\alpha_{c}(t))\rangle_{\dot{\gamma}(t)})^{F},$
(2.16)

where
$c_{0}=\langle\dot{\gamma}(0),V(\gamma(0))\rangle_{\dot{\gamma}(0)}^{F}<-1$.

By multi-variable integral, (2.12) and the homothetic property of $V$,

$\displaystyle\widetilde{\mathrm{vol}}(t)$ $\displaystyle=$
$\displaystyle\frac{\overline{\mathrm{vol}}(t)}{|1+\langle
V(\tilde{\gamma}(t)),\bar{e}_{1}(t)\rangle^{F}_{\bar{e}_{1}(t)}|}=\frac{1}{|1+c_{0}|}e^{2ct}\overline{\mathrm{vol}}(t)$
(2.17) $\displaystyle=$
$\displaystyle\frac{1}{|1+c_{0}|}e^{2ct}e^{2c(n-1)t}\mathrm{vol}(\alpha_{c}(t))=\frac{1}{|1+c_{0}|}e^{2cnt}\mathrm{vol}(\alpha_{c}(t)).$

Summarizing (2.14)-(2.17), we get

$\tau^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\tilde{\gamma}(t),\dot{\tilde{\gamma}}(t))=\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\gamma(\alpha(t)),\dot{\gamma}(\alpha_{c}(t)))+c(n+1)t-\frac{n+1}{2}\ln|1+c_{0}|,$

so

$\displaystyle S^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,\tilde{y})$
$\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tau^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\tilde{\gamma}(t),\dot{\tilde{\gamma}}(t))|_{t=0}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\alpha_{c}(0)\frac{\mathrm{d}}{\mathrm{d}s}\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\gamma(s),\dot{\gamma}(s))|_{s=0}+(n+1)c$
$\displaystyle=$ $\displaystyle
S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,y)+(n+1)c,$

which ends the proof.

##  3\. Isoparametric hypersurface

###  3.1. Isoparametric hypersurface in Finsler geometry

Suppose $f$ is a regular real function on the Finsler manifold $(M,F)$ (so the
smooth one-form $\mathrm{d}f$ is non-vanishing everywhere). Then its gradient
field $\nabla^{F}f$ is a smooth tangent vector field on $M$, non-vanishing
everywhere, determined by $\langle\nabla^{F}f,u\rangle^{F}_{\nabla
f}=(\mathrm{d}f)(u)=u\cdot f$ for each tangent vector $u$, or equivalently,
$\nabla f=\mathcal{L}^{-1}(\mathrm{d}f)$, where $\mathcal{L}:TM\backslash
0\rightarrow T^{*}M\backslash 0$ is the Legendre transformation. The Laplacian
of $f$ with respect to the smooth measure $\mathrm{d}\mu$ is

$\Delta^{F}_{\mathrm{d}\mu}f=\mathrm{div}_{\mathrm{d}\mu}\nabla^{F}f,$ (3.18)

The divergence in (3.18) is defined by
$\mathrm{div}_{\mathrm{d}\mu}X=L_{X}\mathrm{d}\mu$ for any smooth tangent
vector field $X$, in which $L$ is the Lie derivative. In particular, when
$\mathrm{d}\mu$ coincides with the measure for the osculation metric
$g^{F}_{\nabla f}$, the corresponding $\Delta^{F}_{\mathrm{d}\mu}f$ is simply
denoted as $\Delta^{F}f$. $\Delta^{F}f$ is in fact the Laplacian of $f$ with
respect to the osculation metric $g^{F}_{\nabla^{F}f}$.

We call the regular real function $f$ on the Finsler manifold $(M,F)$
transnormal, if $F(\nabla^{F}f)$ only depends on the values of $f$. Further
more, we call the transnormal function $f$ $\mathrm{d}\mu$-isoparametric if
$\Delta^{F}_{\mathrm{d}\mu}f$ only depends on the values of $f$. The level
sets of a $d\mu$-isoparametric function $f$ is called a
$\mathrm{d}\mu$-isoparametric hypersurface. Using $\Delta^{F}f$ instead of
$\Delta^{F}_{d\mu}f$ in above notions, we get isoparametric function and
isoparametric hypersurface instead. In this paper, our discussion for a
$d\mu$-isoparametric or isoparametric function is local where no critical
value or focal set is involved.

Replacing $f$ with $\varphi\circ f$ for some suitable one-variable smooth real
function $\varphi$, we may normalize a transnormal,
$\mathrm{d}\mu$-isoparametric, or isoparametric function at some $x_{0}\in M$
such that $F(\nabla f)\equiv 1$ and $f(x_{0})=0$ (see Lemma 4.1 in [13]).
Notice that the level sets before and after this normalization are essentially
the same.

The relation between $\Delta^{F}_{\mathrm{d}\mu}f$ and $\Delta^{F}f$ is
revealed by Lemma 14.1.2 in [23], which is reformulated as following.

######  Lemma 3.1.

Suppose $f$ is a regular real function on a Finsler manifold $(M,F)$, such
that the integral curves of $\nabla^{F}f$ are $F$-unit speed geodesics, then
we have

$\Delta^{F}f(x)=\Delta^{F}_{\mathrm{d}\mu}f(x)+S^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x)),$
(3.19)

in which the S-curvature $S^{F}_{\mathrm{d}\mu}$ is for $F$ and the smooth
measure $\mathrm{d}\mu$.

######  Remark 3.2.

Lemma 3.1 is enough for our later usage. However, similar argument can prove
(3.19) when integral curves of $\nabla^{F}f$ are geodesics with possibly
nonconstant speed. So by Lemma 4.1 in [27], any transnormal function $f$ on
the Finsler manifold $(M,F)$ satisfies (3.19) in Lemma 3.1.

###  3.2. Isoparametric hypersurface in a Lorentz Finsler manifold

Let $F:\mathcal{A}\rightarrow\mathbb{R}_{>0}$ be a Lorentz Finsler metric
satisfying Condition (C), i.e., for each $x\in M$,
$\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$ is a nonempty conic convex open
subset of $T_{x}M$.

For every $x\in M$, the Legendre transformation
$\mathcal{L}_{x}(y)=\langle\cdot,y\rangle^{F}_{y}$ for $y\in\mathcal{A}_{x}$
is a positively 1-homogeneous smooth map with isomorphic tangent maps
everywhere, so $\mathcal{L}_{x}(\mathcal{A}_{x})$ is a nonempty conic open
subset in $T_{x}^{*}M$.

######  Lemma 3.3.

The Legendre transformation $\mathcal{L}_{x}$ is injective.

Proof. Assume conversely that
$\mathcal{L}_{x}(y^{\prime}_{0})=\mathcal{L}_{x}(y^{\prime}_{1})$ for
different $y^{\prime}_{0},y^{\prime}_{1}\in\mathcal{A}_{x}$. Then
$y_{0}=\tfrac{y^{\prime}_{0}}{F(x,y^{\prime}_{0})}$ and
$y_{1}=\tfrac{y^{\prime}_{1}}{F(x,y^{\prime}_{1})}$ in $S^{\tilde{F}}_{x}M$
are linearly independent and
$\mathrm{ker}\mathcal{L}_{y_{0}}=\mathrm{ker}\mathcal{L}_{y_{1}}$. Since $F$
satisfies Condition (C), $\varphi(t)=F(y_{t})$ with $y_{t}=(1-t)y_{0}+ty_{1}$
is a smooth function for $t\in[0,1]$. Similar calculation as (2.6) shows

$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\varphi(t)=\frac{\langle
y_{1}-y_{0},y_{1}-y_{0}\rangle^{F}_{y_{t}}\langle
y_{t},y_{t}\rangle_{y_{t}}^{F}-(\langle
y_{t},y_{1}-y_{0}\rangle^{F}_{y_{t}})^{2}}{F(x,y_{t})^{3}}$

is negative for every $t\in[0,1]$ because
$\langle\cdot,\cdot\rangle_{y_{t}}^{F}$ has the signature type $(1,n-1)$ and
the two vectors $y_{1}-y_{0}$ and $y_{t}$ are linearly independent. Because
$\varphi(0)=\varphi(1)=1$, the concavity of $\varphi(t)$ implies
$\frac{\mathrm{d}}{\mathrm{d}t}\varphi(0)>0$ and
$\frac{\mathrm{d}}{\mathrm{d}t}\varphi(1)<0$. So the following vectors

$v_{0}=(y_{1}-y_{0})-\langle
y_{1}-y_{0},y_{0}\rangle^{F}_{y_{0}}y_{0}=(y_{1}-y_{0})-\frac{\mathrm{d}}{\mathrm{d}t}\varphi(0)y_{0}=(-1-\frac{\mathrm{d}}{\mathrm{d}t}\varphi(0))y_{0}+y_{1}$

and

$v_{1}=(y_{1}-y_{0})-\langle
y_{1}-y_{0},y_{1}\rangle^{F}_{y_{1}}=(y_{1}-y_{0})-\frac{\mathrm{d}}{\mathrm{d}t}\varphi(1)y_{1}=-y_{0}+(1-\frac{\mathrm{d}}{\mathrm{d}t}\varphi(1))y_{1}$

are linearly independent. However,
$\mathrm{ker}\mathcal{L}_{x}(y_{i})\cap\mathrm{span}\\{y_{0},y_{1}\\}=\mathbb{R}v_{i}$.
This is a contradiction to
$\mathrm{ker}\mathcal{L}_{x}(y_{0})=\mathrm{ker}\mathcal{L}_{x}(y_{0})$.

Denote $\mathcal{L}(x,y)=\mathcal{L}_{x}(y)$ for every $x\in M$ and
$y\in\mathcal{A}_{x}$. Then we see from Lemma 3.3 that $\mathcal{L}$ is a
diffeomorphism from $\mathcal{A}$ to the conic open subset
$\mathcal{L}(\mathcal{A})=\coprod_{x\in
M}\mathcal{L}_{x}(\mathcal{A}_{x})\subset T^{*}M\backslash 0$.

Now we consider a regular real function $f$ defined around $x_{0}\in M$.
Assume that the value of $\mathrm{d}f$ at $x_{0}$ is contained in
$\mathcal{L}_{x_{0}}(\mathcal{A}_{x_{0}})$, then
$\nabla^{F}f=\mathcal{L}^{-1}(\mathrm{d}f)$ uniquely defines a smooth tangent
vector field on some sufficiently small neighborhood $\mathcal{U}$ of $x_{0}$.
Equivalently, $\nabla^{F}f$ can be determined by $\langle
u,\nabla^{F}f\rangle^{F}_{\nabla f}=\mathrm{d}f(u)$, $\forall x\in\mathcal{U}$
and $u\in T_{x}M$. We will call it the gradient field of $f$.

Using $\nabla^{F}f$ around $x_{0}$, the Laplacian
$\Delta^{F}_{\mathrm{d}\mu}f=\mathrm{div}_{\mathrm{d}\mu}\nabla^{F}f$ for the
smooth measure $\mathrm{d}\mu$ can be similarly defined as in Finsler
geometry. When $\mathrm{d}\mu=\sqrt{|\det(g_{ij}(x,\nabla
f))|}\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$, i.e., it is the one induced by the
osculation Lorentz metric $g^{F}_{\nabla f}$, the corresponding Laplacian is
simply denoted as $\Delta^{F}f$, which coincides with the Laplacian of $f$ for
the osculation Lorentz metric $g^{F}_{\nabla^{F}f}$ [20].

We call $f$ transnormal if $F(\nabla^{F}f)$ only depends on the values of $f$.
Further more, we call the transnormal function $f$
$\mathrm{d}\mu$-isoparametric if $\Delta^{F}_{\mathrm{d}\mu}f$ only depends on
the values of $f$. The level set of a $d\mu$-isoparametric function $f$ is
called a $\mathrm{d}\mu$-isoparametric hypersurface. If we replace
$\Delta^{F}_{d\mu}f$ by $\Delta^{F}f$ in above notions, isoparametric function
and isoparametric hypersurface are defined.

The relation between $\Delta^{F}_{\mathrm{d}\mu}f$ and $\Delta^{F}f$ can be
revealed by the following analog of Lemma 3.1.

######  Lemma 3.4.

Suppose $f$ is a transnormal function on a Lorentz Finsler manifold $(M,F)$
such that the integral curves of $\nabla^{F}f$ are $F$-unit speed geodesics,
then we have

$\Delta^{F}f(x)=\Delta^{F}_{\mathrm{d}\mu}f(x)+S^{F}_{d\mu}(x,\nabla^{F}f(x)),$
(3.20)

in which the S-curvature $S^{F}_{d\mu}$ for the smooth measure
$\mathrm{d}\mu$.

Proof. Denote $\sigma(x)$ the positive smooth function in the local
presentation $\mathrm{d}\mu=\sigma(x)\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$ for
the smooth measure $\mathrm{d}\mu$. The divergence operator defining
$\Delta^{F}f$ is with respect to the smooth measure

$\displaystyle\sqrt{|\det(g_{ij}(x,\nabla^{F}f(x)))|}\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{|\det(g_{ij}(x,\nabla^{F}f(x)))|}}{\sigma(x)}\cdot\sigma(x)\mathrm{d}x^{1}\cdots\mathrm{d}x^{n}$
$\displaystyle=$ $\displaystyle
e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}\mathrm{d}\mu,$

in which $\tau^{F}_{\mathrm{d}\mu}$ is the distortion function defining the
S-curvature $S^{F}_{\mathrm{d}\mu}$ for $F$ and $\mathrm{d}\mu$. Using the
definition (3.18) of $\Delta^{F}f$ and the Lebniz property for Lie derivative,
we get

$\displaystyle\Delta^{F}f$ $\displaystyle=$
$\displaystyle\mathrm{div}_{e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}\mathrm{d}\mu}\nabla^{F}f=\frac{L_{\nabla^{F}f}(e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}\mathrm{d}\mu)}{e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}\mathrm{d}\mu}$
$\displaystyle=$
$\displaystyle\frac{[(\nabla^{F}f)e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}]\mathrm{d}\mu+e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))}L_{\nabla^{F}f}\mathrm{d}\mu}{e^{\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x)))}\mathrm{d}\mu}$
$\displaystyle=$
$\displaystyle\Delta^{F}_{\mathrm{d}\mu}f+(\nabla^{F}f)\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x)).$

Let $\gamma(t)$ be the integral curve of $\nabla^{F}f$ with $\gamma(0)=x$. By
our assumption, it is a $F$-unit speed geodesic. So

$\displaystyle(\nabla^{F}f)\tau^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x))=\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\tau^{F}_{\mathrm{d}\mu^{F}_{\mathrm{d}\mu}}(\gamma(t),\dot{\gamma}(t))=S^{F}_{\mathrm{d}\mu}(x,\nabla^{F}f(x)),$

which ends the proof of Lemma 3.1.

######  Remark 3.5.

We believe that similar calculation as in Finsler geometry can show the
gradient field $\nabla^{F}f$ for a transnormal function $f$ on a Lorentz
Finsler manifold generates geodesics with possibly nonconstant speeds, i.e.,
Lemma 3.4 is valid for all transnormal functions.

##  4\. Correspondence between isoparametric hypersurface

###  4.1. Proof of Theorem 1.1

Let $F$ be a Finsler metric on $M$ and $V$ a homothetic vector field for $F$
with dilation $c$ and $F(-V(x))>1$ everywhere. Denote $\Psi_{t}$ the flow of
local diffeomorphisms generated by $V$. Let
$\tilde{F}:\mathcal{A}\rightarrow\mathbb{R}_{>0}$ be the Lorentz Finsler
metric induced by the navigation process with the datum $(F,V)$. For each
$x\in M$, $\mathcal{A}_{x}=\mathcal{A}\cap T_{x}M$ is the conic convex open
subset consisting of all $\tilde{y}=y+F(x,y)V(x)$ with $\langle
y,V(x)\rangle_{y}^{F}<-F(x,y)$.

Let $f$ be a normalized transnormal function locally defined around $x_{0}$ in
the Finsler manifold $(M,F)$ satisfying

$f(x_{0})=0,\quad F(\nabla f)\equiv 1,\quad\mbox{and}\quad\langle\nabla
f(x),V(x)\rangle^{F}_{\nabla f(x)}<-1\mbox{ around }x_{0}.$ (4.21)

Denote $M_{t}=f^{-1}(t)$ the level sets of $f$ in some sufficiently small
neighborhood of $x_{0}$, and define the smooth map $\Psi$ around $x_{0}$ such
that $\Psi|_{M_{\alpha_{c}(t)}}=\Psi_{t}$.

######  Lemma 4.1.

$\Psi$ is an orientation reversing diffeomorphism around $x_{0}$.

Proof. The smooth map $\Psi$ fixes each $x\in M_{0}$. We only need to observe
the tangent map $\Psi_{*}:T_{x}M\rightarrow T_{x}M$ has negative determinant
for $x\in M_{0}$ sufficiently close to $x_{0}$. With respect to the inner
product $\langle\cdot,\cdot\rangle_{\nabla^{F}f(x)}^{F}$, $T_{x}M$ can be
orthogonally decomposed as the sum of $T_{x}M_{0}$ and
$\mathbb{R}\nabla^{F}f(x)$. The restriction of $\Psi_{*}$ to $T_{x}M_{0}$ is
identity, and it maps $\nabla^{F}f(x)$ to $\nabla^{F}f(x)+V(x)$. So
$\det(\Psi_{*})=\langle\nabla^{F}f(x),\nabla^{F}f(x)+V(x)\rangle^{F}_{\nabla^{F}f(x)}=1+\langle\nabla^{F}f(x),V(x)\rangle^{F}_{\nabla^{F}f(x)}$.
When $x\in M_{0}$ is sufficiently close to $x_{0}$,
$\langle\nabla^{F}f(x),V(x)\rangle^{F}_{\nabla^{F}f(x)}<-F(x,\nabla^{F}f(x))=-1$,
so we have $\det(\Psi_{*})<0$.

Lemma 4.1 enables us to define the smooth function $\tilde{f}$ around $x_{0}$
such that $\tilde{f}^{-1}(t)=\widetilde{M}_{t}=\Psi(M_{\alpha_{c}(t)})$.

In the discussion below, we fix an arbitrary $t$ which is sufficiently close
to 0, $x\in M_{\alpha_{c}(t)}$ and $\tilde{x}=\Psi(x)=\Psi_{t}(x)$ which are
sufficiently close to $x_{0}$. Denote $\gamma(\cdot)$ the integral curve of
$\nabla^{F}f$ with $\gamma(\alpha_{c}(t))=x$ and
$\tilde{\gamma}(\cdot)=\Psi(\gamma(\alpha_{c}(\cdot)))$. Then
$\tilde{\gamma}(0)=\gamma(0)\in M_{0}=\widetilde{M}_{0}$ is sufficiently close
to $x_{0}$, $\tilde{\gamma}(t)=\tilde{x}$, $\tilde{f}(\tilde{x})=t$,
$f(x)=\alpha_{c}(t)$ and $2cf(x)+1=e^{2ct}$.

Firstly, we prove the transnormal property for $\tilde{f}$.

By Lemma 4.1 in [27], $\gamma(\cdot)$ is a unit speed geodesic on $(M,F)$.
Then by Lemma 2.5, $\tilde{\gamma}(\cdot)$ is a unit speed geodesic on
$(M,\tilde{F})$, and
$T_{\tilde{\gamma}(t)}\overline{M}_{t}=\Psi_{*}(T_{\gamma(\alpha_{c}(t))}M_{\alpha_{c}(t)})=(\Psi_{t})_{*}(T_{\gamma(\alpha_{c}(t))}M_{\alpha_{c}(t)})$
is $g^{\tilde{F}}_{\dot{\tilde{\gamma}}(t)}$-orthogonal to
$\dot{\tilde{\gamma}}(t)$. So $\nabla^{\tilde{F}}\tilde{f}$ exists around
$x_{0}$. For arbitrary $\tilde{x}$,
$\nabla^{\tilde{F}}\tilde{f}(\tilde{x})=\dot{\tilde{\gamma}}(t)$ which is a
$\tilde{F}$-unit vector. Obviously, $\tilde{f}(x_{0})=f(x_{0})=0$. So we see

######  Lemma 4.2.

The function $\tilde{f}$ is a transnormal function for $\tilde{F}$ and it is
normalized at $x_{0}$.

Secondly, we compare $\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f(x)$ and
$\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}(\tilde{x})=(\Psi^{*}\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f})(x)$.
Since
$\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f=\mathrm{div}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\nabla^{F}f$
and
$\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}=\mathrm{div}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\nabla^{\tilde{F}}\tilde{f}$,
we need to compare $\nabla^{F}f$ and $\mathrm{d}\mu^{F}_{\mathrm{BH}}$ at $x$
with $\nabla^{\tilde{F}}\tilde{f}$ and $\mathrm{d}\mu^{F}_{\mathrm{BH}}$ at
$\tilde{x}$ respectively.

Direct calculation shows

$\nabla^{\tilde{F}}\tilde{f}(\tilde{x})=\dot{\tilde{\gamma}}(t)=\Psi_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))=\Psi_{*}(e^{2ct}\dot{\gamma}(\alpha_{c}(t)))=\Psi_{*}((2cf(x)+1)\nabla^{F}f(x)).$
(4.22)

Denote $\Phi=\Psi_{-t}\circ\Psi$. It is a local diffeomorphism around $x_{0}$
fixing each point in $M_{\alpha_{c}(t)}$ (so it fixes $x$). The tangent map
$\Phi_{*}:T_{x}M\rightarrow T_{x}M$ maps $T_{x}M_{\alpha_{c}(t)}$ identically
to itself. Meanwhile, $\Phi_{*}$ maps
$\frac{\mathrm{d}}{\mathrm{d}t}{\gamma}(\alpha_{c}(t))$ to

$(\Psi_{-t})_{*}(\dot{\tilde{\gamma}}(t))=(\Psi_{-t})_{*}((\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}{\gamma}(\alpha_{c}(t))+V(\tilde{x}))=\frac{\mathrm{d}}{\mathrm{d}t}{\gamma}(\alpha_{c}(t)))+V(x).$

So by (2.8),

$\displaystyle\Phi^{*}(\mathrm{d}\mu^{F}_{\mathrm{BH}}(x))$ $\displaystyle=$
$\displaystyle|\det(\Phi_{*}|_{T_{x}M})|\cdot\mathrm{d}\mu^{F}_{\mathrm{BH}}(x)$
$\displaystyle=$
$\displaystyle|\langle\dot{\gamma}(\alpha_{c}(t))+e^{-2ct}V(\alpha_{c}(t)),\dot{\gamma}(\alpha_{c}(t))\rangle^{F}_{\dot{\gamma}(x)}|\cdot\mathrm{d}\mu^{F}_{\mathrm{BH}}$
$\displaystyle=$
$\displaystyle|1+e^{-2ct}(c_{0}(x)-2c\alpha_{c}(t))|\cdot\mathrm{d}\mu^{F}_{\mathrm{BH}}(x)$
$\displaystyle=$ $\displaystyle
e^{-2ct}|1+c_{0}(x)|\mathrm{d}\mu^{F}_{\mathrm{BH}}(x),$

where
$c_{0}(x)=\langle\dot{\gamma}(0),V(\gamma(0))\rangle^{F}_{\dot{\gamma}(0)}<-1$
is a smooth function around $x_{0}$ which is constant along each integral
curve of $\nabla^{F}f$ (so we have $(\nabla^{F}f)c_{0}(x)=0$).

By the homothetic property of $\Psi_{t}$, i.e.,
$\Psi_{t}^{*}(\mathrm{d}\mu^{F}_{\mathrm{BH}}(\tilde{x}))=e^{-2cnt}\mathrm{d}\mu^{F}_{\mathrm{BH}}(x)$,
we have

$\displaystyle\Psi^{*}(\mathrm{d}\mu^{F}_{\mathrm{BH}}(\tilde{x}))$
$\displaystyle=$
$\displaystyle\Phi^{*}\Psi_{t}^{*}(\mathrm{d}\mu^{F}_{\mathrm{BH}}(\tilde{x}))=\Phi^{*}(e^{-2cnt}\mathrm{d}\mu^{F}_{\mathrm{BH}}(x))$
(4.23) $\displaystyle=$ $\displaystyle
e^{-2c(n+1)t}|1+c_{0}(x)|\mathrm{d}\mu^{F}_{\mathrm{BH}}(x)$ $\displaystyle=$
$\displaystyle|1+c_{0}(x)|(2cf(x)+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}}(x).$

Now we are ready to prove the following lemma comparing
$\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f$ and
$\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}$.

######  Lemma 4.3.

Let $f$ be a normalized transnormal function for $F$ satisfying (4.21), then

$\Psi^{*}\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}=(2cf+1)\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f-2cn.$
(4.24)

Proof. The proof uses (4.22), (4.23) and the same calculation as for Lemma 7.4
in [29], i.e.,

$\displaystyle\Psi^{*}\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}$
$\displaystyle=$
$\displaystyle\Psi^{*}\mathrm{div}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\Psi_{*}((2cf+1)\nabla^{F}f))=\mathrm{div}_{\Psi^{*}\mathrm{d}\mu^{F}_{\mathrm{BH}}}((2cf+1)\nabla^{F}f)$
$\displaystyle=$
$\displaystyle\mathrm{div}_{|1+c_{0}(x)|(2cf+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}}}((2cf+1)\nabla^{F}f)$
$\displaystyle=$
$\displaystyle\frac{L_{(2cf+1)\nabla^{F}f}(|1+c_{0}(x)|(2cf+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}})}{|1+c_{0}(x)|(2cf+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}}}$
$\displaystyle=$
$\displaystyle(\nabla^{F}f)(2cf+1)+(2cf+1)\frac{L_{\nabla^{F}f}(|1+c_{0}(x)|(2cf+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}})}{|1+c_{0}(x)|(2cf+1)^{-n-1}\mathrm{d}\mu^{F}_{\mathrm{BH}}}$
$\displaystyle=$ $\displaystyle
2c+(2cf+1)\nabla^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f+(2cf+1)(\nabla^{F}f)(\ln(2cf+1)^{-n-1})$
$\displaystyle+(2cf+1)(\nabla^{F}f)(\ln|1+c_{0}(x)|)$ $\displaystyle=$
$\displaystyle(2cf+1)\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f-2cn,$

in which $(\nabla^{F}f)(\ln|1+c_{0}(x)|)$ vanishes because $c_{0}(x)$ is
constant on each integral curve of $\nabla^{F}f$.

Thirdly, we compare $\Delta^{F}f(x)$ and
$\Delta^{\tilde{F}}\tilde{f}(\tilde{x})=(\Psi^{*}\Delta^{\tilde{F}}\tilde{f})(x)$.
We have observed in above discussion that the integral curves of $\nabla^{F}f$
and $\nabla^{\tilde{F}}\tilde{f}$ are the unit speed geodesics $\gamma(\cdot)$
and $\tilde{\gamma}(\cdot)$ for $F$ and $\tilde{F}$ respectively, so by Lemma
3.1 and Lemma 3.4, we have

$\displaystyle\Delta^{F}f(x)$ $\displaystyle=$
$\displaystyle\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f(x)+S^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(x,\nabla^{F}f(x)),\mbox{
and}$ (4.25) $\displaystyle\Delta^{\tilde{F}}\tilde{f}(\tilde{x})$
$\displaystyle=$
$\displaystyle\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}(\tilde{x})+S^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}(\tilde{x},\nabla^{\tilde{F}}\tilde{f}(\tilde{x})),$
(4.26)

in which the S-curvatures are for $\mathrm{d}\mu^{F}_{\mathrm{BH}}$. By Lemma
2.4 and Lemma 2.6,

$\displaystyle
S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{\tilde{F}}(\tilde{x},\nabla^{\tilde{F}}\tilde{f}(\tilde{x}))$
$\displaystyle=$ $\displaystyle
S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{\tilde{F}}(\Psi_{t}(\tilde{\gamma}(\alpha_{c}(t))),(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))+V(\tilde{\gamma}(t)))$
(4.27) $\displaystyle=$ $\displaystyle
S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{F}(\Psi_{t}(\tilde{\gamma}(\alpha_{c}(t))),(\Psi_{t})_{*}(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t))))+(n+1)c$
$\displaystyle=$ $\displaystyle
S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{F}(\tilde{\gamma}(\alpha_{c}(t)),\frac{\mathrm{d}}{\mathrm{d}t}\gamma(\alpha_{c}(t)))+(n+1)c$
$\displaystyle=$ $\displaystyle
e^{2ct}S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{F}(x,\nabla^{F}f(x))+(n+1)c$
$\displaystyle=$
$\displaystyle(2cf(x)+1)S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{F}(x,\nabla^{F}f(x))+(n+1)c.$

By Lemma 4.3 and (4.25)-(4.27),

$\displaystyle\Delta^{\tilde{F}}\tilde{f}(\tilde{x})$ $\displaystyle=$
$\displaystyle\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}(\tilde{x})+S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{\tilde{F}}(\tilde{x},\nabla^{\tilde{F}}\tilde{f}(\tilde{x}))$
$\displaystyle=$
$\displaystyle(2cf(x)+1)\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f(x)-2nc+(2cf(x)+1)S_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}^{F}(x,\nabla^{F}f(x))+(n+1)c$
$\displaystyle=$ $\displaystyle(2cf(x)+1)\Delta^{F}f(x)-(n-1)c.$

So we have the following lemma

######  Lemma 4.4.

Let $f$ be a normalized transnormal function for $F$ satisfying (4.21), then

$\Psi^{*}\Delta^{\tilde{F}}\tilde{f}=(2cf+1)\Delta^{F}f-(n-1)c.$ (4.28)

Finally, we summarize above discussion to the proof of Theorem 1.1.

Proof of Theorem 1.1. Suppose that $f$ is a
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric function for $F$ satisfying
(4.21). We see from Lemma 4.1 and Lemma 4.2 that the function $\tilde{f}$
determined by
$\tilde{f}^{-1}(t)=\widetilde{M}_{t}=\Psi_{t}(M_{\alpha_{c}(t)})=\Psi_{t}(f^{-1}(\alpha_{c}(t)))$
around $x_{0}$ is a transnormal function normalized at $x_{0}$. Further more,
since $\Delta^{F}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}f$ is constant on each
$M_{\alpha_{c}(t)}$, Lemma 4.3 indicates that
$\Delta^{\tilde{F}}_{\mathrm{d}\mu^{F}_{\mathrm{BH}}}\tilde{f}$ is constant on
each $\widetilde{M}_{t}$ as well, i.e., $\tilde{f}$ is
$\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric for $\tilde{F}$. In
particular, $\tilde{f}^{-1}(0)=\widetilde{M}_{0}=\widetilde{M}_{0}=f^{-1}(0)$
is a $\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric hypersurface for the
Lorentz Finsler metric $\tilde{F}$.

When $f$ is an isoparametric function for $F$ satisfying (4.21), we use Lemma
4.4 instead of Lemma 4.3. The other argument is similar.

###  4.2. Some explicit examples

Theorem 1.1 can provide many examples of locally defined isoparametric
hypersurfaces for a Lorentz Finsler metric $\tilde{F}$ induced by homothetic
navigation. Here we discuss some examples when $\tilde{F}$ is a Funk metric of
Lorentz Randers type.

Denote $\Omega=\\{x\in\mathbb{R}^{n}\mbox{ with }|x|\leq 1\\}$, in which
$|x|=\langle x,x\rangle^{1/2}=\sum_{i=1}^{n}(x^{i})^{2}$ for
$x=(x^{1},\cdots,x^{n})$ is the standard Euclidean norm on $\mathbb{R}^{n}$.
Then a Lorentz Finsler metric $\tilde{F}$ on
$M=\mathbb{R}^{n}\backslash\Omega$ can be induced by the navigation process
with the datum $(F,V)$, where $F$ is the standard Euclidean metric on
$\mathbb{R}^{n}$ and $V(x)=-x$ is a homothetic field with dilation
$\tfrac{1}{2}$. The metric $\tilde{F}$ generalizes the Funk metric in Finsler
geometry, so we may call it a Lorentz Funk metric. It is of Randers type
because it can be explicitly presented as

$\displaystyle\tilde{F}(x,y)=\frac{\sqrt{\langle
x,y\rangle^{2}-(|x|^{2}-1)|y|^{2}}-\langle x,y\rangle}{|x|^{2}-1},$

where $x\in M=\mathbb{R}^{n}\backslash{\Omega}$, i.e., $|x|>1$ and
$y\in\mathcal{A}_{x}=\\{y|\langle
x,y\rangle^{2}-(|x|^{2}-1)|y|^{2}>0\\}\subset\mathbb{R}^{n}=T_{x}M$. Since $F$
is Riemannian, the $\mathrm{d}\mu^{F}_{\mathrm{BH}}$-isoparametric properties
and isoparametric properties for $\tilde{F}$ coincide.

Let $x_{0}$ by any point in $M=\mathbb{R}^{n}\backslash\Omega$ and $N$ an
isoparametric hypersurface in $(\mathbb{R}^{n},F)$ passing $x_{0}$, i.e., $N$
is a hyperplane, a sphere, or a spherical cylinder [22]. Denote $\mathbf{n}$
the $F$-unit normal vector of $N$ at $x_{0}$. Then Theorem 1.1 implies that
$N$ is isoparametric for $\tilde{F}$ in a sufficiently small neighborhood of
$x_{0}$ when $|\langle x,\mathbf{n}\rangle|>1$. Notice that there are two
isoparametric functions $f_{\pm}$ for $F$ normalized at $x$ with
$f^{-1}_{\pm}(0)=N$ and $\nabla^{F}f_{\pm}(x_{0})=\pm\mathbf{n}$. To get the
isoparametric function $\tilde{f}$ for $\tilde{F}$ in Theorem 1.1, we need to
take $f$ from $f_{\pm}$ such that $\langle x_{0},\nabla^{F}f(x_{0})\rangle>1$.

Some isoparametric hypersurfaces in $\tilde{F}$ exist globally and the
corresponding normalized isoparametric functions can be easily presented. Fix
an arbitrary $a>1$. Then the function $f_{1}(x)=|x|-a$ is an isoparametric
function for $F$ normalized at any point on the sphere
$S_{o}(a)=\\{x||x|=a\\}$. By Theorem 1.1,
$\tilde{f}_{1}(x)=\ln\frac{a-1}{|x|-1}$ is an isoparametric function for
$\tilde{F}$ in some neighborhood of $S_{o}(a)$. So each sphere $S_{o}(a)$ with
$a>1$ is an isoparametric hypersurface for $\tilde{F}$. The function
$f_{2}(x)=x^{1}-a$ for $x=(x^{1},\cdots,x^{n})$ is another isoparametric
function for $F$ normalized at any point on the hyperplane $x^{1}=a$. By
Theorem 1.1, $\tilde{f}_{2}(x)=\ln\tfrac{a-1}{x^{1}-1}$ is an isoparametric
function for $\tilde{F}$ in some neighborhood of this hyperplane. So each
hyperplane $x^{1}=a$ with $a>1$ is an isoparametric hypersurface for
$\tilde{F}$.

Acknowledgement. This paper is supported by Beijing Natural Science Foundation
(Z180004), National Natural Science Foundation of China (11771331, 11821101,
12001007), Capacity Building for Sci-Tech Innovation – Fundamental Scientific
Research Funds (KM201910028021), Natural Science Foundation of Anhui province
(2008085QA03, 1908085QA03). The first author thank Anhui University and
Sichuan University for hospitality during the preparation of this paper. The
authors also thanks Qun He and Miguel Angel Javaloyes for helpful discussions
and suggestions.

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