# On isotropic Berwald scalar curvature

Ming Li1 Ming Li: Mathematical Science Research Center, Chongqing University
of Technology, Chongqing 400054, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this short paper, we establish a closer relation between the Berwald scalar
curvature and the $S$-curvature. In fact, we prove that a Finsler metric has
isotropic Berwald scalar curvature if and only if it has weakly isotropic
$S$-curvature. For Finsler metrics of scalar flag curvature and of weakly
isotropic $S$-curvature, they have almost isotropic $S$-curvature if and only
if the flag curvature is weakly isotropic.

1 Partially supported by NSFC (Grant No. 11871126)

## Introduction

Let $(M,F)$ be an $n$-dimensional Finsler manifold. The unit sphere bundle (or
indicatrix bundle) is defined as $SM=\\{(x,y)\in TM|F(x,y)=1\\}$ with the
natural projection $\pi:SM\to M$. It is well known that the tangent bundle
$T(SM)$ admits a natural horizontal subbundle $H(SM)$ and a Sasaki-type metric
$g^{T(SM)}$. The Hilbert form $\omega^{n}=F_{y^{i}}dx^{i}$ defines a contact
structure on $SM$. The Reeb field is just the spray $\mathbf{G}$.

Let $\tilde{P}$ be the Berwald curvature, which is the hv-part of the
curvature form of the Berwald connection $\nabla^{\rm B}$ on $H(SM)$. Let
$\mathbf{E}=\mathrm{tr}\tilde{P}$ be the mean Berwald curvature or the
$\mathbf{E}$-curvature. We introduce the Berwald scalar curvature as
$\mathsf{e}:=\mathrm{tr}\mathbf{E}$ in [9], which is a function on $SM$ in
general cases. If $\mathsf{e}=\pi^{*}c$ for some $c\in C^{\infty}(M)$, then
$F$ is said to have isotropic Berwald scalar curvature. The aim of this paper
is to establish certain relations between the Berwald scalar curvature and the
$S$-curvature of a Finsler manifold.

######  Theorem 1.

Let $(M,F)$ be an $n$-dimensional Finsler manifold. Then $F$ has isotropic
Berwald scalar curvature $\mathsf{e}$ if and only if $F$ has weakly isotropic
$S$-curvature, and

(0.1) $S=\frac{1}{n-1}\mathsf{e}+\pi^{*}\xi(\mathbf{G}),$

where $\xi\in\Omega^{1}(M)$. In both cases the mean Berwald curvature
$\mathbf{E}$ is isotropic. Moreover, the following equation holds

(0.2)
$\begin{split}\mathrm{tr}[\tilde{R}]=&\pi^{*}d\xi+\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n},\end{split}$

where $\tilde{R}$ is the hh-part of the Berwald curvature form.

A well known fact in Finsler geometry is that vanishing $S$-curvature implies
that $\mathbf{E}=0$. Whether or not the converse is true is a natural question
(c.f. [14], p.67). We show that the converse is true under certain conditions.

######  Theorem 2.

Let $(M,F)$ be an $n$-dimensional Finsler manifold. If the Berwald scalar
curvature is constant and the mean stretch curvature $\bar{\Sigma}=0$, then
the $S$-curvature of $F$ satisfies (0.1) with $d\xi=0$, i.e., $S$ is almost
constant.

The following corollary can be obtained from Theorem 2 immediately.

######  Corollary 1.

Let $M$ be a smooth manifold satisfies $H^{1}_{\rm dR}(M;\mathbf{R})=0$. If a
Finsler structure $F$ on $M$ satisfies $\mathbf{J}=0$ and $\mathsf{e}=0$, then
$S=0$.

For weakly Landsberg manifold, we have the following relations among the
Berwald scalar curvature, $S$-curvature and the Cartan type 1-form.

######  Corollary 2.

Let $(M,F)$ be an $n$-dimensional Finsler manifold with vanishing mean
Landsberg curvature $\mathbf{J}=0$. Then the following statements are
equivalent:

  1. (1)

$F$ has constant Berwald scalar curvature $\mathsf{e}$;

  2. (2)

$F$ has almost constant $S$-curvature;

  3. (3)

The Cartan type 1-form $d\eta=cd\omega^{n}$ for certain constant $c$.

To study the Finsler metrics of scalar flag curvature, i.e.,
$\mathbf{K}=\mathbf{K}(x,y)$, is an important topic in Finsler geometry. Many
authors contribute on this subject. One refers [4, 5, 14, 16, 17] and finds
the references therein. Applying Theorem 1 to the Finsler metrics of scalar
flag curvature and of weakly isotropic $S$-curvature, we obtain the following
result.

######  Corollary 3.

Let $(M,F)$ be a $n$-dimensional Finsler manifold of scalar flag curvature and
weakly isotropic $S$-curvature. Then $F$ has almost isotropic $S$-curvature if
and only if the flag curvature is weakly isotropic and satisfies

(0.3) $\mathbf{K}=\frac{3}{n^{2}-1}(\pi^{*}dc)(\mathbf{G})+\pi^{*}\sigma,$

where the $c$ and $\sigma$ are function on $M$, and $c$ is the Berwald scalar
curvature $\mathsf{e}$ up to a constant.

The necessary part of Corollary 3 first appeared in [3] and plays an important
role in the study of Finsler geometry.

In this paper we adopt the index range and notations:

$1\leq i,j,k,\ldots\leq n,\quad 1\leq\alpha,\beta,\gamma,\ldots\leq n-1,\quad
1\leq A,B,C,\ldots\leq 2n-1,$

and
$\bar{\alpha}:=n+\alpha,\bar{\beta}:=n+\beta,\bar{\gamma}:=n+\gamma,\ldots.$

Acknowledgements. The author would like to thanks Professor Huitao Feng for
his consistent support and encouragement.

##  1\. Preliminary results

We will introduce the necessary definitions and background of Finsler geometry
in this section (c.f. [1, 5, 10, 14, 16, 17] for more details).

Let $(M,F)$ be an $n$-dimensional Finsler manifold. Let $SM=\\{F(x,y)=1\\}$ be
the unit sphere bundle (or indicatrix bundle) with the natural projection
$\pi:SM\to M$. Let
$\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{n},\mathbf{e}_{n+1},\ldots,\mathbf{e}_{2n-1}\\}$
be a local adapted orthonormal frame with respect to the Sasaki-type
Riemannian metric

$\displaystyle
g^{T(SM)}=\sum_{i}\omega^{i}\otimes\omega^{i}+\sum_{\bar{\alpha}}\omega^{\bar{\alpha}}\otimes\omega^{\bar{\alpha}}=:g+\dot{g}$

on $T(SM)$, where $\mathbf{e}_{n}=\mathbf{G}$ is the Reeb vector field. Let
$\theta=\left\\{\omega^{1},\ldots,\omega^{n},\omega^{n+1},\ldots,\omega^{2n-1}\right\\}$
be the dual frame, then $\omega^{n}=F_{y^{i}}dx^{i}$ is the Hilbert form.
$\omega^{n}$ defines a contact structure on $SM$. Let $\mathfrak{l}$ be the
trivial bundle generated by $\mathbf{G}$. Let $\mathcal{D}$ be the contact
distribution $\\{\omega^{n}=0\\}$. We define an almost complex structure on
$\mathcal{D}$ by

$J=-\omega^{\alpha}\otimes\mathbf{e}_{\bar{\alpha}}+\omega^{\bar{\alpha}}\otimes\mathbf{e}_{\alpha}.$

We will extend $J$ to be an endomorphism $T(SM)$ by defining
$J(\mathbf{e}_{n})=0$. According to the definition of a contact metric
structure in [2], one only need to verify that
$(SM,J,\mathbf{G},\omega,g^{T(SM)})$ gives rise a contact metric structure.

Let $\mathscr{F}:=V(SM)$ be the integrable distribution given by the tangent
spaces of the fibers of $SM$. Let $p:T(SM)\to\mathscr{F}$ be the natural
projection. Then the tangent bundle $T(SM)$ admits a splitting

(1.1) $T(SM)=\mathfrak{l}\oplus
J\mathscr{F}\oplus\mathscr{F}=:H(SM)\oplus\mathscr{F}.$

The horizontal subbundle $H(SM)=\mathfrak{l}\oplus J\mathscr{F}$ is spanned by
$\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{n}\\}$ on which the Chern connection is
defined. And $\\{\mathbf{e}_{n+1},\ldots,\mathbf{e}_{2n-1}\\}$ gives a local
frame of $\mathscr{F}$.

Let $\nabla^{\rm Ch}:\Gamma(H(SM))\to\Omega^{1}(SM;H(SM))$ be the Chern
connection, which can be extended to a map

$\displaystyle\nabla^{\rm
Ch}:\Omega^{*}(SM;H(SM))\rightarrow\Omega^{*+1}(SM;H(SM)),$

where $\Omega^{*}(SM;H(SM)):=\Gamma(\Lambda^{*}(T^{*}SM)\otimes H(SM))$
denotes the horizontal valued differential forms on $SM$. It is well known
that the symmetrization of Chern connection $\hat{\nabla}^{\rm Ch}$ is the
Cartan connection. The difference between $\hat{\nabla}^{\rm Ch}$ and
$\nabla^{\rm Ch}$ will be referred as the Cartan endomorphism,

$\displaystyle H=\hat{\nabla}^{\rm Ch}-\nabla^{\rm
Ch}\quad\in\Omega^{1}(SM,{\rm End}(H(SM))).$

Set $H=H_{ij}\omega^{j}\otimes\mathbf{e}_{i}$. By Lemma 3 and Lemma 4 in [6],
$H_{ij}=H_{ji}=H_{ij\gamma}\omega^{\bar{\gamma}}$ has the following form under
natural coordinate systems

$\displaystyle H_{ij\gamma}=-A_{pqk}u_{i}^{p}u_{j}^{q}u_{\gamma}^{k},$

where $A_{ijk}=\frac{1}{4}F[F^{2}]_{y^{i}y^{j}y^{k}}$ and $u_{i}^{j}$ are the
transformation matrix from adapted orthonormal frames to natural frames.

Set $\eta={\rm tr}[H]~{}\in\Omega^{1}(SM).$ It is referred as the Cartan-type
form in [6]. The Cartan-type form has the following local formula

$\displaystyle\eta=\sum_{i=1}^{n}H_{ii\gamma}\omega^{\bar{\gamma}}=:H_{\gamma}\omega^{\bar{\gamma}}.$

Let $\bm{\omega}=(\omega_{j}^{i})$ be the connection matrix of the Chern
connection with respect to the local adapted orthonormal frame field, i.e.,

$\displaystyle\nabla^{\rm
Ch}\mathbf{e}_{i}=\omega_{i}^{j}\otimes\mathbf{e}_{j}.$

######  Lemma 1 ([1, 5, 10, 14]).

The connection matrix $\bm{\omega}=(\omega_{j}^{i})$ of $\nabla^{\rm Ch}$ is
determined by the following structure equations,

(1.2) $\left\\{\begin{aligned} &d\vartheta=-\bm{\omega}\wedge\vartheta,\\\
&\bm{\omega}+\bm{\omega}^{t}=-2H,\end{aligned}\right.$

where $\vartheta=(\omega^{1},\ldots,\omega^{n})^{t}$. Furthermore,

$\omega_{\alpha}^{n}=-\omega^{\alpha}_{n}=\omega^{\bar{\alpha}},\quad{\rm
and}\quad\omega^{n}_{n}=0.$

######  Remark 1.

In [6], we proved that the Chern connection is just the Bott connection on
$H(SM)$ in the theory of foliation (c.f. [19]).

Let $R^{\rm Ch}=\left(\nabla^{\rm Ch}\right)^{2}$ be the curvature of
$\nabla^{\rm Ch}$. Let $\Omega=\left(\Omega_{j}^{i}\right)$ be the curvature
forms of $R^{\rm Ch}$. From the torsion freeness, the curvature form has no
pure vertical differential form

(1.3)
$\displaystyle\Omega_{j}^{i}:=d\omega_{j}^{i}-\omega_{j}^{k}\wedge\omega_{k}^{i}=\frac{1}{2}R_{j~{}kl}^{~{}i}\omega^{k}\wedge\omega^{l}+P_{j~{}k\gamma}^{~{}i}\omega^{k}\wedge\omega^{\bar{\gamma}}.$

The Landsberg curvature is defined as

$\mathbf{L}:=L^{~{}i}_{j~{}\gamma}\omega^{\bar{\gamma}}\otimes\omega^{j}\otimes\mathbf{e}_{i}=-P_{j~{}n\gamma}^{~{}i}\omega^{\bar{\gamma}}\otimes\omega^{j}\otimes\mathbf{e}_{i},$

the mean Landsberg curvature is defined by

$\mathbf{J}={\rm
tr}\mathbf{L}=J_{\gamma}\omega^{\bar{\gamma}}=-P_{i~{}n\gamma}^{~{}i}\omega^{\bar{\gamma}}.$

The following formula is well known (c.f. [1, 5, 10, 14])

(1.4) $\displaystyle P_{n~{}k\gamma}^{~{}i}=-H_{ki\gamma|n}=-L_{ik\gamma}.$

If a Finsler manifold satisfies $P=0$, $\mathbf{L}=0$ or $\mathbf{J}=0$, then
it is called a Berwald, Landsberg or weak Landsberg manifold, respectively.

We will need parts of the Riemannian geometry of the sphere bundle, especially
the geometry of the fibres. The following lemma can be found in [10].

######  Lemma 2.

Let $\Theta=(\Theta_{B}^{A})$ be the connection form of the Levi-Civita
connection $\nabla^{T(SM)}$ of $g^{T(SM)}$ with respect to the adapted frame
$\\{\mathbf{e}_{A}\\}$. Then we have

(1.5)
$\Theta_{\bar{\beta}}^{\bar{\gamma}}=\omega_{\beta}^{\gamma}+H_{\beta\gamma\alpha}\omega^{\bar{\alpha}},$

where $\omega_{\beta}^{\gamma}$ are the connection forms of the Chern
connection and $H_{\beta\gamma\alpha}\omega^{\bar{\alpha}}$ are the
coefficients of the Cartan endomorphism.

As the restriction the Levi-Civita connection $\nabla^{T(SM)}$ on
$\mathscr{F}$, $\nabla^{\mathscr{F}}:=p\nabla^{T(SM)}p$ is the Euclidean
connection of the bundle $(\mathscr{F},\dot{g})$. It is clear that along each
fiber of $S_{x}M$, $x\in M$, $\nabla^{\mathscr{F}}$ is just the Levi-Civita
connection of the Riemannian manifold $(S_{x}M,\dot{g}_{x})$. We also define a
connection $\bar{\nabla}^{\rm Ch}$ on $\mathscr{F}$ by

(1.6) $\bar{\nabla}^{\rm
Ch}\mathbf{e}_{\bar{\beta}}=\omega_{\beta}^{\alpha}\otimes\mathbf{e}_{\bar{\alpha}}.$

We introduces some symbols to denote different covariant differentials for
conveniens . For example, let $T=T^{i}_{j}\mathbf{e}_{i}\otimes\omega^{j}$ be
an arbitrary smooth local section of the bundle $H(SM)\otimes H^{*}(SM)$ over
$SM$. Then

$\nabla^{\rm
Ch}T=\left(dT^{i}_{j}+T^{k}_{j}\omega^{i}_{k}-T^{i}_{k}\omega^{k}_{j}\right)\otimes\mathbf{e}_{i}\otimes\omega^{j}$

If we expand the coefficients as one forms on $SM$ in terms of the adapted
coframe, then we denote

$(\nabla^{\rm
Ch}T)^{i}_{j}:=dT^{i}_{j}+T^{k}_{j}\omega^{i}_{k}-T^{i}_{k}\omega^{k}_{j}=:T^{i}_{j|\alpha}\omega^{\alpha}+T^{i}_{j|n}\omega^{n}+T^{i}_{j;\gamma}\omega^{\bar{\gamma}}.$

Therefore we obtain

$T^{i}_{j|\alpha}=i_{\mathbf{e}_{\alpha}}(\nabla^{\rm Ch}T)^{i}_{j},\quad
T^{i}_{j|n}=i_{\mathbf{e}_{n}}(\nabla^{\rm Ch}T)^{i}_{j},\quad
T^{i}_{j;\gamma}=i_{\mathbf{e}_{\bar{\gamma}}}(\nabla^{\rm Ch}T)^{i}_{j},$

where $i_{v}$ is the notation for the interior multiplication on differential
forms by any vector $v$.

According to the splitting (1.1) and using the almost complex structure $J$,
we obtain a section of $\mathscr{F}\otimes\mathscr{F}^{*}$ from $T$ as
following

$\bar{T}=T^{\alpha}_{\beta}\mathbf{e}_{\bar{\alpha}}\otimes\omega^{\bar{\beta}}.$

By (1.6), the covariant differential of $\bar{T}$ by using $\bar{\nabla}^{\rm
Ch}$ is given by

$\bar{\nabla}^{\rm
Ch}\bar{T}=\left(dT^{\alpha}_{\beta}+T^{\mu}_{\beta}\omega^{\alpha}_{\mu}-T^{\alpha}_{\mu}\omega^{\mu}_{\beta}\right)\otimes\mathbf{e}_{\bar{\alpha}}\otimes\omega^{\bar{\beta}}$

Similarly, we denote that

$(\bar{\nabla}^{\rm
Ch}\bar{T})^{\alpha}_{\beta}:=dT^{\alpha}_{\beta}+T^{\mu}_{\beta}\omega^{\alpha}_{\mu}-T^{\alpha}_{\mu}\omega^{\mu}_{\beta}=T^{\alpha}_{\beta\parallel\gamma}\omega^{\gamma}+T^{\alpha}_{\beta\parallel
n}\omega^{n}+T^{\alpha}_{\beta,\gamma}\omega^{\bar{\gamma}}.$

Therefore, by Lemma 1, one has the following relations

(1.7) $\begin{split}T^{\alpha}_{\beta\parallel
i}&=i_{\mathbf{e}_{i}}(dT^{\alpha}_{\beta}+T^{\mu}_{\beta}\omega^{\alpha}_{\mu}-T^{\alpha}_{\mu}\omega^{\mu}_{\beta})=i_{\mathbf{e}_{i}}(dT^{\alpha}_{\beta}+T^{k}_{\beta}\omega^{\alpha}_{k}-T^{\alpha}_{k}\omega^{k}_{\beta})=T^{\alpha}_{\beta|i}\\\
T^{\alpha}_{\beta,\gamma}&=i_{\mathbf{e}_{\bar{\gamma}}}(dT^{\alpha}_{\beta}+T^{\mu}_{\beta}\omega^{\alpha}_{\mu}-T^{\alpha}_{\mu}\omega^{\mu}_{\beta})\\\
&=i_{\mathbf{e}_{\bar{\gamma}}}(dT^{\alpha}_{\beta}+T^{k}_{\beta}\omega^{\alpha}_{k}-T^{\alpha}_{k}\omega^{k}_{\beta})-i_{\mathbf{e}_{\bar{\gamma}}}T^{n}_{\beta}\omega_{n}^{\alpha}+i_{\mathbf{e}_{\bar{\gamma}}}T^{\alpha}_{n}\omega^{n}_{\beta}\\\
&=T^{\alpha}_{\beta;\gamma}+T^{n}_{\beta}\delta^{\alpha}_{\gamma}-T_{n}^{\alpha}\delta_{\beta\gamma}.\end{split}$

It is obvious that if $T$ is a section
$J\mathscr{F}\otimes(J\mathscr{F})^{*}$, then
$T^{\alpha}_{\beta;\gamma}=T^{\alpha}_{\beta,\gamma}$. These relations will be
used in the following study.

The Berwald connection can be represented as $\nabla^{\rm B}=\nabla^{\rm
Ch}+J^{*}\mathbf{L}$, where $J^{*}$ is dual of the almost complex structure
$J$. Let $\tilde{\bm{\omega}}=(\tilde{\omega}_{j}^{i})$ denote the Berwald
connection form, then

$\displaystyle\tilde{\omega}_{j}^{i}=\omega_{j}^{i}-L^{i}_{j\alpha}\omega^{\alpha}.$

From the torsion freeness the Chern connection and (1.4), the Berwald
connection is torsion free,

$d\omega^{i}=\omega^{j}\wedge\tilde{\omega}_{j}^{i}.$

Let $\tilde{\Omega}_{j}^{i}$ be the curvature forms of the Berwald connection.
By the torsion freeness of the Berwald connection, we have

$\displaystyle\tilde{\Omega}_{j}^{i}:=d\tilde{\omega}_{j}^{i}-\tilde{\omega}_{j}^{k}\wedge\tilde{\omega}_{k}^{i}=:\frac{1}{2}\tilde{R}_{j~{}kl}^{~{}i}\omega^{k}\wedge\omega^{l}+\tilde{P}_{j~{}k\gamma}^{~{}i}\omega^{k}\wedge\omega^{\bar{\gamma}},$

where $\tilde{R}_{j~{}kl}^{~{}i}=-\tilde{R}_{j~{}lk}^{~{}i}.$ By using Lemma
1, the curvature relations between the Chern connection and the Berwald
connection are as following. The $hh$-curvatures satisfy

(1.8)
$\displaystyle\begin{split}&\tilde{R}^{~{}\alpha}_{\beta~{}\gamma\mu}=R^{~{}\alpha}_{\beta~{}\gamma\mu}-(L^{\nu}_{\beta\gamma}L^{\alpha}_{\nu\mu}-L^{\nu}_{\beta\mu}L^{\alpha}_{\nu\gamma})+(L^{\alpha}_{\beta\gamma|\mu}-L^{\alpha}_{\beta\mu|\gamma}),\\\
&\tilde{R}^{~{}\alpha}_{\beta~{}\gamma n}=R^{~{}\alpha}_{\beta~{}\gamma
n}+L^{\alpha}_{\beta\gamma|n},\quad\tilde{R}^{~{}\alpha}_{n~{}kl}=-\tilde{R}^{~{}n}_{\alpha~{}kl}=R^{~{}\alpha}_{n~{}kl}.\end{split}$

The mean stretch cuvature

(1.9) $\bar{\Sigma}:=2\mathrm{tr}[\tilde{R}-R]$

is introduced and studied in [11].

The relations between the $hv$-curvatures of the Berwald connection and the
Chern connection are given by

(1.10)
$\displaystyle\begin{split}&\tilde{P}^{~{}\alpha}_{\beta~{}\gamma\mu}=P^{~{}\alpha}_{\beta~{}\gamma\mu}+L^{\alpha}_{\beta\gamma;\mu},\quad\tilde{P}^{~{}n}_{\alpha~{}\gamma\mu}=2L_{\alpha\gamma\mu},\\\
&\tilde{P}^{~{}\alpha}_{\beta~{}n\mu}=0,\quad\tilde{P}^{~{}\alpha}_{n~{}k\gamma}=0,\quad\tilde{P}^{~{}n}_{\alpha~{}n\gamma}=0.\end{split}$

The mean Berwald curvature or the $\mathbf{E}$-curvature is defined by

$\mathbf{E}:=\mathrm{tr}\tilde{P}=:E_{\gamma\mu}\omega^{\gamma}\wedge\omega^{\bar{\mu}}.$

Under the local adapted frame, the coefficients of the $\mathbf{E}$-curvature
is represented as

(1.11)
$E_{\gamma\mu}=\tilde{P}^{~{}i}_{i~{}\gamma\mu}=\tilde{P}^{~{}\alpha}_{\alpha~{}\gamma\mu}=P^{~{}\alpha}_{\alpha~{}\gamma\mu}+L^{\alpha}_{\alpha\gamma;\mu}=P^{~{}\alpha}_{\alpha~{}\gamma\mu}+J_{\gamma;\mu}=P^{~{}\alpha}_{\alpha~{}\gamma\mu}+J_{\gamma,\mu}.$

We deliberately omit the factors $\frac{1}{2}$ and $F^{-1}$ in the usual
definition of $\mathbf{E}$-curvature as a tensor on $TM_{0}$ in literatures.

##  2\. The $S$-curvature and the Berwald scalar curvature

In this section we develop the relations between the $S$-curvature and the
Berwald scalar curvature $\mathsf{e}$. We first state some necessary
definitions.

Let $dV_{M}$ be any volume form of $M$. On a local coordinate chart
$(U;x^{i})$, $dV_{M}=\sigma(x)dx^{1}\wedge\cdots\wedge dx^{n}$. The following
important function on $SM$ is well defined,

$\tau=\ln\frac{\sqrt{\det{(g_{ij})}}}{\sigma(x)}.$

$\tau$ is called the distortion of $(M,F)$. The derivative of $\tau$ along the
vector field $\mathbf{G}$ will be denoted by $S:=\mathbf{G}(\tau)$. The
function $\mathbf{S}:=F\cdot S$ defined on $TM_{0}$ is called the
$S$-curvature. In this paper, we will also call $S$ the $S$-curvature for
convenient. These two important concepts were first introduced by Zhongmin
Shen in [15].

######  Remark 2.

In the study [8], we show that the centro-affine differential geometric
structure of the fibres of $SM$ naturally fit into the overall geometry of the
fibre bundle $SM$. In affine geometry, the Tchebychev potential is the
logarithm of the Radon-Nikodym derivative of the one induced by the affine
metric and the measure induced by the embedding. One easily shows that the
restriction of $-\tau$ on each fiber of $SM$ is just the Tchebychev potential
of the fiber. Therefore the distortion can be naturally considered as the
family version of the Tchebychev potential. One consults [7, 12, 18] for more
details about the affine differential geometry.

In this paper, we will use the following definitions.

######  Definition 1 ([4, 14]).

Let $(M,F)$ be an $n$-dimensional Finsler manifold.

  1. (i)

If the $S$-curvature satisfies

(2.1) $S=\frac{1}{n-1}\pi^{*}c+\pi^{*}\xi(\mathbf{G})$

for some $c\in C^{\infty}(M)$ and $\xi\in\Omega^{1}(M)$, then $F$ is said to
have weakly isotropic $S$-curvature;

  2. (ii)

If $S$ satisfies (2.1) and $d\xi=0$, then it is called to be almost isotropic;

  3. (iii)

If $S$ satisfies (2.1) and $\xi=0$, then it is called to be isotropic.

Let $\widetilde{dV_{M}}=e^{-f}dV_{M}$ be any other volume form on $M$, then
the related distortion $\tilde{\tau}=\tau+\pi^{*}f$, where $f\in
C^{\infty}(M)$. Thus the $S$-curvature related to $\widetilde{dV_{M}}$ is
given by

(2.2) $\tilde{S}=S+\pi^{*}df(\mathbf{G}).$

Therefore, the $S$-curvatures of a Finsler manifold are determined up to the
exact 1-forms on the base manifold $M$. If $H^{1}_{\rm dR}(M;\mathbf{R})=0$,
then the conditions almost isotropic $S$-curvature and isotropic $S$-curvature
are equivalent.

Let $f\in C^{\infty}(SM)$ be any smooth function on $SM$, then $df$ admits the
following decomposition with respect to (1.1),

(2.3)
$df=f_{|\alpha}\omega^{\alpha}+f_{|n}\omega^{n}+f_{,\alpha}\omega^{\bar{\alpha}},$

where we denote $f_{|i}:=\mathbf{e}_{i}(f)$, and
$f_{,\alpha}:=\mathbf{e}_{\bar{\alpha}}(f)$.

######  Lemma 3.

From the fact $d^{2}f=0$, one has the following identities,

(2.4) $\displaystyle
f_{|\alpha|\beta}-f_{|\beta|\alpha}+f_{,\gamma}R^{~{}\gamma}_{n~{}\alpha\beta}=0,$
(2.5) $\displaystyle
f_{|\alpha|n}-f_{|n|\alpha}+f_{,\gamma}R^{~{}\gamma}_{n~{}\alpha n}=0,$ (2.6)
$\displaystyle
f_{|\alpha,\beta}-f_{|n}\delta_{\alpha\beta}+f_{,\gamma}P^{~{}\gamma}_{n~{}\alpha\beta}-f_{,\beta|\alpha}=0,$
(2.7) $\displaystyle f_{|\alpha}+f_{|n,\alpha}-f_{,\alpha|n}=0,$ (2.8)
$\displaystyle f_{,\alpha,\beta}-f_{,\beta,\alpha}=0.$

The following useful equations can be derived from the above identities

(2.9)
$\displaystyle(f_{|n})_{,\alpha|n}+(f_{|n})_{|\alpha}=f_{,\alpha|n|n}+f_{,\gamma}R^{~{}\gamma}_{n~{}\alpha
n},$ (2.10)
$\displaystyle(f_{|n})_{,\alpha,\beta}+(f_{|n})\delta_{\alpha\beta}=(f_{,\gamma}P^{~{}\gamma}_{n~{}\alpha\beta}-f_{,\beta|\alpha})+f_{,\alpha|n,\beta}.$

###### Proof.

By using the properties of the Chern connection and the notations in Section
1, we have

(2.11)
$\begin{split}&d(f_{|\alpha}\omega^{\alpha})=(d(f_{|\alpha})-f_{|\beta}\omega^{\beta}_{\alpha})\wedge\omega^{\alpha}-f_{|\beta}\omega^{\beta}_{n}\wedge\omega^{n}\\\
=&f_{|\alpha|\beta}\omega^{\beta}\wedge\omega^{\alpha}+f_{|\alpha|n}\omega^{n}\wedge\omega^{\alpha}+f_{|\alpha,\beta}\omega^{\bar{\beta}}\wedge\omega^{\alpha}+f_{|\alpha}\omega^{\bar{\alpha}}\wedge\omega^{n}\\\
=&-\frac{1}{2}(f_{|\alpha|\beta}-f_{|\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}-f_{|\alpha|n}\omega^{\alpha}\wedge\omega^{n}-f_{|\alpha,\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-f_{|\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}},\end{split}$
(2.12)
$\begin{split}&d(f_{|n}\omega^{n})=f_{|n|\alpha}\omega^{\alpha}\wedge\omega^{n}+f_{|n,\alpha}\omega^{\bar{\alpha}}\wedge\omega^{n}+f_{|n}\delta_{\alpha\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}\\\
=&f_{|n|\alpha}\omega^{\alpha}\wedge\omega^{n}+f_{|n}\delta_{\alpha\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-f_{|n,\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}},\end{split}$

and

(2.13)
$\begin{split}&d(f_{,\alpha}\omega^{\bar{\alpha}})=d(f_{,\alpha})\wedge\omega^{\bar{\alpha}}+f_{,\alpha}(-\Omega_{n}^{\alpha}-\omega_{n}^{\beta}\wedge\omega_{\beta}^{\alpha})\\\
=&(d(f_{,\alpha})-f_{,\beta}\omega_{\alpha}^{\beta})\wedge\omega^{\bar{\alpha}}-f_{,\alpha}\Omega_{n}^{\alpha}\\\
=&f_{,\alpha|\beta}\omega^{\beta}\wedge\omega^{\bar{\alpha}}+f_{,\alpha|n}\omega^{n}\wedge\omega^{\bar{\alpha}}+f_{,\alpha,\beta}\omega^{\bar{\beta}}\wedge\omega^{\bar{\alpha}}\\\
&-\frac{1}{2}f_{,\alpha}R^{~{}\alpha}_{n~{}\beta\gamma}\omega^{\beta}\wedge\omega^{\gamma}-f_{,\alpha}R^{~{}\alpha}_{n~{}\beta
n}\omega^{\beta}\wedge\omega^{n}-f_{,\alpha}P^{~{}\alpha}_{n~{}\beta\gamma}\omega^{\beta}\wedge\omega^{\bar{\gamma}}\\\
=&-\frac{1}{2}f_{,\alpha}R^{~{}\alpha}_{n~{}\beta\gamma}\omega^{\beta}\wedge\omega^{\gamma}-f_{,\alpha}R^{~{}\alpha}_{n~{}\beta
n}\omega^{\beta}\wedge\omega^{n}+(f_{,\gamma|\beta}-f_{,\alpha}P^{~{}\alpha}_{n~{}\beta\gamma})\omega^{\beta}\wedge\omega^{\bar{\gamma}}\\\
&+f_{,\alpha|n}\omega^{n}\wedge\omega^{\bar{\alpha}}-\frac{1}{2}(f_{,\alpha,\beta}-f_{,\beta,\alpha})\omega^{\bar{\alpha}}\wedge\omega^{\bar{\beta}}.\end{split}$

By taking the sum of (2.11), (2.12) and (2.13), we obtain

(2.14)
$\begin{split}0=&-\frac{1}{2}(f_{|\alpha|\beta}-f_{|\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}-f_{|\alpha|n}\omega^{\alpha}\wedge\omega^{n}-f_{|\alpha,\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-f_{|\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}}\\\
&+f_{|n|\alpha}\omega^{\alpha}\wedge\omega^{n}+f_{|n}\delta_{\alpha\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-f_{|n,\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}}\\\
&-\frac{1}{2}f_{,\alpha}R^{~{}\alpha}_{n~{}\beta\gamma}\omega^{\beta}\wedge\omega^{\gamma}-f_{,\alpha}R^{~{}\alpha}_{n~{}\beta
n}\omega^{\beta}\wedge\omega^{n}+(f_{,\gamma|\beta}-f_{,\alpha}P^{~{}\alpha}_{n~{}\beta\gamma})\omega^{\beta}\wedge\omega^{\bar{\gamma}}\\\
&+f_{,\alpha|n}\omega^{n}\wedge\omega^{\bar{\alpha}}-\frac{1}{2}(f_{,\alpha,\beta}-f_{,\beta,\alpha})\omega^{\bar{\alpha}}\wedge\omega^{\bar{\beta}}\\\
=&-\frac{1}{2}(f_{|\alpha|\beta}-f_{|\beta|\alpha}+f_{,\gamma}R^{~{}\gamma}_{n~{}\alpha\beta})\omega^{\alpha}\wedge\omega^{\beta}-(f_{|\alpha|n}-f_{|n|\alpha}+f_{,\gamma}R^{~{}\gamma}_{n~{}\alpha
n})\omega^{\alpha}\wedge\omega^{n}\\\
&-(f_{|\alpha,\beta}-f_{|n}\delta_{\alpha\beta}+f_{,\gamma}P^{~{}\gamma}_{n~{}\alpha\beta}-f_{,\beta|\alpha})\omega^{\alpha}\wedge\omega^{\bar{\beta}}-(f_{|\alpha}+f_{|n,\alpha}-f_{,\alpha|n})\omega^{n}\wedge\omega^{\bar{\alpha}}\\\
&-\frac{1}{2}(f_{,\alpha,\beta}-f_{,\beta,\alpha})\omega^{\bar{\alpha}}\wedge\omega^{\bar{\beta}}.\end{split}$

Therefore, the coefficients in (2.14) with respect to the local adapted bases
of $\Omega^{2}(SM)$ vanish. The identities (2.4)-(2.8) are proved in this way.

Taking covariant derivative of (2.7) along $\mathbf{e}_{n}$ and substituting
in (2.5) yields (2.9). Taking covariant derivative of (2.7) along
$\mathbf{e}_{\bar{\alpha}}$ and substituting in (2.6) yields (2.10). ∎

Applying Lemma 3 to the distortion $\tau$ yields the following lemma.

######  Lemma 4.

The differential of $\tau$ is given by

(2.15)
$d\tau=\tau_{|i}\omega^{i}+\eta=\tau_{|\alpha}\omega^{\alpha}+S\omega^{n}+\eta.$

The following identities are valid

(2.16)
$\displaystyle\tau_{|\alpha|\beta}-\tau_{|\beta|\alpha}+R_{i~{}\alpha\beta}^{~{}i}=0,$
(2.17) $\displaystyle S_{|\alpha}-\tau_{|\alpha|n}-R_{i~{}\alpha n}^{~{}i}=0,$
(2.18)
$\displaystyle\tau_{|\gamma,\mu}-S\delta_{\gamma\mu}+P_{i~{}\gamma\mu}^{~{}i}=0,$
(2.19) $\displaystyle S_{,\mu}+\tau_{|\mu}+P_{i~{}n\mu}^{~{}i}=0.$

Furthermore, one has

(2.20) $S_{|\alpha}-S_{,\alpha|n}=J_{\alpha|n}+R^{~{}i}_{i~{}\alpha n},$

and

(2.21) $S_{,\gamma,\mu}+S\delta_{\gamma\mu}=E_{\gamma\mu}.$

###### Proof.

These identities (2.16)-(2.21) are direct consequences of Lemma 3 by using
(2.15), (1.11) and some well known Bianchi identities or the following fact

(2.22)
$d\eta=d(H_{\gamma}\omega^{\bar{\gamma}})=-\frac{1}{2}R_{i~{}kl}^{~{}i}\omega^{k}\wedge\omega^{l}-P_{i~{}k\gamma}^{~{}i}\omega^{k}\wedge\omega^{\bar{\gamma}}.$

∎

Let $\xi\in\Omega^{1}(M)$ be a 1-form. In a local coordinate system,
$\xi=\xi_{i}(x)dx^{i}$. Let

(2.23)
$\pi^{*}dx^{i}=u^{i}_{j}\omega^{j}=u^{i}_{\alpha}\omega^{\alpha}+\frac{y^{i}}{F}\omega^{n}$

be the relationship between the natural bases and the orthonormal ones. By
using (2.23), $\pi^{*}\xi$ can be written as

(2.24)
$\pi^{*}\xi=\xi_{i}u^{i}_{\alpha}\omega^{\alpha}+\frac{\xi_{i}y^{i}}{F}\omega^{n}=:\xi_{\alpha}\omega^{\alpha}+\xi_{n}\omega^{n}.$

The following lemma gives a representation of $\pi^{*}d\xi$ on $SM$.

######  Lemma 5.

For any $\xi\in\Omega^{1}(M)$, one can express $\pi^{*}d\xi$ in the following
form

(2.25)
$d\pi^{*}\xi=-\frac{1}{2}(\xi_{\alpha|\beta}-\xi_{\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}-(\xi_{\alpha|n}-\xi_{n|\alpha})\omega^{\alpha}\wedge\omega^{n}.$

Moreover, one has

(2.26) $\xi_{\alpha,\beta}-\xi_{n}\delta_{\alpha\beta}=0.$

###### Proof.

For any sections $\xi_{\alpha}\omega^{\alpha}\in\Gamma(J\mathscr{F}^{*})$ and
$\xi_{n}\omega^{n}\in\Gamma(\mathfrak{l}^{*})$, one has

(2.27)
$\begin{split}&d(\xi_{\alpha}\omega^{\alpha})=(d(\xi_{\alpha})-\xi_{\beta}\omega^{\beta}_{\alpha})\wedge\omega^{\alpha}-\xi_{\beta}\omega^{\beta}_{n}\wedge\omega^{n}\\\
=&\xi_{\alpha|\beta}\omega^{\beta}\wedge\omega^{\alpha}+\xi_{\alpha|n}\omega^{n}\wedge\omega^{\alpha}+\xi_{\alpha,\beta}\omega^{\bar{\beta}}\wedge\omega^{\alpha}+\xi_{\alpha}\omega^{\bar{\alpha}}\wedge\omega^{n}\\\
=&-\frac{1}{2}(\xi_{\alpha|\beta}-\xi_{\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}-\xi_{\alpha|n}\omega^{\alpha}\wedge\omega^{n}-\xi_{\alpha,\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-\xi_{\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}},\end{split}$

and

(2.28)
$\begin{split}&d(\xi_{n}\omega^{n})=\xi_{n|\alpha}\omega^{\alpha}\wedge\omega^{n}+\xi_{n,\alpha}\omega^{\bar{\alpha}}\wedge\omega^{n}+\xi_{n}\delta_{\alpha\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}\\\
=&\xi_{n|\alpha}\omega^{\alpha}\wedge\omega^{n}+\xi_{n}\delta_{\alpha\beta}\omega^{\alpha}\wedge\omega^{\bar{\beta}}-\xi_{n,\alpha}\omega^{n}\wedge\omega^{\bar{\alpha}}.\end{split}$

When $\xi\in\Omega^{1}(M)$, the derivative of $\xi_{n}$ along the direction
$\mathbf{e}_{\bar{\alpha}}=-u^{i}_{\alpha}F\frac{\partial}{\partial y^{i}}$ is
given by

(2.29)
$\xi_{n,\alpha}=\mathbf{e}_{\bar{\alpha}}(\xi_{n})=-u^{j}_{\alpha}F\frac{\partial}{\partial
y^{j}}(\frac{\xi_{i}y^{i}}{F})=-u^{j}_{\alpha}(\delta_{j}^{i}-\frac{F_{y^{j}}y^{i}}{F})\xi_{i}=-\xi_{i}u^{i}_{\alpha}=-\xi_{\alpha}.$

Thus the facts (2.27)-(2.29) yield

(2.30)
$d\pi^{*}\xi=-\frac{1}{2}(\xi_{\alpha|\beta}-\xi_{\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}-(\xi_{\alpha|n}-\xi_{n|\alpha})\omega^{\alpha}\wedge\omega^{n}-(\xi_{\alpha,\beta}-\xi_{n}\delta_{\alpha\beta})\omega^{\alpha}\wedge\omega^{\bar{\beta}}.$

Since $d\pi^{*}\xi=\pi^{*}d\xi$ does not involve vertical differential forms,
we obtain the desired formulae from (2.30). ∎

Now we present a proof of the theorems and corollaries.

###### Proof of Theorem 1.

The necessary part of the theorem has been proved in [9]. The key ingredient
is that along each fibre of $SM$, the elliptic equation

$\Delta_{\dot{g}}f+\dot{g}(\eta,d^{SM/M}f)+(n-1)f=0,$

has only linear solutions (c.f.[13, 12]). By using this fact and (2.21), in
[9], we prove that if $\mathsf{e}$ is constant along each fiber of $SM$, then
$S$ is weakly isotropic and

(2.31) $S=\frac{1}{n-1}\mathsf{e}+\xi_{i}\frac{y^{i}}{F},$

where $\xi=\xi_{i}(x)dx^{i}\in\Omega^{1}(M)$.

Now we prove the sufficient part. Assume that the $S$-curvature is weakly
isotropic and satisfies (2.1) for some $c\in C^{\infty}(M)$ and
$\xi\in\Omega^{1}(M)$. The condition (2.1) and the representation (2.24) of
$\xi$ shows that

(2.32) $\xi_{n}=S-\frac{1}{n-1}\pi^{*}c.$

By taking derivative of (2.32) along the fibres, the formula (2.29) implies
that

(2.33) $\xi_{\alpha}=-S_{,\alpha}.$

Substituting (2.33) and (2.32) in the formula (2.26) yields

(2.34) $S_{,\alpha,\beta}+(S-\frac{1}{n-1}\pi^{*}c)\delta_{\alpha\beta}=0.$

By (2.21) and (2.34), one obtains that the $\mathbf{E}$-curvature is
isotropic, i.e.,

(2.35) $E_{\alpha\beta}=\frac{1}{n-1}\pi^{*}c\delta_{\alpha\beta}.$

The equation (2.35) obviously implies the Berwald scalar curvature

(2.36) $\mathsf{e}=\pi^{*}c.$

Now applying (2.33),(2.32) and (2.36) to the equation (2.25) yields

(2.37)
$\pi^{*}d\xi=\frac{1}{2}(S_{,\alpha|\beta}-S_{,\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}+(S_{,\alpha|n}+(S-\frac{1}{n-1}\mathsf{e})_{|\alpha})\omega^{\alpha}\wedge\omega^{n}.$

Applying (2.16), (2.17), (2.19) and (2.20) to (2.37), we obtain

(2.38)
$\begin{split}\pi^{*}d\xi=&-\frac{1}{2}(\tau_{|\alpha|\beta}-\tau_{|\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}+\frac{1}{2}(J_{\alpha|\beta}-J_{\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}\\\
&+(S_{|\alpha}-\tau_{|\alpha|n}+J_{\alpha|n})\omega^{\alpha}\wedge\omega^{n}-\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n}\\\
=&\frac{1}{2}R^{~{}i}_{i~{}\alpha\beta}\omega^{\alpha}\wedge\omega^{\beta}+\frac{1}{2}(J_{\alpha|\beta}-J_{\beta|\alpha})\omega^{\alpha}\wedge\omega^{\beta}\\\
&+(R^{~{}i}_{i~{}\alpha
n}+J_{\alpha|n})\omega^{\alpha}\wedge\omega^{n}-\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n}.\end{split}$

The relations (1.8) between the hh-curvature forms of the Berwald connection
and the Chern connection and (2.38) yields

(2.39)
$\begin{split}\pi^{*}d\xi=\mathrm{tr}[\tilde{R}]-\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n}=\mathrm{tr}[R]+\frac{1}{2}\bar{\Sigma}-\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n},\end{split}$

where $\bar{\Sigma}$ is the mean stretch curvature (1.9). ∎

###### The proof of Theorem 2.

Assume that the Finsler metric $F$ satisfies $d\mathsf{e}=0$ and
$\bar{\Sigma}=0$. From (2.39) we obtain

(2.40)
$\begin{split}\pi^{*}d\xi=\mathrm{tr}[R]=\frac{1}{2}H_{\gamma}R^{~{}\gamma}_{n~{}ab}\omega^{a}\wedge\omega^{b}.\end{split}$

Since that $\eta=d^{SM/M}\tau$ and the fibres of $SM$ are compact, $\eta$ has
zero points on each fibre of $SM$. However $\pi^{*}d\xi$ is constant along
each fibre. We conclude that the both sides of the equation (2.40) vanish.
Therefore $d\xi=0$ as $\pi$ is surjective. And $\mathrm{tr}[R]=0$. ∎

###### The proof of Corollary 1.

We are going to prove the chain of statements:
(1)$\implies$(2)$\implies$(3)$\implies$(1).

(1)$\implies$(2): By the assumption $d\mathsf{e}=0$ and $\mathbf{J}=0$, a
similar argument as the proof of Theorem 2 shows that the $S$-curvature is
almost constant

(2.41) $S=\frac{1}{n-1}\mathsf{e}+\pi^{*}\xi(\mathbf{G}),$

with $d\xi=0$.

(2)$\implies$(3): Assume that $S$ is given by (2.41) with
$d\mathsf{e}=d\xi=0$. The hypothesis $\mathbf{J}=0$ and (2.39) yields
$\mathrm{tr}[R]=0$. By Theorem 1, the hypothesis $\mathbf{J}=0$ and (2.22), we
obtain $d\eta=-\mathrm{tr}[P]=-\mathbf{E}=-\mathsf{e}d\omega^{n}$.

(3)$\implies$(1): By (2.22), the proof is clear. ∎

###### The proof of Corollary 3.

Assume that $F$ is of scalar curvature, the following formula (c.f. [10]) is
well known

(2.42) $R^{~{}i}_{i~{}\alpha
n}+J_{\alpha|n}=-\frac{n+1}{3}\mathbf{K}_{,\alpha}.$

Assume that $F$ has almost isotropic $S$-curvature. By (2.42) and (2.38), one
obtains

(2.43)
$0=\frac{n+1}{3}\mathbf{K}_{,\alpha}\omega^{\alpha}\wedge\omega^{n}+\frac{1}{n-1}d\mathsf{e}\wedge\omega^{n}.$

Since $\mathsf{e}$ is a function on $M$, (2.43) and (2.7) yields

(2.44)
$0=(\frac{n+1}{3}\mathbf{K}-\frac{1}{n-1}\mathsf{e}_{|n})_{,\alpha}\omega^{\alpha}\wedge\omega^{n}.$

Therefore
$\frac{n+1}{3}\mathbf{K}-\frac{1}{n-1}d\mathsf{e}(\mathbf{G})=\pi^{*}\sigma$,
for some $\sigma\in C^{\infty}(M)$.

Conversely, $F$ has weakly isotropic flag curvature if and only if

(2.45)
$-\frac{n+1}{3}\mathbf{K}_{,\alpha}=-\frac{1}{n-1}c_{|n,\alpha}=\frac{1}{n-1}c_{|\alpha},$

where $c\in C^{\infty}(M)$. Assume that $F$ has weakly isotropic
$S$-curvature, (2.38) and (2.45) implies that

(2.46)
$\begin{split}\pi^{*}d\xi=&\frac{1}{2}(R^{~{}i}_{i~{}\alpha\beta}+(J_{\alpha|\beta}-J_{\beta|\alpha}))\omega^{\alpha}\wedge\omega^{\beta}+\frac{1}{n-1}d(c-\mathsf{e})\wedge\omega^{n}\\\
=&\frac{1}{2}\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{\alpha}\wedge\omega^{\beta}+\frac{1}{n-1}d(c-\mathsf{e})\wedge\omega^{n}.\end{split}$

Taking derivative of (2.46), one obtains

$\begin{split}0=&\frac{1}{2}(d\tilde{R}^{~{}i}_{i~{}\alpha\beta}\wedge\omega^{\alpha}\wedge\omega^{\beta}+\tilde{R}^{~{}i}_{i~{}\alpha\beta}d\omega^{\alpha}\wedge\omega^{\beta}-\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{\alpha}\wedge
d\omega^{\beta})\\\
&+\frac{1}{n-1}d(c-\mathsf{e})\wedge\sum_{\alpha}(\omega^{\alpha}\wedge\omega^{\bar{\alpha}})\\\
=&\frac{1}{2}(d\tilde{R}^{~{}i}_{i~{}\alpha\beta}\wedge\omega^{\alpha}\wedge\omega^{\beta}+\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{\gamma}\wedge\omega_{\gamma}^{\alpha}\wedge\omega^{\beta}-\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{\alpha}\wedge\omega^{\gamma}\wedge\omega_{\gamma}^{\beta})\\\
&+(\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{n}\wedge\omega_{n}^{\alpha}\wedge\omega^{\beta}-\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{\alpha}\wedge\omega^{n}\wedge\omega_{n}^{\beta})\\\
&+\frac{1}{n-1}[(c-\mathsf{e})_{|\beta}\omega^{\beta}+(c-\mathsf{e})_{|n}\omega^{n}]\wedge\sum_{\alpha}(\omega^{\alpha}\wedge\omega^{\bar{\alpha}})\\\
=&\frac{1}{2}(\tilde{R}^{~{}i}_{j~{}\alpha\beta|i}\omega^{j}+\tilde{R}^{~{}i}_{i~{}\alpha\beta,\gamma}\omega^{\bar{\gamma}})\wedge\omega^{\alpha}\wedge\omega^{\beta}-2\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{n}\wedge\omega^{\alpha}\wedge\omega^{\bar{\beta}}\\\
&+\frac{1}{n-1}[(c-\mathsf{e})_{|\beta}\omega^{\beta}+(c-\mathsf{e})_{|n}\omega^{n}]\wedge\sum_{\alpha}(\omega^{\alpha}\wedge\omega^{\bar{\alpha}}),\\\
\end{split}$

therefore

(2.47)
$2\tilde{R}^{~{}i}_{i~{}\alpha\beta}\omega^{n}\wedge\omega^{\alpha}\wedge\omega^{\bar{\beta}}=\frac{1}{n-1}(c-\mathsf{e})_{|n}\omega^{n}\wedge\sum_{\alpha}(\omega^{\alpha}\wedge\omega^{\bar{\alpha}}).$

Since
$\tilde{R}^{~{}i}_{i~{}\alpha\beta}=-\tilde{R}^{~{}i}_{i~{}\beta\alpha}$, both
sides of (2.47) vanish. By this fact, (2.46) is simplified as

(2.48) $\pi^{*}d\xi=-\frac{1}{n-1}d(c-\mathsf{e})\wedge\omega^{n}.$

Hence

(2.49)
$0=d(c-\mathsf{e})\wedge(\omega^{1}\wedge\omega^{n+1}+\cdots+\omega^{n-1}\wedge\omega^{2n-1}),$

which is equivalent to the following system

$\displaystyle 0$
$\displaystyle=d(c-\mathsf{e})\wedge\omega^{1}\wedge\omega^{n+1},$
$\displaystyle\cdots\cdots$ $\displaystyle 0$
$\displaystyle=d(c-\mathsf{e})\wedge\omega^{n-1}\wedge\omega^{2n-1}.$

Therefore $d(c-\mathsf{e})=0$ and thus $d\xi=0$.

∎

## References

  * [1] David Bao, Shiing-Shen Chern and Zhongmin Shen, An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, Inc., 2000.
  * [2] David E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds. (2nd Edition). Birkhäuser, New York, 2010.
  * [3] Xinyue Chen(g), Xiaohuan Mo and Zhongmin Shen, On the flag curvature of Finsler metrics of scalar curvature. J. of London Math. Soc., Vol. 68, No. 2, 2003: 762-780.
  * [4] Xinyue Cheng and Zhongmin Shen, Finsler Geometry-an approach via Randers spaces. Science Press Beijing and Springer-Verlag Berlin Heidelberg, 2012.
  * [5] Shiing-Shen Chern and Zhongmin Shen, Riemann-Finsler Geometry. Nankai Tracts in Mathematics, Vol. 6, World Scientific, 2005.
  * [6] Huitao Feng and Ming Li, Adiabatic limit and connections in Finsler geometry. Communications in Analysis and Geometry, Vol. 21, No. 3, 2013: 607-624.
  * [7] Anmin Li, Udo Simon and Guosong Zhao, Global Affine Differential Geometry of Hypersurfaces. W. de Gruyter, Berlin-New York, 1993.
  * [8] Ming Li, Equivalence theorems of Minkowski spaces and applications in Finsler geometry.(in Chinese) Acta Math. Sinica (Chin. Ser.) Vol. 62, No.2, 2019: 177-190. (see arXiv:1504.04475v2 for the English version.)
  * [9] Ming Li and Lihong Zhang, Properties of Berwald scalar curvature. Front. Math. China, 15(6), 2020: 1143-1153.
  * [10] Xiaohuan Mo, An Introduction to Finsler Geometry. Peking University series in Math., Vol. 1, World Scientific Publishing Co. Pte. Ltd., 2006.
  * [11] Behzad Najafi and Akbar Tayebi, Weakly stretch Finsler metrics. Publ. Math. Debrecen, Vol. 91, 2017: 1-12.
  * [12] Katsumi Nomizu and Takeshi Sasaki, Affine Differential Geometry. Cambridge University Press, 1994.
  * [13] Rolf Schneider, Zur affinen Differentialgeometrie im Großen. I. Math. Zeitschr., Vol. 101, 1967: 375-406.
  * [14] Yibing Shen and Zhongmin Shen, Introduction to Modern Finsler Geometry. World Scientific, Singapore, 2016.
  * [15] Zhongmin Shen, Volume comparison and its applications in Riemann-Finsler geometry. Adv. in Math., Vol. 128, 1997: 306-328.
  * [16] Zhongmin Shen, Differential Geometry of Spray and Finsler Spaces. Kluwer Acad. Publ., 2001.
  * [17] Zhongmin Shen, Lectures on Finsler Geometry. World Scientific, 2001.
  * [18] Udo Simon, Angela Schwenk-Schellschmidt and Helmut Viesel, Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture notes, Science University Tokyo, 1991.
  * [19] Weiping Zhang, Lectures on Chern-Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, Vol. 4, World Scientific Publishing Co. Pte. Ltd., 2001.


# News on strangeness production from NA61/SHINE ††thanks: Presented at Quark
Matter 2022, 5 April 2022, Kraków, Poland

Maciej Lewicki
for the NA61/SHINE Collaboration  Institute of Nuclear Physics Polish Academy
of Sciences

###### Abstract

NA61/SHINE is a fixed target experiment at the CERN Super Proton Synchrotron.
The main goals of the experiment are to discover the critical point of
strongly interacting matter and to study the properties of the onset of
deconfinement. In order to reach these goals, a study of hadron production
properties is performed in nucleus-nucleus, proton-proton and proton-nucleus
interactions as a function of collision energy and size of the colliding
nuclei. In this talk, the new results on identified charged kaon production in
the intermediate size system (40Ar+45Sc and 7Be+9Be) collisions at SPS beam
momentum range (13A-150A GeV/c) will be shown. Additionally, the new
measurements of strange mesons ($K^{0}_{S}$), resonances ($K^{0*}$(892),
$\Xi(1530)^{0}$ and $\bar{\Xi}(1530)^{0}$) and baryons ($\Xi^{-}$(1321),
$\Xi^{+}$(1321)) produced in $p$+$p$ interactions are presented.

##  1 NA61/SHINE facility

NA61/SHINE is a fixed target spectrometer [1] located in CERN’s North Area,
utilizing the SPS proton, ion and hadron beams. Tracking capabilities are
provided by four large volume Time Projection Chambers (TPC), two of which are
located in magnetic fields. The Projectile Spectator Detector (PSD), a zero
degree, modular calorimeter, is used to determine the centrality of the
collisions. The aim of the experiment is to explore the QCD phase diagram
($\mu_{B},T$) by a two-dimensional scan in collision energy and system size.
The yields of hadrons produced in the collisions are studied for indications
of the onset of deconfinement [2] and the critical point of the phase
transition [3]. This article discusses the new NA61/SHINE results on the
strange hadron production in two main categories: charged kaon spectra in
intermediate size systems of Ar+Sc and Be+Be [4, 5, 7, 8, 9], and the high
precision measurements of strange mesons, baryons and resonances from $p$+$p$
interactions [14, 15, 16, 21]. The data on strange hadron production
properties are a rich source of knowledge on the dynamics of high energy
collisions, particularly in the region of deconfinement phase transition.

##  2 Studies of the onset of deconfinement by NA61/SHINE

The Statistical Model of the Early Stage (SMES) [2] predicts a first order
phase transition from the hadronic matter to the QGP phase between top AGS and
top SPS energies. In the transition region, constant temperature and pressure
in the mixed phase and an increase of the number of internal degrees of
freedom are expected. The most prominent predicted signature of the
deconfinement transition is a rapid, non-monotonic change of the
$K^{+}/\pi^{+}$ ratio as a function of collision energy (“the horn”). Such
signature was observed in central Pb+Pb collisions by the NA49 experiment,
both at mid-rapidity, as well as in mean multiplicities. The NA61/SHINE
extends the set of experimental data by additional measurements of lower mass
systems: $p$+$p$, Be+Be and Ar+Sc.

Figure 1: The $K^{+}/\pi^{+}$ charged meson ratio in midrapidity (left) and
for total yields (right). The plots show data on $p$+$p$, Be+Be and Ar+Sc
collisions measured by the NA61/SHINE (full symbols) compared with available
World data (open symbols).

The Fig. 1 shows the compilation of results obtained by both Collaborations,
as well as other available World data. A plateau-like structure is visible in
$p$+$p$ interactions at SPS energy range and similar values are observed for
Be+Be collisions—both at mid-rapidity as well as in the total yields. For all
five analysed energies of Ar+Sc collisions, the ratio $K^{+}/\pi^{+}$ is
significantly higher than in collisions of small systems ($p$+$p$, Be+Be),
although with clear qualitative difference with respect to Pb+Pb data—no horn
structure is visible. Figure 2 shows a detailed look into the system size
dependence of $K^{+}/\pi^{+}$ ratio at midrapidity and inverse slope parameter
$T$ fitted to the transverse momentum spectra of $K^{+}$ mesons. Both shown
quantities display a similar, threshold-like behaviour, which cannot be
reproduced by any of considered models. Ar+Sc is the lightest of measured
systems at which a significant enhancement of the $K^{+}/\pi^{+}$ ratio and
increase in the $T$ parameter is observed.

The observed rapid change of hadron production properties that starts when
moving from Be+Be to Ar+Sc collisions at top SPS energies (’’the jump‘‘) hints
a beginning of the creation of large clusters of strongly interacting
matter—the onset of fireball. The similarities of $p$+$p$ and Be+Be systems
suggest that interactions of these systems could happen in clusters of binary
collisions of nucleons, in which produced hadrons originate from the
formation, evolution and fragmentation of strings [10]. On the other hand, in
case of larger systems (Pb+Pb, but also Ar+Sc), the density of strings may
become sufficiently high, that the strings are close enough to interact and
change their properties, forming a collectively evolving QGP fireball [11].
The NA61/SHINE‘s studies of system size dependence of hadron production
properties allows for the first time to sketch the diagrams of high energy
nuclear collisions, displaying the domains in which different hadron
production mechanisms dominate [12, 13].

Figure 2: The system size dependence of the $K^{+}/\pi^{+}$ ratio at
midrapidity measured in collisions at 150$A$ GeV/$c$ compared with dynamical
(left) and statistical (middle) models. Right: Measurement of the inverse
slope parameter $T$, fitted to $K^{+}$ transverse momentum spectra at the same
collision energy.

##  3 New results on strange mesons produced in $p$+$p$ interactions

The studies of $K$ meson production in $p$+$p$ collisions is interesting not
only as a reference for A+A systems, but also for understanding the
strangeness production in elementary interactions. The NA61/SHINE has
collected data on almost 30 million $p$+$p$ collisions, allowing for
measurements of unprecedented precision at SPS energy range. The first results
on $K^{0}_{S}$ mesons produced in $p$+$p$ collisions at 158 GeV/$c$ [16] are
also the first step of the energy scan of $K^{0}_{S}$ production in $p$+$p$
interactions, which will provide a baseline for the future analysis in
collisions of heavier systems. Additionally, thanks to high statistics, large
acceptance and good resolution, the new results on $K^{0}_{S}$, $K^{+}$ and
$K^{-}$ have significantly higher precision than previously published data at
the SPS energies and will become an important discrimination and tuning tool
for the models of high energy collisions. The measured spectra of discussed
mesons are displayed in Fig. 3.

K${}_{S}^{0}$ (158 GeV/$c$)

Figure 3: New, high-statistics NA61/SHINE results on $K$ mesons produced in
$p$+$p$ inelastic interactions at 158 GeV/$c$: the two first panels show the
comparison of $K^{+}$ and $K^{-}$ $p_{T}$ spectra (preliminary) compared to
the previously published data and the plot on the right displays the rapidity
distribution of the $K_{S}^{0}$ meson [16].

Strange hadrons may also be produced in the form of short-lived excited
states. The study of such resonances is a unique insight into understanding of
the time evolution of the fireball created at high energy collisions. The
transverse mass and rapidity spectra, as well as total yields of
$K^{*}(892)^{0}$ mesons measured by the NA61/SHINE [14, 15] are displayed in
Fig. 4. The measurement may help to distinguish between two possible scenarios
for the fireball freeze-out: the sudden and the gradual one [19]. Moreover,
the ratio of $K^{*}(892)^{0}$ to the charged kaon yield (Fig. 4) may allow to
determine the time between chemical and kinetic freeze-outs [19, 20]. Together
with future NA61/SHINE measurements in Be+Be, Ar+Sc, and Xe+La collisions, it
will allow estimating the time between freeze-outs for these systems at three
SPS energies. The spectra are also important inputs for Blast-Wave
parametrization and Hadron Resonance Gas models, broader discussed in Ref.
[15] and [17, 18], respectively. The described high precision results on
$K^{*}(892)^{0}$ production will also provide an important reference for
tuning Monte Carlo string-hadronic models.

$K^{*}(892)^{0}$ (40–158 GeV/$c$)

Figure 4: Left: Results on rapidity spectra of $K^{*}(892)^{0}$ mesons
produced in inelastic $p$+$p$ collisions at 40, 80, and 158 GeV/$c$. Middle:
Ratio of mean multiplicity of $\langle K^{*}(892)^{0}\rangle$ meson to the
mean multiplicity of charged kaons $\langle K^{\pm}\rangle$. Right: Comparison
to Hadron Gas Model predictions [17]

##  4 Multi-strange hyperon production in $p$+$p$ interactions

The high statistics sample of $p$+$p$ collisions at 158 GeV/c beam momentum,
obtained by the NA61/SHINE, allowed the more challenging measurement of
$\Xi^{-}$, $\bar{\Xi}^{+}$, $\Xi(1530)^{0}$ and $\bar{\Xi}(1530)^{0}$ hyperon
production [21]—the only results at the SPS energy. Multi-strange hyperon
candidates are identified by their decay typologies. The $\Xi(1530)^{0}$ is
produced in the primary interaction and decays strongly into $\Xi^{-}$ and
$\pi^{+}$. Then the $\Xi^{-}$ travels certain distance, after which it decays
into a $\Lambda$ and a $\pi^{-}$. Subsequently, the $\Lambda$ decays into a
proton and a $\pi^{-}$. The results of described procedure are the two-
dimensional spectra in $y$ and $p_{T}$, which are consequently corrected for
detector geometrical acceptance, reconstruction efficiency, secondary
interactions and finally also branching ratios to unmeasured channels. The
$p_{T}$-integrated rapidity distribution of $\Xi^{-}$, $\bar{\Xi}^{+}$,
$\Xi(1530)^{0}$ and $\bar{\Xi}(1530)^{0}$ is shown in Fig. 5. The Figure also
compiles the data on charged $\Xi$ yields at midrapidity from various
experiments, showing the collision energy dependence. The figure shows a
strong suppression of antibaryon yield at SPS, with
$\langle\bar{\Xi}^{+}\rangle/\langle\Xi^{-}\rangle=0.395\pm 0.012\pm 0.030$,
which disappears already at RHIC.

Figure 5: $\Xi$ hyperon production in inelastic $p$+$p$ collisions at 158
GeV/$c$: d$n/$d$y$ distributions of $\Xi^{-}$ and $\bar{\Xi}^{+}$ (left),
$\Xi(1530)^{0}$ and $\bar{\Xi}(1530)^{0}$ (middle) and collision energy
dependence $\Xi^{-}$ and $\bar{\Xi}^{+}$ yields at midrapidity, including new
results from SPS.

The $\Xi$ mean multiplicities measured by NA61/SHINE in inelastic $p$+$p$
interactions are used to calculate the enhancement factors of $\Xi$ hyperons
observed in centrality selected Pb+Pb, in semicentral C+C, and in Si+Si
collisions as measured by NA49 [22] at the CERN SPS. The results for
midrapidity densities are shown in Fig. 6 as a function of $\langle
N_{W}\rangle$. The enhancement factor increases approximately linearly from
3.5 in C+C to 9 in central Pb+Pb collisions.

Figure 6: Strangeness enhancement factors determined using $\Xi^{-}$ and
$\bar{\Xi}^{+}$ baryon production measured by the NA61/SHINE as the baseline.

Acknowledgments: This work was supported by the Polish Minister of Education
and Science (contract No. 2021/WK/10) and partially also by the Polish
National Science Centre grant 2015/18/M/ST2/00125.

## References

  * [1] NA61/SHINE Collaboration, JINST 9, P06005 (2014)
  * [2] M. Gazdzicki, M.I. Gorenstein, Acta Phys. Pol. B 30, 2705 (1999)
  * [3] NA61/SHINE, Eur. Phys. J. C 77, 671 (2017)
  * [4] P. Podlaski, SQM 2019, Bari, Italy, indico.cern.ch/event/755366/contributions/3357938
  * [5] S. Pulawski, SQM 2019, Bari, Italy, indico.cern.ch/event/755366/contributions/3427002/
  * [6] NA61/SHINE, Tech. Rep. CERN-SPSC-2018-008, SPSC-P-330-ADD-10
  * [7] NA61/SHINE, Eur.Phys.J.C 80 (2020) 10, 961, Eur.Phys.J.C 81 (2021) 2, 144 (erratum)
  * [8] NA61/SHINE, Eur.Phys.J.C 81 (2021) 1, 73
  * [9] NA61/SHINE, Eur.Phys.J.C 81 (2021) 5, 397
  * [10] K. Werner, Phys. Rept., vol. 232, pp. 87–299, 1993.
  * [11] T. Kalaydzhyan, E. Shuryak, Phys. Rev. C, vol. 90, no. 1, p. 014901, 2014;
T. Kalaydzhyan and E. Shuryak, Phys. Rev. D, vol. 90, no. 2, p. 025031, 2014;
I. Iatrakis, A. Ramamurti, and E. Shuryak, Phys. Rev. D, vol. 92, no. 1, p.
014011, 2015

  * [12] M. P. Lewicki, L. Turko,  arXiv:2002.00631
  * [13] E. Andronov, M. Kuich, M. Gaździcki, arXiv:2205.06726
  * [14] NA61/SHINE, Eur.Phys.J.C 80 (2020) 5, 460
  * [15] NA61/SHINE, Eur.Phys.J.C 82 (2022) 4, 322
  * [16] NA61/SHINE, Eur.Phys.J.C 82 (2022), 96
  * [17] V. V. Begun et al., Phys. Rev. C 98, no. 5, 054909 (2018)
  * [18] J. Phys. G: Nucl. Part. Phys. 48 085004 (2021)
  * [19] C. Markert, G. Torrieri and J. Rafelski, AIP Conf. Proc. 631, 533 (2002)
  * [20] C. Blume, Acta Phys. Polon. B 43, 577 (2012)
  * [21] NA61/SHINE, Eur.Phys.J.C 81 (2021) 10, 911
  * [22] NA49, Phys. Rev. C 80, 034906 (2009), 0906.0469