# On a Class of Generalized Berwald Manifolds

A. Tayebi and F. Eslami

###### Abstract

The class of generalized Berwald metrics contains the class of Berwald
metrics. In this paper, we characterize two-dimensional generalized Berwald
$(\alpha,\beta)$-metrics with vanishing S-curvature. Let $F=\alpha\phi(s)$,
$s=\beta/\alpha$, be a two-dimensional generalized Berwald
$(\alpha,\beta)$-metric on a manifold $M$. Suppose that $F$ has vanishing
S-curvature. We show that one of the following holds: (i) if $F$ is a regular
metric, then it reduces to a Riemannian metric of isotropic sectional
curvature or a locally Minkowskian metric; (ii) if $F$ is an almost regular
metric that is not Riemannian nor locally Minkowskian, then we find the
explicit form of $\phi=\phi(s)$ which obtains a generalized Berwald metric
that is neither a Berwald nor Landsberg nor a Douglas metric. This provides a
generalization of Szabó rigidity theorem for the class of
$(\alpha,\beta)$-metrics. In the following, we prove that left invariant
Finsler surfaces with vanishing S-curvature must be Riemannian surfaces of
constant sectional curvature. Finally, we construct a family of odd-
dimensional generalized Berwald Randers metrics.

Keywords: Generalized Berwald metric, Berwald metric, $(\alpha,\beta)$-metric,
S-curvature.111 2010 Mathematics subject Classification: 53C60, 53C25.

##  1 Introduction

A Finsler metric $F$ on a manifold $M$ is called a generalized Berwald metric
if there exists a covariant derivative $\nabla$ on $M$ such that the parallel
translations induced by $\nabla$ preserve the Finsler function $F$
[18][19][20][26]. In this case, $F$ is called a generalized Berwald metric on
$M$ and $(M,F)$ is called a generalized Berwald manifold. If the covariant
derivative $\nabla$ is also torsion-free, then $F$ reduces to a Berwald
metric. Therefore, the class of Berwald metrics belongs to the class of
generalized Berwald metrics. The importance of the class of generalized
Berwald manifolds lies in the fact that these manifolds may have a rich
isometry group [16][17]. For some progress about the class of generalized
Berwald manifolds, see [2], [4], [20], [25], [26], [27] and [28].

To find generalized Berwald metrics, one can consider the class of
$(\alpha,\beta)$-metrics. An $(\alpha,\beta)$-metric is a Finsler metric
defined by $F:=\alpha\phi(s)$, $s={\beta}/{\alpha}$, where $\phi=\phi(s)$ is a
smooth function on a symmetric interval $(-b_{0},b_{0})$ with certain
regularity, $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and
$\beta=b_{i}(x)y^{i}$ is a 1-form on the base manifold. The simplest
$(\alpha,\beta)$-metrics are the Randers metrics $F=\alpha+\beta$ which were
discovered by G. Randers when he studied 4-dimensional general relativity.
These metrics have been widely applied in many areas of natural sciences,
including physics, biology, psychology, etc [5]. In [26], Vincze proved that a
Randers metric $F=\alpha+\beta$ is a generalized Berwald metric if and only if
dual vector field $\beta^{\sharp}$ is of constant Riemannian length. In [20],
Tayebi-Barzegari generalized Vincze’s result for $(\alpha,\beta)$-metrics and
showed that an $(\alpha,\beta)$-metric satisfying the so-called sign property
is a generalized Berwald metric if and only if $\beta^{\sharp}$ is of constant
Riemannian length. Then, Vincze showed that an $(\alpha,\beta)$-metric
satisfying the regularity property $\phi^{\prime}(0)\neq 0$ is a generalized
Berwald metric if and only if $\beta^{\sharp}$ is of constant Riemannian
length [25]. In [4], Bartelme${\ss}$-Matveev proved that a Finsler metric is a
generalized Berwald metric if and only if it is monochromatic.

Generally, two-dimensional Finsler metrics have some different and special
Riemannian and non-Riemannian curvature properties from the higher dimensions.
For example, Bartelme${\ss}$-Matveev showed that except for torus and the
Klein bottle, the other closed 2-dimensional manifolds can not have non-
Riemannian generalized Berwald metrics [4]. In [28], Vincze et al. showed that
a connected generalized Berwald surface is a Landsberg surface if and only if
it is a Berwald surface. The S-curvature is constructed by Shen for given
comparison theorems on Finsler manifolds [5]. An interesting problem is to
study generalized Berwald metrics with vanishing S-curvature. Here, we
characterize the class of two-dimensional generalized Berwald
$(\alpha,\beta)$-metrics with vanishing S-curvature and prove the following.

######  Theorem 1.1.

Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be a two-dimensional generalized
Berwald $(\alpha,\beta)$-metric on a connected manifold $M$. Suppose that $F$
has vanishing S-curvature. Then one of the following holds:

(i) If $F$ is a regular metric, then it reduces to a Riemannian metric of
isotropic sectional curvature or a locally Minkowskian metric;

(ii) If $F$ is an almost regular metric that is not Riemannian nor locally
Minkowskian, then $\phi$ is given by

$\displaystyle\phi=c\exp\Bigg{[}\int_{0}^{s}{\frac{kt+q\sqrt{b^{2}-t^{2}}}{1+kt^{2}+qt\sqrt{b^{2}-t^{2}}}dt}\Bigg{]},$
(1.1)

where $c>0$, $q>0$, and $k$ are real constants, and $\beta$ satisfies

$r_{ij}=0,\ \ \ \ s_{i}=0.$ (1.2)

In this case, $F$ is neither a Berwald nor Landsberg nor a Douglas metric.

It is remarkable that, for a generalized Berwald $(\alpha,\beta)$-metric
$F=\alpha\phi(s)$, $s=\beta/\alpha$, on an $n$-dimensional manifold $M$, we
show that ${\bf S}=0$ if and only if $F$ is Riemannian or $\beta$ is a Killing
form with constant length (see Lemma 3.2).

The celebrated Szabó rigidity theorem states that every 2-dimensional Berwald
surface is either locally Minkowskian or Riemannian (see [13]). Every Berwald
metric has vanishing S-curvature [22]. Then, Theorem 1.1 is an extension of
Szabó’s result for $(\alpha,\beta)$-metrics.

The condition of generalized Berwaldness can not be dropped from the
assumption of Theorem 1.1. There are many two-dimensional
$(\alpha,\beta)$-metrics with vanishing S-curvature which are not Riemannian
nor locally Minkowskian. For example, let us consider Shen’s Fish-Tank Randers
metric $F=\alpha+\beta$ on $\mathbb{R}^{2}$ given by following

$\displaystyle F={\sqrt{(xv-yu)^{2}+(u^{2}+v^{2})(1-x^{2}-y^{2})}\over
1-x^{2}-y^{2}}+{yu-xv\over 1-x^{2}-y^{2}}.$

$F$ has vanishing S-curvature [14] while it is not a generalized Berwald
metric ($\|\beta\|_{\alpha}=\sqrt{x^{2}+y^{2}}$). This metric is not
Riemannian. Also, $F$ has vanishing flag curvature while it is not locally
Minkowskian.

In Theorem 1.1, the vanishing of S-curvature is necessary. For example, see
the following.

######  Example 1.

Let us consider $G:=\\{(x,y)\in\mathbb{R}^{2}|y>0\\}$ and define a
multiplication on $G$ by
$(x_{1},y_{1})\ast(x_{2},y_{2}):=(x_{2}y_{1}+x_{1},y_{1}y_{2})$, for
$(x_{i},y_{i})\in G$, $i=1,2$. $(G,\ast)$ is a Lie group [8]. One can
introduce a Randers metric $F$ on $G$ by

$F=\frac{\sqrt{2dx^{2}+2dxdx+2dy^{2}}}{y}+\frac{dx+dy}{y}.$ (1.3)

Then

$\displaystyle a_{11}=a_{22}=\frac{2}{y^{2}},\ \ \
a_{21}=a_{12}=\frac{1}{y^{2}},\ \ \ b_{1}=b_{2}=\frac{1}{y},$ $\displaystyle
a^{11}=a^{22}=\frac{2}{3}y^{2},\ \ \ a^{12}=a^{21}=-\frac{1}{3}y^{2},\ \ \
b^{1}=b^{2}=\frac{1}{3}y.$

It is easy to see that $\alpha$ is a positive-definite Riemannian metric.
Also, we get

$b:=\|\beta\|_{\alpha}=\sqrt{a^{ij}b_{i}b_{j}}=\sqrt{b_{i}b^{i}}=\sqrt{b_{1}b^{1}+b_{2}b^{2}}=\sqrt{\frac{2}{3}}$

It follows that $F$ is a positive-definite Randers metric on $G$. In [26],
Vincze showed that a Randers metric $F=\alpha+\beta$ is a generalized Berwald
metric if and only if $\beta$ is of constant length with respect to $\alpha$.
Then, the Randers metric defined by (1.3) is a generalized Berwald metric. We
have

$d\beta=\frac{1}{y^{2}}dx\wedge dy\neq 0.$

Thus $\beta$ is not closed which implies that $F$ can not be a Berwald metric.
We have

$s_{11}=s_{22}=0,\ \ \ s_{12}=\frac{1}{y^{2}}=-s_{21},\ \ \
s_{1}=-\frac{1}{3y},\ \ \ s_{2}=\frac{1}{3y}.$ (1.4)

where $s_{ij}$ and $s_{i}$ defined by (3.1) and (3.2). We claim that $F$ has
not vanishing S-curvature. On contrary, let ${\bf S}=0$. Then by Lemma 3.2.1
in [5], we have $r_{ij}+b_{i}s_{j}+b_{j}s_{i}=0$. Contracting it with $b^{i}$
yields

$r_{j}+b^{2}s_{j}=0.$

By considering $r_{i}+s_{i}=0$ and $b^{2}=2/3$, we must have $s_{i}=0$ which
contradicts with (1.4). Then, the Randers metric (1.3) is a generalized
Berwald metric with ${\bf B}\neq 0$, ${\bf L}\neq 0$, ${\bf D}\neq 0$, ${\bf
S}\neq 0$ and ${\bf E}\neq 0$. We claim that $F$ is not R-quadratic. On the
contrary, let the Finsler metric $F$ defined by (1.3) be R-quadratic. By
Theorem 6.2.1 in [5], if the two-dimensional Randers metric (1.3) is
R-quadratic then it has constant S-curvature ${\bf S}=3cF$, for some real
constant $c$. In [23], Xu-Deng proved that every homogeneous Finsler metric of
isotropic S-curvature has vanishing S-curvature. Thus, we must have ${\bf
S}=0$ which is a contradiction. Therefore, the Randers metric (1.3) is not
R-quadratic.

Theorem 1.1 may not be held for an $(\alpha,\beta)$-metric of constant
S-curvature. For example, at a point $x=(x^{1},x^{2})\in\mathbb{R}^{2}$ and in
the direction $y=(y^{1},y^{2})\in T_{x}\mathbb{R}^{2}$, consider the
Riemannian metric $\alpha=\alpha(x,y)$ and one form $\beta=\beta(x,y)$ as
follows

$\displaystyle\alpha(x,y):=\sqrt{(y^{1})^{2}+e^{2x^{1}}(y^{2})^{2}},\ \ \
\beta(x,y):=y^{1}.$ (1.5)

Then $s_{ij}=0$ and $r_{ij}=b^{2}a_{ij}-b_{i}b_{j}$ which yield
$r_{i}+s_{i}=0$. If $\phi=\phi(s)$ satisfies

$\displaystyle\Phi=-6k\frac{\phi\Delta^{2}}{b^{2}-s^{2}},$ (1.6)

for some constant $k$, then $F=\alpha\phi(\beta/\alpha)$ has constant
S-curvature ${\bf S}=3kF$ (see [6]). Here, $\Delta=\Delta(b,s)$ and
$\Phi=\Phi(b,s)$ defined by (3.3) and (3.4), respectively. The existence of
regular solution of (1.6) for arbitrary $k\in\mathbb{R}$, when $\alpha$ and
$\beta$ are given by (1.5), is proved in [10]. It is easy to see that $F$ is a
generalized Berwald metric while it is not a Berwald metric.

We must mention that Theorem 1.1 does not hold for Finsler metrics of
$dim(M)\geq 3$. Denote generic tangent vectors on $\mathbb{S}^{3}$ as
$u{\partial}/{\partial x}+v{\partial}/{\partial y}+w{\partial}/{\partial z}$.
The Finsler functions for Bao-Shen’s Randers metrics $F=\alpha+\beta$ are
given by the following

$F=\frac{\,\sqrt{\mathcal{K}(cu-zv+yw)^{2}+(zu+cv-xw)^{2}+(xv+cw-
yu)^{2}}\,}{1+x^{2}+y^{2}+z^{2}}\pm\frac{\,\sqrt{\,\mathcal{K}-1\,}\ (cu-
zv+yw)\,}{\,1+x^{2}+y^{2}+z^{2}\,},$

where $\mathcal{K}>1$ is a real constant. For these metrics, we have

$b:=\|\beta\|_{\alpha}=\sqrt{1-\frac{1}{\mathcal{K}}}$

One can see that, the one form $\beta$ is not closed, and then $F$ can not be
a Douglas metric. This family of Randers metrics is generalized Berwald
metrics with ${\bf S}=0$ which are not Berwaldian.

In [4], it is proved that every 3-dimensional closed manifold admits a non-
Riemannian generalized Berwald metric. However, the closeness condition is
very restrictive. Indeed, for every manifold $M$ with $dim(M)\geq 3$, one can
construct generalized Berwald $(\alpha,\beta)$-metrics that are not
Berwaldian.

######  Example 2.

The projective spherical metric on $\mathbb{R}^{3}$ is given by the following

$\alpha:=\frac{\sqrt{(1+||X||^{2})||Y||^{2}-\langle X,Y\rangle^{2}}}{1+\langle
X,X\rangle},\,\,X\in\mathbb{R}^{3},\,\,Y\in T_{X}\mathbb{R}^{3},$ (1.7)

where $\langle,\rangle$ and $||.||$ denote the Euclidean inner product and
norm on $\mathbb{R}^{3}$, respectively. Put $X=(x,y,z)$ and $Y=(u,v,w)$.
Suppose that $\beta=\kappa(b_{1}u+b_{2}v+b_{3}w)$ is a Killing 1-form of
$\alpha$, where $0<\kappa<1$. By a simple calculation, we get

$\displaystyle b_{1}=\frac{A^{1}_{2}y+A^{1}_{3}z+C^{1}}{1+\langle
X,X\rangle},\ \ b_{2}=\frac{A^{2}_{1}x+A^{2}_{3}z+C^{2}}{1+\langle
X,X\rangle},\ \ b_{3}=\frac{A^{3}_{1}x+A^{3}_{2}y+C^{3}}{1+\langle
X,X\rangle},\ $ (1.8)

where $A=(A^{i}_{j})$ is an antisymmetric real matrix, and $C=(C^{i})$ is a
constant vector in $\mathbb{R}^{3}$. Let us put

$C=(0,1,0),\ \ \ A^{1}_{2}=A^{2}_{3}=0,\ \ A^{1}_{3}=1.$ (1.9)

In this case, $\beta$ is a Killing 1-form with $\|\beta\|_{\alpha}=\kappa<1$
such that is not closed. According to Shen’s theorem in [12], a regular
$(\alpha,\beta)$-metric is a Berwald metric if and only if $\beta$ is parallel
with respect to $\alpha$. Using the Riemannian metric (1.7) and 1-form $\beta$
satisfying (1.8) and (1.9), one can construct generalized Berwald
$(\alpha,\beta)$-metrics which are not Berwaldian.

Homogeneous Finsler manifolds are those Finsler manifolds $(M,F)$ that the
orbit of the natural action of $I(M,F)$ on $M$ at any point of $M$ is the
whole $M$. For the class of homogeneous generalized Berwald
$(\alpha,\beta)$-metrics, we prove the following.

######  Corollary 1.1.

Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be a two-dimensional homogeneous
generalized Berwald $(\alpha,\beta)$-metric on a manifold $M$. Then $F$ has
isotropic S-curvature if and only if it is a Riemannian metric of constant
sectional curvature or a locally Minkowskian metric.

Let $G$ be a connected Lie group with a bi-invariant Finsler metric
${\bar{F}}$, and $H$ a closed subgroup of $G$. Denote the Lie algebras of $G$
and $H$ as $\mathfrak{g}$ and $\mathfrak{h}$, respectively. Let $\rho$ be the
natural projection from $G$ to $M=G/H$. Then there exists a uniquely defined
$G$-invariant metric $F$ on $M$ such that for any $g\in G$, the tangent map
$\rho_{*}:\big{(}T_{g}G,{\bar{F}}(g,.)\big{)}\rightarrow\big{(}T_{\pi(g)}M,F(\pi(g),.)\big{)}$
is a submersion (see Lemma 3.1 in [24]). The $G$-invariant Finsler metric $F$
is called the normal homogeneous metric induced by ${\bar{F}}$. The pair
$(M,F)$ is said the normal homogeneous space induced by $\rho:G\rightarrow
M=G/H$ and ${\bar{F}}$. For normal homogeneous generalized Berwald metrics, we
have the following.

######  Corollary 1.2.

Every two-dimensional normal homogeneous generalized Berwald
$(\alpha,\beta)$-metric is a Riemannian metric of non-negative constant
sectional curvature or a locally Minkowskian metric.

A Finsler metric $F$ on a manifold $M$ is called a generalized normal
homogeneous metric if for every $x,y\in M$, there exists a
$\delta(x)$-translation of $(M,F)$ sending $x$ to $y$ (see [30]). In this
paper, we consider two-dimensional homogeneous generalized Berwald Randers
metric of generalized normal-type and prove the following.

######  Corollary 1.3.

Every two-dimensional generalized normal homogeneous generalized Berwald
Randers metric must be a Riemannian metric of non-negative constant sectional
curvature or a locally Minkowskian metric.

Let $(G,\cdot)$ be a connected Lie group with identity element $e$, and
$\lambda_{g}$ denotes the left translation by $g\in G$. A Finsler metric $F$
on $G$ is called a left invariant Finsler metric if it satisfies
$F\circ(\lambda_{\bf g})_{*}=F$ for any $g\in G$. In [2], Aradi proved that
left invariant Finsler metrics are generalized Berwald metrics. For this class
of Finsler metrics, we have the following.

######  Theorem 1.2.

Every left invariant Finsler surface has vanishing S-curvature if and only if
it is a Riemannian metric of constant sectional curvature.

By considering Theorems 1.1 and 1.2, it seems that two-dimensional generalized
Berwald $(\alpha,\beta)$-metrics with vanishing S-curvature may be only
Riemannian. But we do not find a short proof for this conjecture.

A Finsler metric $F$ on a manifold $M$ is called an isotropic Berwald metric,
if its Berwald curvature is given by

$\displaystyle{\bf B}_{y}(u,v,w)=\\!\\!\\!$
$\displaystyle\\!\\!\\!cF^{-1}\Big{\\{}{\bf
h}(u,v)\big{(}w-\textbf{g}_{y}(w,\ell)\ell\big{)}+{\bf
h}(v,w)\big{(}u-\textbf{g}_{y}(u,\ell)\ell\big{)}$ (1.10)
$\displaystyle\\!\\!\\!\qquad\qquad\ \ \ \ \ \ \ \ \qquad+{\bf
h}(w,u)\big{(}v-\textbf{g}_{y}(v,\ell)\ell\big{)}+2F{\bf
C}_{y}(u,v,w)\ell\Big{\\}}.$

where ${\bf h}_{y}(u,v)={\bf g}_{y}(u,v)-F^{-2}(y){\bf g}_{y}(y,u){\bf
g}_{y}(y,v)$ is the angular form in direction $y$, ${\bf C}$ denotes the
Cartan torsion of $F$ and $c\in C^{\infty}(M)$. Every Berwald metric is an
isotropic Berwald metric with $c=0$. The Funk metrics are isotropic Berwald
metrics with $c=1/2$. As a straightforward conclusion of Theorem 1.2, one can
get the following.

######  Corollary 1.4.

Every left invariant isotropic Berwald surface must be a Riemannian surface of
constant sectional curvature.

##  2 Preliminaries

Let $M$ be an $n$-dimensional $C^{\infty}$ manifold, $TM=\bigcup_{x\in
M}T_{x}M$ the tangent space and $TM_{0}:=TM-\\{0\\}$ the slit tangent space of
$M$. Let $(M,F)$ be a Finsler manifold. The following quadratic form
$\textbf{g}_{y}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ is called
fundamental tensor

$\textbf{g}_{y}(u,v):={1\over 2}\frac{\partial^{2}}{\partial s\partial
t}\Big{[}F^{2}(y+su+tv)\Big{]}_{s=t=0},\ \ u,v\in T_{x}M.$

Let $x\in M$ and $F_{x}:=F|_{T_{x}M}$. To measure the non-Euclidean feature of
$F_{x}$, one can define ${\bf C}_{y}:T_{x}M\times T_{x}M\times
T_{x}M\rightarrow\mathbb{R}$ by

${\bf C}_{y}(u,v,w):={1\over
2}\frac{d}{dt}\left[\textbf{g}_{y+tw}(u,v)\right]_{t=0},\ \ u,v,w\in T_{x}M.$

The family ${\bf C}:=\\{{\bf C}_{y}\\}_{y\in TM_{0}}$ is called the Cartan
torsion.

For a Finsler manifold $(M,F)$, its induced spray on $TM$ is denoted by ${\bf
G}={\bf G}(x,y)$ which in a standard coordinate $(x^{i},y^{i})$ for $TM_{0}$
is given by ${\bf G}=y^{i}{{\partial}/{\partial
x^{i}}}-2G^{i}(x,y){{\partial}/{\partial y^{i}}}$, where

$G^{i}:=\frac{1}{4}g^{il}\Big{[}\frac{\partial^{2}F^{2}}{\partial
x^{k}\partial y^{l}}y^{k}-\frac{\partial F^{2}}{\partial x^{l}}\Big{]}.$

For a Finsler metric $F$ on an $n$-dimensional manifold $M$, the Busemann-
Hausdorff volume form $dV_{F}=\sigma_{F}(x)dx^{1}\cdots dx^{n}$ is defined by

$\sigma_{F}(x):={{\rm Vol}\big{(}\mathbb{B}^{n}(1)\big{)}\over{\rm
Vol}\Big{\\{}(y^{i})\in R^{n}\ \big{|}\ F\big{(}y^{i}\frac{\partial}{\partial
x^{i}}|_{x}\big{)}<1\Big{\\}}}.$

Let $G^{i}$ denote the geodesic coefficients of $F$ in the same local
coordinate system. Then for ${\bf y}=y^{i}{\partial/\partial x^{i}}|_{x}\in
T_{x}M$, the S-curvature is defined by

${\bf S}({\bf y}):={\partial G^{i}\over\partial
y^{i}}(x,y)-y^{i}{\partial\over\partial x^{i}}\Big{[}\ln\sigma_{F}(x)\Big{]}.$
(2.1)

For a vector $y\in T_{x}M_{0}$, the Berwald curvature ${\bf
B}_{y}:T_{x}M\times T_{x}M\times T_{x}M\rightarrow T_{x}M$ is defined by ${\bf
B}_{y}(u,v,w):=B^{i}_{\ jkl}(y)u^{j}v^{k}w^{l}{{\partial}/{\partial
x^{i}}}|_{x}$, where

$B^{i}_{\ jkl}:={{\partial^{3}G^{i}}\over{\partial y^{j}\partial y^{k}\partial
y^{l}}}.$

$F$ is called a Berwald metric if $\bf{B}=0$. Every Berwald metric satisfies
${\bf S}=0$ (see [22]).

For $y\in T_{x}M$, define the Landsberg curvature ${\bf L}_{y}:T_{x}M\times
T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ by

${\bf L}_{y}(u,v,w):=-\frac{1}{2}{\bf g}_{y}\big{(}{\bf
B}_{y}(u,v,w),y\big{)}.$

A Finsler metric $F$ is called a Landsberg metric if ${\bf L}=0$.

Taking a trace of Berwald curvature ${\bf B}$ give us the mean of Berwald
curvature ${\bf E}$ which is defined by ${\bf E}_{y}:T_{x}M\times
T_{x}M\rightarrow\mathbb{R}$, where

${\bf E}_{y}(u,v):={1\over 2}\sum_{i=1}^{n}g^{ij}(y){\bf g}_{y}\big{(}{\bf
B}_{y}(u,v,\partial_{i}),\partial_{j}\big{)}.$ (2.2)

where $\\{\partial_{i}\\}$ is a basis for $T_{x}M$ at $x\in M$. In local
coordinates, ${\bf E}_{y}(u,v):=E_{ij}(y)u^{i}v^{j}$, where

$E_{ij}:=\frac{1}{2}B^{m}_{\ mij}.$

Taking a horizontal derivation of the mean of Berwald curvature ${\bf E}$
along Finslerian geodesics give us the $H$-curvature ${\bf H}={\bf H}(x,y)$
which is defined by ${\bf H}_{y}=H_{ij}dx^{i}\otimes dx^{j}$, where

$H_{ij}:=E_{ij|m}y^{m}.$

Here, $``|"$ denotes the horizontal covariant differentiation with respect to
the Berwald connection of $F$.

For a non-zero vector $y\in T_{x}M_{0}$, one can define ${\bf
D}_{y}:T_{x}M\times T_{x}M\times T_{x}M\rightarrow T_{x}M$ by ${\bf
D}_{y}(u,v,w):=D^{i}_{\ jkl}(y)u^{i}v^{j}w^{k}{\partial}/{\partial
x^{i}}|_{x}$, where

$D^{i}_{\ jkl}:=\frac{\partial^{3}}{\partial y^{j}\partial y^{k}\partial
y^{l}}\Bigg{[}G^{i}-\frac{2}{n+1}\frac{\partial G^{m}}{\partial
y^{m}}y^{i}\Bigg{]}.$ (2.3)

$\bf D$ is called the Douglas curvature. $F$ is called a Douglas metric if
$\bf{D}=0$.

For a non-zero vector $y\in T_{x}M_{0}$, the Riemann curvature is a family of
linear transformation $\textbf{R}_{y}:T_{x}M\rightarrow T_{x}M$ which is
defined by $\textbf{R}_{y}(u):=R^{i}_{k}(y)u^{k}{\partial/{\partial x^{i}}}$,
where

$R^{i}_{k}(y)=2{\partial G^{i}\over{\partial
x^{k}}}-{\partial^{2}G^{i}\over{{\partial x^{j}}{\partial
y^{k}}}}y^{j}+2G^{j}{\partial^{2}G^{i}\over{{\partial y^{j}}{\partial
y^{k}}}}-{\partial G^{i}\over{\partial y^{j}}}{\partial G^{j}\over{\partial
y^{k}}}.$ (2.4)

The family $\textbf{R}:=\\{\textbf{R}_{y}\\}_{y\in TM_{0}}$ is called the
Riemann curvature.

For a flag $P:={\rm span}\\{y,u\\}\subset T_{x}M$ with the flagpole $y$, the
flag curvature ${\bf K}={\bf K}(P,y)$ is defined by

${\bf K}(x,y,P):={{\bf g}_{y}\big{(}u,{\bf R}_{y}(u)\big{)}\over{\bf
g}_{y}(y,y){\bf g}_{y}(u,u)-{\bf g}_{y}(y,u)^{2}}.$ (2.5)

The flag curvature ${\bf K}(x,y,P)$ is a function of tangent planes $P={\rm
span}\\{y,v\\}\subset T_{x}M$. A Finsler metric $F$ is of scalar flag
curvature if ${\bf K}(x,y,P)={\bf K}(x,y)$ is independent of $P$. Also, $F$ is
called of isotropic and constant flag curvature if ${\bf K}={\bf K}(x)$ and
${\bf K}=constant$, respectively.

Throughout this paper, we use the Berwald connection on Finsler manifolds. The
pullback bundle $\pi^{*}TM$ admits a unique linear connection, called the
Berwald connection. Let $(M,F)$ be an $n$-dimensional Finsler manifold. Let
$\\{e_{j}\\}$ be a local frame for $\pi^{*}TM$,
$\\{\omega^{i},\omega^{n+i}\\}$ be the corresponding local coframe for
$T*(TM_{0})$ and $\\{\omega^{i}_{j}\\}$ be the set of local Berwald connection
forms with respect to $\\{e_{j}\\}$. Then the connection forms are
characterized by the structure equations as follows

  * •

Torsion freeness:

$\displaystyle d\omega^{i}=\omega^{j}\wedge\omega^{i}_{\ j}.$ (2.6)

  * •

Almost metric compatibility:

$\displaystyle dg_{ij}-g_{kj}\omega^{k}_{\ i}-g_{ik}\omega^{k}_{\
j}=-2L_{ijk}\omega^{k}+2C_{ijk}\omega^{n+k},$ (2.7)

where $\omega^{i}:=dx^{i}$ and $\omega^{n+k}:=dy^{k}+y^{j}\omega^{k}_{\ j}$.

The horizontal and vertical covariant derivations with respect to the Berwald
connection respectively are denoted by “$|$” and “, ”. For more details, one
can see [13].

##  3 Proof of Theorems 1.1 and 1.2

For an $(\alpha,\beta)$-metric, let us define $b_{i;j}$ by
$b_{i;j}\theta^{j}:=db_{i}-b_{j}\theta^{j}_{i}$, where $\theta^{i}:=dx^{i}$
and $\theta^{j}_{i}:=\gamma^{j}_{ik}dx^{k}$ denote the Levi-Civita connection
form of $\alpha$. Let

$\displaystyle r_{ij}:=\frac{1}{2}(b_{i;j}+b_{j;i}),\ \ \
s_{ij}:=\frac{1}{2}(b_{i;j}-b_{j;i}),\ \ \ r_{i0}:=r_{ij}y^{j},\ \
r_{00}:=r_{ij}y^{i}y^{j},\ \ r_{j}:=b^{i}r_{ij},$ (3.1) $\displaystyle
s_{i0}:=s_{ij}y^{j},\ \ \ s_{j}:=b^{i}s_{ij},\ \ \ s^{i}_{\ j}=a^{im}s_{mj},\
\ \ s^{i}_{\ 0}=s^{i}_{\ j}y^{j},\ \ \ r_{0}:=r_{j}y^{j},\ \ \ \
s_{0}:=s_{j}y^{j}.$ (3.2)

Put

$\displaystyle Q\\!\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\!\frac{\phi^{\prime}}{\phi-s\phi},\ \ \ \ \
\Delta:=1+sQ+(b^{2}-s^{2})Q^{\prime},\ \ \ \ \
\Theta:=\frac{Q-sQ^{\prime}}{2\Delta},$ (3.3) $\displaystyle\Phi\\!\\!\\!\\!$
$\displaystyle:=$
$\displaystyle\\!\\!\\!\\!-(Q-sQ^{\prime})(n\Delta+1+sQ)-(b^{2}-s^{2})(1+sQ)Q^{\prime\prime},$
(3.4) $\displaystyle\Psi\\!\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!\\!\frac{\phi^{\prime\prime}}{2\big{[}(\phi-s\phi^{\prime})+(b^{2}-s^{2})\phi^{\prime\prime}\big{]}},$

where $b:=\|\beta\|_{\alpha}$. Let $G^{i}=G^{i}(x,y)$ and
$\bar{G}^{i}_{\alpha}=\bar{G}^{i}_{\alpha}(x,y)$ denote the coefficients of
$F$ and $\alpha$, respectively, in the same coordinate system. By definition,
we have

$\displaystyle G^{i}=G^{i}_{\alpha}+\alpha
Qs^{i}_{0}+\alpha^{-1}(r_{00}-2Q\alpha s_{0})(\Theta y^{i}+\alpha\Psi b^{i}).$
(3.5)

Clearly, if $\beta$ is parallel with respect to $\alpha$, that is
$r_{ij}=s_{ij}=0$, then $G^{i}={G}^{i}_{\alpha}=\gamma^{i}_{jk}(x)y^{j}y^{k}$
are quadratic in $y$. In this case, $F$ reduces to a Berwald metric.

For an $(\alpha,\beta)$-metric $F=\alpha\phi(s)$, $s=\beta/\alpha$, on an
$n$-dimensional manifold $M$, the $S$-curvature is given by

${\bf
S}=\Big{[}2\Psi-\frac{f^{\prime}(b)}{bf(b)}\Big{]}(r_{0}+s_{0})-\frac{\Phi}{2\alpha\Delta^{2}}(r_{00}-2\alpha
Qs_{0}),$ (3.6)

where

$\displaystyle f(b):=\frac{\int^{\pi}_{0}\sin^{n-2}t\ T(b\cos
t)dt}{\int^{\pi}_{0}\sin^{n-2}tdt},\ \ \ \
T(s):=\phi(\phi-s\phi^{\prime})^{n-2}\big{[}(\phi-s\phi^{\prime})+(b^{2}-s^{2})\phi^{\prime\prime}\big{]}.$

For more details, see [6].

To prove Theorem 1.1, we need the following key lemma.

######  Lemma 3.1.

([4][20]) Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be an
$(\alpha,\beta)$-metric on a manifold $M$. Then $F$ is a generalized Berwald
metric if and only if $\beta$ has constant length with respect to $\alpha$.

###### Proof.

In [20], Tayebi-Barzegari showed that every $(\alpha,\beta)$-metric
$F=\alpha\phi(s)$, $s=\beta/\alpha$, with sign property is a generalized
Berwald metric if and only if the dual vector field $\beta^{\sharp}$ is of
constant Riemannian length. In [9], Ivanov proved that a two-dimensional
Finsler metric is a generalized Berwald metric if and only if it is
monochromatic, i.e., Finsler metrics such that each two tangent spaces are
isomorphic as normed spaces. In [4], Bartelme${\ss}$-Matveev extended his
result for $n$-dimensional Finsler metric. It follows that an
$(\alpha,\beta)$-metric is a generalized Berwald metric if and only if dual
vector field $\beta^{\sharp}$ is of constant Riemannian length. ∎

In [6], Cheng-Shen characterized $(\alpha,\beta)$-metrics with isotropic
S-curvature on a manifold $M$ of dimension $n\geq 3$. Soon, they found that
their result holds for the class of $(\alpha,\beta)$-metrics with constant
length one-forms, only. Here, we give a characterization of the class of
generalized Berwald metrics with vanishing S-curvature.

######  Lemma 3.2.

Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be a generalized Berwald
$(\alpha,\beta)$-metric on an $n$-dimensional manifold $M$. Then ${\bf S}=0$
if and only if $F$ is Riemannian or $\beta$ is a Killing form with constant
length.

###### Proof.

Let $b:=\|\beta\|_{\alpha}=\sqrt{a^{mj}b_{j}b_{m}}=\sqrt{b_{m}b^{m}}$. Then,
the following holds

$\frac{\partial b}{\partial
x^{i}}=\frac{1}{b}b^{m}b_{m|i}=\frac{1}{b}(r_{i}+s_{i}).$ (3.7)

By Lemma 3.1, we have $b=constant$. Then, by (3.7) we obtain $r_{i}+s_{i}=0$.
In this case, (3.6) reduces to following

${\bf S}=-\frac{\Phi}{2\alpha\Delta^{2}}(r_{00}-2\alpha Qs_{0}),$ (3.8)

By (3.8), ${\bf S}=0$ if and only if $\Phi=0$ or $\beta$ satisfies

$r_{00}=2\alpha Qs_{0}.$ (3.9)

In the case of $\Phi=0$, $F$ reduces to a Riemannian metric (see Proposition
2.2 in [11]). Now, let (3.9) holds. We are going to simplify (3.9). For this
aim, one can change the $y$-coordinates $(y^{i})$, $i=1,\cdots,n$, at a point
to the polar coordinates $(s,u^{A})$, where $A=2,\cdots,n$ (see [6]). For an
arbitrary and fix point $x\in M$, let us take an orthonormal basis $e_{i}$ at
$x$ such that the Riemannian metric is written as
$\alpha=\sqrt{\sum_{i=1}^{n}(y^{i})^{2}}$ and its related one-form is given by
$\beta=by^{1}$, where $b:=||\beta||_{\alpha}$. Let us fix an arbitrary number
$s$ such that $|s|<b$. Define

${\bar{\alpha}}=\sqrt{\sum_{A=2}^{n}(y^{A})^{2}}.$

Then, by $\beta=s\alpha$ we get

$y^{1}=\frac{s}{\sqrt{b^{2}-s^{2}}}{\bar{\alpha}},\ \ \ \ y^{A}=u^{A}.$ (3.10)

Also, we have

$\alpha=\frac{b}{\sqrt{b^{2}-s^{2}}}\bar{\alpha},\ \ \ \ \
\beta=\frac{bs}{\sqrt{b^{2}-s^{2}}}\bar{\alpha}.$ (3.11)

Let us put

$\displaystyle\bar{r}_{10}:=\sum_{A=2}^{n}r_{1A}y^{A},\ \
\bar{s}_{10}:=\sum_{A=2}^{n}s_{1A}y^{A},\ \
\bar{r}_{00}:=\sum_{A,B=2}^{n}r_{AB}y^{A}y^{B},\ \
\bar{r}_{0}:=\sum_{A=2}^{n}r_{A}y^{A},\ \
\bar{s}_{0}:=\sum_{A=2}^{n}s_{A}y^{A}.$

Then we get the following useful relations

$\displaystyle r_{1}=br_{11},\ \ \ \ r_{A}=br_{1A},\ \ \ s_{1}=0,\ \ \ \
s_{A}=bs_{1A},$ (3.12) $\displaystyle
r_{00}=\frac{s^{2}}{b^{2}-s^{2}}\bar{\alpha}^{2}r_{11}+\frac{2s}{\sqrt{b^{2}-s^{2}}}\bar{\alpha}\bar{r}_{10}+\bar{r}_{00},$
(3.13) $\displaystyle
r_{10}=\frac{s}{\sqrt{b^{2}-s^{2}}}\bar{\alpha}r_{11}+\bar{r}_{10},\ \ \ \ \ \
\ s_{0}=\bar{s}_{0}=b\bar{s}_{10}.$ (3.14)

Using (3.11)-(3.14), the equation (3.9) can be written as follows

$\displaystyle\bar{r}_{00}+\frac{s^{2}}{b^{2}-s^{2}}\bar{\alpha}^{2}\
r_{11}=\frac{2}{\sqrt{b^{2}-s^{2}}}\left[b^{2}Q\bar{s}_{10}-s\bar{r}_{10}\right]\bar{\alpha}.$
(3.15)

By (3.15), we get two following relations

$\displaystyle\bar{r}_{00}+\frac{s^{2}}{b^{2}-s^{2}}\bar{\alpha}^{2}r_{11}=0,$
(3.16) $\displaystyle b^{2}Q\bar{s}_{10}-s\bar{r}_{10}=0.$ (3.17)

On the other hand, (3.11) implies that

$\displaystyle\frac{s^{2}}{b^{2}-s^{2}}\bar{\alpha}^{2}-\frac{1}{b^{2}}\beta^{2}=0.$
(3.18)

By (3.16) and (3.18), we get

$\displaystyle b^{2}\bar{r}_{00}+\beta^{2}r_{11}=0.$ (3.19)

The following hold

$\frac{\partial\bar{r}_{00}}{\partial y^{1}}=0,\ \ \ \ \ \
\frac{\partial\beta}{\partial y^{1}}=b.$

Then by differentiating (3.19) with respect to $y^{1}$ we have
${\beta}/{b}r_{11}=0$. Thus $r_{11}=0$ and by putting it in (3.19), we get
$\bar{r}_{00}=0$. Putting these relations in (3.13) and (3.14) imply that

$\displaystyle
r_{00}=\frac{2s\bar{\alpha}}{\sqrt{b^{2}-s^{2}}}\bar{r}_{10}=\frac{2\beta}{b}\bar{r}_{10},$
(3.20) $\displaystyle r_{10}=\bar{r}_{10}.$ (3.21)

Now, by considering (3.20) and (3.21), we divide the problem into two cases:
(a) $\bar{r}_{10}=0$ and (b) $\bar{r}_{10}\neq 0$.

Case (a): $\bar{r}_{10}=0$. In this case, by (3.20) we get $r_{ij}=0$. Putting
it in (3.9) implies that $s_{i}=0$. In this case, $\beta$ reduces to a Killing
one-form of constant length with respect to $\alpha$.

Case (b): $\bar{r}_{10}\neq 0$. We have ${\partial\bar{r}_{10}}/{\partial
y^{1}}=0$ and ${\partial\bar{s}_{10}}/{\partial y^{1}}=0$. Thus,
differentiating (3.17) with respect to $y^{1}$ yields

$\displaystyle(s)_{y^{1}}\bar{r}_{10}=b^{2}(Q)_{y^{1}}\bar{s}_{10}.$ (3.22)

Contracting (3.17) with $(s)_{y^{1}}$ give us

$\displaystyle s(s)_{y^{1}}\bar{r}_{10}=b^{2}Q(s)_{y^{1}}\bar{s}_{10}.$ (3.23)

By (3.22) and (3.23), we get

$\displaystyle\Big{[}(s)_{y^{1}}Q-s(Q)_{y^{1}}\Big{]}\bar{s}_{10}=0.$ (3.24)

According to (3.24), we get $\bar{s}_{10}=0$ or $Q(s)_{y^{1}}=s(Q)_{y^{1}}$.
Let $\bar{s}_{10}=0$ holds. Then (3.17) reduces to $s=0$, which is impossible.
Then, we have

$Q(s)_{y^{1}}=s(Q)_{y^{1}},$

which is equal to

$\displaystyle\frac{(Q)_{y^{1}}}{Q}=\frac{(s)_{y^{1}}}{s}.$ (3.25)

Using $s_{y^{1}}\neq 0$ and $(Q)_{y^{1}}=s_{y^{1}}(Q)_{s}$, then (3.25) give
us

$\frac{(Q)_{s}}{Q}=\frac{1}{s}$

which yields

$\ln(Q)-\ln(s)=c,$

where $c$ is a real constant. Thus $Q=ks$, where $k$ is a non-zero real
constant. In this case, we get $\phi=\sqrt{1+ks^{2}}$ which shows that $F$ is
a Riemannian metric. This completes the proof. ∎

Here, we solve an ODE which will appear in the proof of Theorem 1.1.

######  Lemma 3.3.

Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be an $(\alpha,\beta)$-metric on a
manifold $M$. Suppose that $\phi$ satisfies following

$\displaystyle\alpha\Theta_{2}+2\Lambda_{1}\Theta_{1}+\Lambda_{2}Q=0,$ (3.26)

where

$\displaystyle\Lambda_{1}:=b^{i}\alpha_{y^{i}},\ \ \
\Lambda_{2}:=b^{i}b^{j}\alpha_{y^{i}y^{j}},\ \ \ \
\Theta_{1}:=b^{i}Q_{y^{i}},\ \ \Theta_{2}:=b^{i}b^{j}Q_{y^{i}y^{j}}.$

Then $F$ is a singular Finsler metric given by

$\displaystyle\phi=c\exp\Bigg{[}\int_{0}^{s}{\frac{k_{1}t+k_{2}\sqrt{b^{2}-t^{2}}}{1+t\big{(}k_{1}t+k_{2}\sqrt{b^{2}-t^{2}}\big{)}}dt}\Bigg{]},$
(3.27)

where $k_{1}$ and $k_{2}$ are real constants, and $c>0$ is a non-zero
constant.

###### Proof.

Let us put

$A_{jk}:=\alpha^{2}a_{jk}-y_{j}y_{k}.$

Then, the followings hold

$\displaystyle\alpha_{y^{i}}=\frac{1}{\alpha}y_{i},$
$\displaystyle\alpha_{y^{j}y^{k}}=\frac{1}{\alpha^{3}}A_{jk},$
$\displaystyle\alpha_{y^{j}y^{k}y^{l}}=-\frac{1}{\alpha^{5}}\Big{[}A_{jk}y_{l}+A_{jl}y_{k}+A_{lk}y_{j}\Big{]}.$

Also, one can obtain the following

$\displaystyle\Lambda_{1}=s,$ (3.28)
$\displaystyle\Lambda_{2}=\frac{1}{\alpha}(b^{2}-s^{2}),$ (3.29)
$\displaystyle\Theta_{1}=\frac{1}{\alpha}(b^{2}-s^{2})Q^{\prime},$ (3.30)
$\displaystyle\Theta_{2}=\frac{1}{\alpha^{2}}(b^{2}-s^{2})\big{[}(b^{2}-s^{2})Q^{\prime\prime}-3sQ^{\prime}\big{]}.$
(3.31)

Suppose that (3.26) holds. Putting (3.28)-(3.31) into (3.26) yield

$\displaystyle
Q^{\prime\prime}-\frac{s}{b^{2}-s^{2}}Q^{\prime}+\frac{1}{b^{2}-s^{2}}Q=0,$

which implies that

$Q=k_{1}s+k_{2}\sqrt{b^{2}-s^{2}},$ (3.32)

where $k_{1}$ and $k_{2}$ are real constants. Considering (3.3), the equation
(3.32) is equal to following

$\displaystyle\Big{[}1+k_{1}s^{2}+k_{2}s\sqrt{b^{2}-s^{2}}\Big{]}\phi^{\prime}=\Big{[}k_{1}s+k_{2}\sqrt{b^{2}-s^{2}}\Big{]}\phi.$
(3.33)

By (3.33), we get (3.27). It is an almost regular $(\alpha,\beta)$-metric,
namely, it is singular in two directions ${\bf y}=(\pm b,0,0)\in T_{x}M$ at
any point $x$ (for more details, see [12]). ∎

The function $\phi$ in (3.27) is specifically has been seen for the first time
in Asanov’s paper [3] as follows

$\displaystyle\phi=c\exp\Bigg{[}\int_{0}^{s}{\frac{k\sqrt{b^{2}-t^{2}}}{1+tk\sqrt{b^{2}-t^{2}}}dt}\Bigg{]}.$
(3.34)

Then, its more general form (3.27) found by Shen in [12], where he looked for
unicorn metrics, namely the Landsberg metrics which are not Berwaldian. Shen
realized that the function $\phi=\phi(b,s)$ in (3.27) can be expressed in
terms of elementary functions. See (7.2) in [12].

Here, we show that the Douglas curvature of Finsler surfaces satisfies a
special relation. More precisely, we prove the following.

######  Lemma 3.4.

The Douglas curvature of any Finsler surface $(M,F)$ satisfies

$D^{i}_{\ jkl|s}y^{s}=\mathfrak{D}_{jkl}y^{i}$ (3.35)

for some tensor $\mathfrak{D}=\mathfrak{D}_{ijk}dx^{i}\otimes dx^{j}\otimes
dx^{k}$ which are homogeneous of degree -1 in $y$.

###### Proof.

By definition, the Douglas curvature of a two-dimensional Finsler metric is
given by

$D^{i}_{\ jkl}=B^{i}_{\ jkl}-\frac{2}{3}\Big{\\{}E_{jk}\delta^{i}_{\
l}+E_{kl}\delta^{i}_{\ j}+E_{lj}\delta^{i}_{\ k}+E_{jk,l}y^{i}\Big{\\}}.$
(3.36)

Taking a horizontal derivation of (3.36) along Finslerian geodesics and using
$y^{i}_{\ |s}=0$ give us the following

$D^{i}_{\ jkl|m}y^{m}=B^{i}_{\
jkl|m}y^{m}-\frac{2}{3}\Big{\\{}H_{jk}\delta^{i}_{\ l}+H_{kl}\delta^{i}_{\
j}+H_{lj}\delta^{i}_{\ k}+E_{jk,l|m}y^{m}y^{i}\Big{\\}}.$ (3.37)

Every Finsler surface is of scalar flag curvature ${\bf K}={\bf K}(x,y)$.
Then, by (11.24) in [13] we have

$\displaystyle B^{i}_{\
jml|k}y^{k}=2\textbf{K}C_{jlm}y^{i}-\frac{1}{3}\Big{\\{}y_{l}\delta^{i}_{m}+y_{m}\delta^{i}_{l}-2g_{lm}y^{i}\Big{\\}}\textbf{K}_{j}-\frac{1}{3}\Big{\\{}y_{j}\delta^{i}_{m}+y_{m}\delta^{i}_{j}-2g_{jm}y^{i}\Big{\\}}\textbf{K}_{l}$
$\displaystyle-\frac{1}{3}\Big{\\{}y_{j}\delta^{i}_{l}+y_{l}\delta^{i}_{j}-2g_{jl}y^{i}\Big{\\}}\textbf{K}_{m}-\frac{1}{3}F^{2}\Big{\\{}\textbf{K}_{jm}h^{i}_{l}+\textbf{K}_{jl}h^{i}_{m}+\textbf{K}_{lm}h^{i}_{j}\Big{\\}},$
(3.38)

where $y_{i}=FF_{y^{i}}$, $\textbf{K}_{j}:=\textbf{K}_{y^{j}}$ and
$\textbf{K}_{jk}:=\textbf{K}_{y^{j}y^{k}}$. Taking a trace of (3.38) yields

$H_{jl}=-\frac{1}{2}\Big{\\{}y_{l}\textbf{K}_{j}+y_{j}\textbf{K}_{l}+F^{2}\textbf{K}_{jl}\Big{\\}}.$
(3.39)

By putting (3.38) and (3.39) in (3.37) we get

$\displaystyle D^{i}_{\ jkl|m}y^{m}=\\!\\!\\!$
$\displaystyle\\!\\!\\!\frac{1}{3}\Big{\\{}6\textbf{K}C_{jkl}+2\big{(}g_{kl}\textbf{K}_{j}+g_{kj}\textbf{K}_{l}+g_{jl}\textbf{K}_{k}\big{)}+\big{(}y_{j}\textbf{K}_{kl}+y_{k}\textbf{K}_{jl}+y_{l}\textbf{K}_{kj}\big{)}$
(3.40)
$\displaystyle\\!\\!\\!\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-2E_{jk,l|m}y^{m}\Big{\\}}y^{i}.$

(3.40) give us (3.35). ∎

Now, we show that the covariant derivative of Berwald curvature of
2-dimensional generalized Berwald $(\alpha,\beta)$-metric with vanishing
S-curvature satisfies an interesting relation which will play an important
role in the proof of Theorem 1.1.

######  Lemma 3.5.

The Berwald curvature of any non-Riemannian 2-dimensional generalized Berwald
$(\alpha,\beta)$-metric with vanishing S-curvature satisfies following

$\displaystyle h^{m}_{p}B^{p}_{\ jkl|s}y^{s}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\ s^{m}_{\
0|0}(\alpha_{jkl}Q+\alpha_{jk}Q_{l}+\alpha_{lk}Q_{j}+\alpha_{lj}Q_{k}+\alpha
Q_{jkl}+\alpha_{l}Q_{jk}+\alpha_{j}Q_{lk}+\alpha_{k}Q_{jl})$ (3.41)
$\displaystyle\\!\\!\ +s^{m}_{\
0}(\alpha_{jkl}Q_{|0}+\alpha_{jk}Q_{l|0}+\alpha_{lk}Q_{j|0}+\alpha_{lj}Q_{k|0}+\alpha
Q_{jkl|0}+\alpha_{l}Q_{jk|0}+\alpha_{j}Q_{lk|0}$ $\displaystyle\\!\\!\
+\alpha_{k}Q_{jl|0})+A^{m}_{\ l}X_{jk}+A^{m}_{\ j}X_{lk}+A^{m}_{\
k}X_{jl}+B^{m}_{\ l}Y_{jk}+B^{m}_{\ j}Y_{lk}+B^{m}_{\ k}Y_{jl}=0,\ \ \ \ \ \ \
$

where

$\displaystyle
X_{jk}:=Q_{|0}\alpha_{jk}+Q_{k|0}\alpha_{j}+Q_{j|0}\alpha_{k}+Q_{jk|0}\alpha,$
$\displaystyle Y_{jk}:=Q\alpha_{jk}+Q_{k}\alpha_{j}+Q_{j}\alpha_{k}+\alpha
Q_{jk},$ $\displaystyle A^{m}_{\ l}:=s^{m}_{\ l}-F^{-2}s^{0}_{\ l}y^{m},$
$\displaystyle B^{m}_{\ l}:=A^{m}_{\ l|s}y^{s}=s^{m}_{\ l|0}-F^{-2}s^{0}_{\
l|0}y^{m}.$

###### Proof.

By Lemma 3.2, we have $r_{ij}=0$ and $s_{j}=0$. In this case, (3.5) reduces to
following

$\displaystyle G^{i}=G^{i}_{\alpha}+\alpha Qs^{i}_{\ 0}.$ (3.42)

Taking three vertical derivations of (3.42) with respect to $y^{j}$, $y^{l}$
and $y^{k}$ give us

$\displaystyle B^{i}_{\ jkl}=\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\\!s^{i}_{\
0}\big{\\{}\alpha
Q_{jkl}+\alpha_{l}Q_{jk}+\alpha_{j}Q_{lk}+\alpha_{k}Q_{jl}+\alpha_{jkl}Q+\alpha_{jk}Q_{l}+\alpha_{lk}Q_{j}+\alpha_{lj}Q_{k}\big{\\}}$
(3.43) $\displaystyle+$ $\displaystyle\\!\\!\\!\\!s^{i}_{\
j}\big{\\{}Q\alpha_{lk}+Q_{k}\alpha_{l}+Q_{l}\alpha_{k}+\alpha
Q_{lk}\big{\\}}+s^{i}_{\
k}\big{\\{}Q\alpha_{jl}+Q_{j}\alpha_{l}+Q_{l}\alpha_{j}+\alpha
Q_{jl}\big{\\}}$
$\displaystyle\\!\\!\\!\\!\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
+s^{i}_{\ l}\big{\\{}Q\alpha_{jk}+Q_{k}\alpha_{j}+Q_{j}\alpha_{k}+\alpha
Q_{jk}\big{\\}}.$

Contracting (3.43) with $h^{m}_{i}$ implies that

$\displaystyle h^{m}_{i}B^{i}_{\
jkl}=\big{\\{}\alpha_{jkl}Q+\alpha_{jk}Q_{l}\\!\\!\\!\\!$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\alpha_{lk}Q_{j}+\alpha_{lj}Q_{k}+\alpha
Q_{jkl}+\alpha_{l}Q_{jk}+\alpha_{j}Q_{lk}+\alpha_{k}Q_{jl}\big{\\}}s^{m}_{\
0}$ (3.44) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\big{\\{}Q\alpha_{jk}+Q_{k}\alpha_{j}+Q_{j}\alpha_{k}+\alpha
Q_{jk}\big{\\}}(s^{m}_{\ l}-F^{-2}s^{0}_{\ l}y^{m})$ $\displaystyle+$
$\displaystyle\\!\\!\\!\big{\\{}Q\alpha_{lk}+Q_{k}\alpha_{l}+Q_{l}\alpha_{k}+\alpha
Q_{lk}\big{\\}}(s^{m}_{\ j}-F^{-2}s^{0}_{\ j}y^{m})$ $\displaystyle+$
$\displaystyle\\!\\!\\!\big{\\{}Q\alpha_{jl}+Q_{j}\alpha_{l}+Q_{l}\alpha_{j}+\alpha
Q_{jl}\big{\\}}(s^{m}_{\ k}-F^{-2}s^{0}_{\ k}y^{m}).$

On the other hand, by taking a horizontal derivation of Douglas curvature
along Finslerian geodesics and contracting the result with $h^{m}_{i}$, we get
the following

$h^{m}_{i}D^{i}_{\ jkl|s}y^{s}=h^{m}_{i}B^{i}_{\
jkl|s}y^{s}-\frac{2}{3}\Big{\\{}H_{jk}h^{m}_{l}+H_{kl}h^{m}_{j}+H_{lj}h^{m}_{k}\Big{\\}}.$
(3.45)

Contracting (3.35) with $h^{m}_{i}$ yields

$h^{m}_{i}D^{i}_{\ jkl|s}y^{s}=0.$ (3.46)

Since ${\bf S}=0$, then by definition we get ${\bf H}=0$. Thus, (3.45) and
(3.46) imply that

$h^{m}_{i}B^{i}_{\ jkl|s}y^{s}=0.$ (3.47)

We have $h^{m}_{\ i|s}=0$. Then, by considering (3.47), we have

$\big{(}h^{m}_{i}B^{i}_{\ jkl}\big{)}_{|s}y^{s}=h^{m}_{i}B^{i}_{\
jkl|s}y^{s}=0.$ (3.48)

Therefore, taking a horizontal derivation of (3.44) along Finslerian geodesic
and considering (3.48) give us (3.41). ∎

Proof of Theorem 1.1: Taking a horizontal derivation of $y_{m}s^{m}_{\ 0}=0$
with respect to the Berwald connection of $F$ implies that

$y_{m|0}s^{m}_{\ 0}+y_{m}s^{m}_{\ 0|0}=0.$ (3.49)

Since $y_{m|0}=0$, then (3.49) reduces to following

$y_{m}s^{m}_{\ 0|0}=0.$ (3.50)

By contracting (3.41) with $y_{m}$ and considering (3.50), we get

$\displaystyle s^{0}_{\ l}X_{jk}+s^{0}_{\ j}X_{lk}+s^{0}_{\ k}X_{jl}+s^{0}_{\
l|0}Y_{jk}+s^{0}_{\ j|0}Y_{lk}+s^{0}_{\ k|0}Y_{jl}=0.$ (3.51)

Since $s_{j}=0$, then one can get

$0=(s_{j})_{|0}=(r_{m0}+s_{m0})s^{m}_{\ j}+b_{m}s^{m}_{\ j|0}$

which considering $r_{ij}=0$, it reduces to following

$\displaystyle b_{m}s^{m}_{\ j|0}=-s_{m0}s^{m}_{\ j}.$ (3.52)

Multiplying (3.52) with $y^{j}$ yields

$b_{i}s^{i}_{\ 0|0}+s_{i0}s^{i}_{\ 0}=0.$ (3.53)

Also, contracting (3.52) with $b^{j}$ implies that

$b_{i}s^{i}_{\ j|0}b^{j}=0.$ (3.54)

Multiplying (3.51) with $b^{j}b^{k}b^{l}$ and considering (3.53) and (3.54)
give us

$(\alpha\Theta_{2}+2\Lambda_{1}\Theta_{1}+\Lambda_{2}Q)s_{m0}s^{m}_{\ 0}=0.$
(3.55)

By (3.55), we have two cases: if $s^{m}_{\ 0}s_{m0}=0$, since $\alpha$ is a
positive-definite metric, then we find that $\beta$ is closed. Therefore, by
(3.43) we conclude that $F$ reduces to a Berwald metric. By Szabo’s rigidity
result for Finsler surfaces, $F$ reduces to a locally Minkowskian metric or a
Riemannian metric. On the other hand, every Finsler surface has scalar flag
curvature ${\bf K}={\bf K}(x,y)$. According to Akbar-Zadeh theorem in [1], a
Finsler manifold $(M,F)$ of scalar flag curvature ${\bf K}={\bf K}(x,y)$ has
isotropic flag curvature ${\bf K}={\bf K}(x)$ if and only if it has vanishing
H-curvature ${\bf H}=0$. Thus, the obtained Riemannian metric has isotropic
sectional curvature.

Now, suppose that $F$ is not a Riemannian metric nor a locally Minkowskian
metric. Then, by (3.55) we have
$\alpha\Theta_{2}+2\Lambda_{1}\Theta_{1}+\Lambda_{2}Q=0$. By Lemma 3.3 we
obtain (1.1). In this case, since ${\bf S}=0$, then $F$ can not be a Douglas
metric. On the other hand, the Berwald curvature of 2-dimensional Finsler
manifold is given by

$B^{i}_{\
jkl}=-\frac{2}{F^{2}}L_{jkl}y^{i}+\frac{2}{3}\Big{\\{}E_{jk}h^{i}_{l}+E_{kl}h^{i}_{j}+E_{jl}h^{i}_{k}\Big{\\}}.$
(3.56)

See the relation (15) in [21]. If $F$ is a Landsberg metric then by
considering ${\bf S}=0$, (3.56) implies that $F$ is a Berwald metric. This is
a contradiction. Then, (1.1) is a generalized Berwald metric which is not
Berwald, Landsberg nor Douglas metric. ∎

Proof of Corollary 1.1: In [23], Xu-Deng proved that every homogeneous Finsler
metric of isotropic S-curvature has vanishing S-curvature. Then, by assumption
we get ${\bf S}=0$. The Akbar-Zadeh theorem in [1] stated that a Finsler
manifold $(M,F)$ of scalar flag curvature ${\bf K}={\bf K}(x,y)$ has isotropic
flag curvature ${\bf K}={\bf K}(x)$ if and only if it has vanishing
H-curvature ${\bf H}=0$. On the other hand, every Finsler surface has scalar
flag curvature ${\bf K}={\bf K}(x,y)$. Thus, by Akbar-Zadeh theorem we get
${\bf K}={\bf K}(x)$. Every scalar function on $M$ which is invariant under
isometries of $(M,F)$ is a constant function. The homogeneity of $(M,F)$ and
invariancy of the flag curvature under isometries of $F$ imply that ${\bf
K}=constant$. Then, by Theorem 1.1 we get the proof. ∎

Proof of Corollary 1.2: Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be a two-
dimensional normal homogeneous generalized Berwald $(\alpha,\beta)$-metric. In
[24], Xu-Deng proved that every normal homogeneous manifold has vanishing
S-curvature and non-negative flag curvature. By Theorem 1.1 and the same
method used in the proof of Corollary 1.1, it follows that $F$ is a Riemannian
metric of non-negative constant sectional curvature or a locally Minkowskian
metric. ∎

Proof of Corollary 1.3: Let $(G/H,F)$ be a generalized normal homogeneous
Randers manifold. In [30], Zhang-Deng proved that $F$ has vanishing
S-curvature (Corollary 3.11). Also, they showed that any generalized normal
homogeneous Randers metric has non-negative flag curvature (see Proposition
3.13 in [30]). Then, by Theorem 1.1 we get the proof. ∎

According to Theorem 1.1, every two-dimensional generalized Berwald
$(\alpha,\beta)$-metric with vanishing S-curvature is Riemannian or locally
Minkowskian. Every left invariant Finsler metric is a generalized Berwald
metric [2]. Here, we prove Theorem 1.2 which states that left invariant
Finsler metrics with vanishing S-curvature reduce to Riemannian metrics, only.
The approach of the proof of Theorem 1.2 is completely different from Theorem
1.1.

Proof of Theorem 1.2: To prove a homogeneous surface with ${\bf S}=0$ is
Riemannian, we only need to consider the nontrivial case, namely, a
2-dimensional non-Abelian Lie group $G$ with a left invariant Finsler metric
$F$. At each $y\in\mathfrak{g}$ with $F(y)=1$, there is a ${\bf g}_{y}$
orthonormal basis $e_{1}=y$ and $e_{2}$ tangent to $F=1$. At almost all non-
zero $y$, the spray vector field $\eta$ is nonzero, i.e., $\eta(y)$ is a
nonzero multiple of $e_{2}$. By the homogenous S-curvature formula, we have

${\bf S}(y)=-{\bf I}(\eta)=-{\bf C}_{y}(\eta,e_{2},e_{2})=0.$

Then, ${\bf C}_{y}(e_{2},e_{2},e_{2})=0$ everywhere. The speciality of
2-dimensional spaces implies ${\bf C}=0$ everywhere, so $F$ is Riemannian. By
the same method used to prove Corollary 1.1, one can conclude that the
Riemannian metric is of constant sectional curvature. This completes the
proof. ∎

Proof of Corollary 1.4: By assumption, $F$ has isotropic Berwald curvature

$B^{i}_{\
jkl}=cF^{-1}\Big{\\{}h_{jk}h^{i}_{l}+h_{jl}h^{i}_{k}+h_{kl}h^{i}_{j}+2C_{jkl}y^{i}\Big{\\}}.$
(3.57)

where $c=c(x)$ is a scalar function on $M$. In [22], it is proved that every
Finsler surface of isotropic Berwald curvature (3.57) metric has isotropic
S-curvature ${\bf S}=3cF$. By Xu-Deng’s result in [24], $F$ has vanishing
S-curvature ${\bf S}=0$. Then, by Theorem 1.2, $F$ reduces to a Riemannian
metric. ∎

##  4 Some Examples of Generalized Berwald Manifolds

In this section, we are going to give some important examples of the class of
generalized Berwald manifolds. First, by using trans-Sasakian structure, we
construct a family of odd-dimensional generalized Berwald Randers metrics.

######  Example 3.

$\big{(}\textrm{Odd-dimensional generalized Berwald Randers metrics}\big{)}$
Let $M$ be a differentiable manifold of dimension $2n+1$. Suppose that
$\eta=\eta_{i}(x)dx^{i}$, $\xi=\xi^{i}\partial/\partial x^{i}$ and
$\varphi=\varphi^{i}_{j}\partial/\partial x^{i}\otimes dx^{j}$ are a $1$-form,
a vector field, and a $(1,1)$-tensor on $M$, respectively. The triple
$(\eta,\xi,\varphi)$ is called an almost contact structure on $M$ if it
satisfies

$\varphi(\xi)=0,\ \ \ \ \eta(\xi)=1,\ \ \ \varphi^{2}=-I+\eta\otimes\xi.$

A differentiable manifold of odd dimension $2n+1$ with an almost contact
structure is called an almost contact manifold. Let a manifold $M$ with the
$(\eta,\xi,\varphi)$ structure admits a Riemannian metric ${\bf g}$ such that

${\bf g}(\varphi X,\varphi Y)={\bf g}(X,Y)-\eta(X)\eta(Y).$

Then $M$ is called an almost contact metric structure and ${\bf g}$ is called
a compatible metric. In this case, $(\eta,\xi,\varphi,{\bf g})$ is called
almost contact metric structure. An almost contact metric structure
$(\eta,\xi,\varphi,{\bf g})$ on $M$ is called a trans-Sasakian structure if it
satisfies

$(\nabla_{X}\varphi)Y=c_{1}\big{\\{}{\bf
g}(X,Y)\xi-\eta(Y)X\big{\\}}+c_{2}\big{\\{}{\bf g}(\varphi
X,Y)\xi-\eta(Y)\varphi X\big{\\}}$

for some scalar functions $c_{1}=c_{1}(x)$ and $c_{2}=c_{2}(x)$ on $M$, where
$\nabla$ denotes the Levi-Civita connection of ${\bf g}$.

Now, let $(\eta,\xi,\varphi,{\bf g})$ be a trans-Sasakian structure on $M$.
Define $\alpha,\beta:TM\rightarrow[0,\infty)$ by

$\forall(x,y)\in TM,\ \ \ \
\alpha(x,y):=\sqrt{{\bf{g}}_{x}(y,y)},\,\,\,\,\,\,\,\beta(x,y):=\epsilon\,\eta_{x}(y),$
(4.1)

where $0<\epsilon<1$ be a constant. Then, for the Randers metric
$F:=\alpha+\beta$, we have

$||\beta||_{\alpha}=\epsilon.$

It follows that the class of Randers metrics induced by trans-Sasakian
manifolds $(M,\eta,\xi,\varphi,{\bf g})$ are $(2n+1)$-dimensional generalized
Berwald metrics on $M$.

Here, we give a two-dimensional Randers metric $F=\alpha+\beta$ with vanishing
S-curvature. We show that if $F$ is a generalized Berwald metric then it
reduces to a Riemannian metric.

######  Example 4.

Let ${\bf y}=u{{\partial}/{\partial}x}+v{{\partial}/{\partial}y}\in
T_{(x,y)}\mathbb{R}^{2}$. Consider the Randers metric $F=\alpha+\beta$, where
$\alpha=\alpha({\bf y})$ and $\beta=\beta({\bf y})$ are given by

$\displaystyle{\alpha}\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!{\sqrt{\Big{(}1+(1-{\epsilon}^{2})(x^{2}+y^{2})\Big{)}(u^{2}+v^{2})+\Big{(}1+{\epsilon}^{2}+x^{2}+y^{2}\Big{)}(xv-
yu)^{2}}\over\Big{(}1+(1-{\epsilon}^{2})(x^{2}+y^{2})\Big{)}\sqrt{1+x^{2}+y^{2}}},$
$\displaystyle{\beta}\\!\\!\\!$ $\displaystyle:=$
$\displaystyle\\!\\!\\!-{{\epsilon}(xv-yu)\over
1+(1-{\epsilon}^{2})(x^{2}+y^{2})},$

and ${\epsilon}$ is a real constant (see [15]). $F$ is defined on the whole
sphere for $|{\epsilon}|<1$. It is remarkable that, one can rewrite $F$ in a
polar coordinate system, $x=r\cos(\theta)$, $y=r\sin(\theta)$. Express

$\alpha=\sqrt{a_{11}\mu^{2}+a_{12}\mu\nu+a_{21}\nu\mu+a_{22}\nu^{2}},\ \ \
\beta=b_{1}\mu+b_{2}\nu,$

where

$\displaystyle
a_{11}={1\over(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}},\ \ \
a_{12}=a_{21}=0,\ \ \
a_{22}={r^{2}(1+r^{2})\over\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{2}},$
$\displaystyle b_{1}=0,\ \ \ \ \ \ \ \ b_{2}=-{{\epsilon}r^{2}\over
1+(1-{\epsilon}^{2})r^{2}}.$

Let us put $A:=det(a_{ij})$. Then, we get

$\displaystyle a^{11}={r^{2}(1+r^{2})\over
A\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{2}},\ \ \ a^{22}={1\over
A(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}},\ \ \ \ a^{12}=a^{21}=0,$
$\displaystyle b^{1}=0,\ \ \ \ b^{2}=-{{\epsilon}r^{2}\over
A(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{2}}.$

Therefore, we obtain

$\displaystyle\|\beta\|_{\alpha}^{2}=a^{ij}b_{i}b_{j}=b_{i}b^{i}={{\epsilon}^{2}r^{4}\over
A(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{3}}\neq constant.$ (4.2)

(4.2) means that $F$ is not a generalized Berwald metric. On the other hand, a
direct computation yields

$\displaystyle r_{11}=r_{22}=0,\ \ \ \ \ \
r_{12}=r_{21}={{\epsilon}^{3}r^{3}\over(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{2}},$
$\displaystyle s_{11}=s_{22}=0,\ \ \ \ \
s_{12}={{\epsilon}r\over\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{2}}=-s_{21}$
$\displaystyle
s_{1}={{\epsilon}^{2}r\over(1+r^{2})\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}},\
\ \ \ \ \ \ s_{2}=0,$ $\displaystyle r_{1}=-{{\epsilon}^{4}r^{5}\over
A(1+r^{2})^{2}\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{4}},\ \ r_{2}=0.$

It is easy to find that $r_{ij}+b_{i}s_{j}+b_{j}s_{i}=0$. Then by Lemma 3.2.1
in [5], we get ${\bf S}=0$. Also, one can see that the following holds

$r_{i}+s_{i}=\frac{{\epsilon}^{2}r\big{[}A(1+r^{2})(1+(1-{\epsilon}^{2})r^{2})^{3}-{\epsilon}^{2}r^{4}\big{]}}{A(1+r^{2})^{2}\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{4}}.$
(4.3)

According to (4.3), $F$ is a generalized Berwald metric (equivalently,
$r_{i}+s_{i}=0$) if and only if ${\epsilon}=0$ or the following holds

$\big{(}1+(1-{\epsilon}^{2})r^{2}\big{)}^{4}=0.$ (4.4)

(4.4) contradicts with $|{\epsilon}|<1$. Therefore, $F$ is a generalized
Berwald metric if and only if ${\epsilon}=0$ or equivalently $\beta=0$. In
this case, $F$ reduces to the standard Riemannian metric $F=\alpha$.

######  Example 5.

(Xu) It is proved that a Finsler metric $F=F(x,y)$ is of Randers type
$F=\alpha+\beta$ if and only if it is a solution of the navigation problem on
a Riemannian manifold $(M,{\bf h})$ (see [5]). Zermelo navigation is an
efficient method to study of Randers metrics with certain Riemannian and non-
Riemannian curvature properties. More precisely, any Randers metric
$F=\alpha+\beta$ on a manifold $M$ is a solution of the following Zermelo
navigation problem

$\textbf{h}\Big{(}x,\frac{y}{F}-\mathcal{W}_{x}\Big{)}=1,$

where $\textbf{h}=\sqrt{h_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and
$\mathcal{W}=\mathcal{W}^{i}(x){\partial}/{\partial x^{i}}$ is a vector field
such that

$\textbf{h}(x,-\mathcal{W}_{x})=\sqrt{h_{ij}(x)\mathcal{W}^{i}(x)\mathcal{W}^{j}(x)}<1.$

In fact, $\alpha$ and $\beta$ are given by

$\alpha=\frac{\sqrt{\lambda
h^{2}+\mathcal{W}_{0}}}{\lambda},\qquad\beta=-\frac{\mathcal{W}_{0}}{\lambda},$

respectively and moreover,

$\lambda:=1-\|\mathcal{W}\|^{2}_{h},\ \ \ \ \
W_{0}:=h_{ij}\mathcal{W}^{i}y^{j}.$

For more details, see [5]. Then, $F$ can be written as follows

$F=\frac{\sqrt{\lambda\textbf{h}^{2}+\mathcal{W}_{0}^{2}}}{\lambda}-\frac{\mathcal{W}_{0}}{\lambda}.$
(4.5)

In this case, the pair $({\bf h},\mathcal{W})$ is called the navigation data
of $F$.

Now, let $G/H$ be any homogeneous manifold and
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is its reductive decomposition.
Suppose $\mathfrak{m}=\mathfrak{m}_{0}+\mathfrak{m}_{1}$ be an
$\mathrm{Ad}(H)$-invariant decomposition, in which $\mathfrak{m}_{0}$ is
1-dimensional and the $\mathrm{Ad}(H)$-action on $\mathfrak{m}_{0}$ is
trivial. Let ${\bf h}$ be a $G$-invariant Riemannian metric on $G/H$, such
that $\mathfrak{m}_{0}$ and $\mathfrak{m}_{1}$ are $h$-orthogonal to each
other. Let $\mathcal{W}$ be a $G$-invairant vector field on $G/H$, such that
$\mathcal{W}(o)\in\mathfrak{m}_{0}\backslash\\{0\\}$. Then, the navigation
process with the data $({\bf h},\mathcal{W})$ provides a $G$-invariant
generalized Berwald Randers metric with ${\bf S}=0$ (see [7]).

######  Example 6.

(Xu) As we mentioned in Introduction, the Bao-Shen’s Randers metrics on
$\mathbb{S}^{3}$ are concrete generalized Berwald metrics, namely they are not
Berwaldian. Any non-Riemannian homogeneous Randers sphere
$\mathbb{S}^{3}=SU(3)/SU(2)$ (including Bao-Shen’s Randers metrics) satisfies
${\bf S}=0$ with constant pointwise $\|\beta\|_{\alpha}$-norms. Then, every
non-Riemannian homogeneous Randers sphere is a generalized Berwald metric. An
$\mathbb{S}^{3}\times\mathbb{S}^{1}$, in which $\mathbb{S}^{3}=SU(3)/SU(2)$
and the navigation field is tangent to the $\mathbb{S}^{3}$-factor, is a
4-dimensional generalized Berwald Randers metric (see [29]).

Acknowledgments: The authors are so grateful to Ming Xu for his valuable
comments on this manuscript. Likewise, we thank him for providing us with
examples 5 and 6 which improve the quality of our manuscript. Also, we are
thankful to Behzad Najafi, Mansoor Barzegari and Libing Huang for their
reading of this manuscript and their comments.

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Akbar Tayebi and Faezeh Eslami
Department of Mathematics, Faculty of Science
University of Qom
Qom. Iran
Email<EMAIL_ADDRESS>
Email<EMAIL_ADDRESS>