# On Sprays of Scalar Curvature and Metrizability

Guojun Yang

###### Abstract

Every Finsler metric naturally induces a spray but not so for the converse.
The notion for sprays of scalar (resp. isotropic) curvature has been known as
a generalization for Finsler metrics of scalar (resp. isotropic) flag
curvature. In this paper, a new notion, sprays of constant curvature, is
introduced and especially it shows that a spray of isotropic curvature is not
necessarily of constant curvature even in dimension $n\geq 3$. Further,
complete conditions are given for sprays of isotropic (resp. constant)
curvature to be Finsler-metrizabile. As applications of such a result, the
local structure is determined for locally projectively flat Berwald sprays of
isotropic (resp. constant) curvature which are Finsler-metrizable, and some
more sprays of isotropic curvature are discussed for their metrizability.
Besides, the metrizability problem is also investigated for sprays of scalar
curvature under certain curvature conditions.

Keywords: Finsler Metric, Spray, Berwald Spray, Metrizability,
Scalar/Isotropic/Constant Curvature, Projective Flatness

2010 Mathematics Subject Classification:  53C60, 53B40

##  1 Introduction

Spray geometry studies the properties of spray manifolds, a kind of path
spaces. Spray geometry is more general than Finsler geometry, because every
Finsler metric induces a natural spray but there are a lot of sprays which
cannot be induced by any Finsler metric ([6, 10, 19]). A spray ${\bf G}$ on a
manifold $M$ is a family of compatible second order ODEs which define a
special vector filed on a conical region $\mathcal{C}$ of $TM\setminus\\{0\\}$
(an important case is $\mathcal{C}=TM\setminus\\{0\\}$). The integral curves
of ${\bf G}$ projected onto $M$ are called geodesics of G. Many basic
curvatures, such as Riemann curvature, Ricci curvature, Weyl curvature,
Berwald curvature and Douglas curvature, appearing in Finsler geometry, are
actually defined in spray geometry via the spray coefficients. Geodesics and
these basic curvatures play an important role in the study of spray geometry.

The metrizability problem for a spray ${\bf G}$ seeks for a Finsler metric
whose spray is just G, or whose geodesics coincide with that of ${\bf G}$. A
weaker problem is to consider the projective metrizability of a given spray G,
which aims to look for a Finsler metric projectively related to G. So a
natural question is to determine whether a given spray is (projectively)
Finsler-metrizable or not under certain curvature conditions.

In [13], M. Matsumoto proves that any two-dimensional spray is locally
projectively Finsler-metrizable. More generally, any spray of scalar curvature
is locally projectively Finsler-metrizable ([4, 5]). In [14], Z. Muzsnay gives
some sprays which are not Finsler-metrizable under some conditions satisfied
by the holonomy distribution generated from the horizontal vector fields of a
spray. In [19], the present author constructs a class of sprays whose
metrizable and non-metrizable conditions are completely determined
respectively. Inspired by the sprays constructed in [19], S.G. Elgendi and Z.
Muzsnay discuss a more general class of sprays and prove the non-metrizability
of such sprays by computing the dimension of the holonomy distribution under
certain conditions ([6]). In [12], B. Li and Z. Shen introduce the notion of
sprays with isotropic curvature and give some non-metrizability conditions for
locally projectively flat sprays, and shows that a locally projectively flat
spray with vanishing Riemann curvature is metrizable (cf. [17]). In [10], Y.
Li, X. Mo and Y. Yu give a class of locally projectively flat Berwald sprays
which are non-metrizable.

In this paper, we are going to study some special properties and the
metrizability problem of some special classes of sprays: locally projectively
flat Berwald sprays, sprays of scalar curvature (resp. isotropic curvature,
constant curvature). A locally projectively flat spray, which is always of
scalar curvature, means that its geodesics are locally straight lines. A
Berwald spray means that its spray coefficients $G^{i}=G^{i}(x,y)$ are
quadratic in $y$. A spray ${\bf G}$ is said to be of scalar curvature if its
Riemann curvature $R^{i}_{\ k}$ satisfies

$R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i},$ (1)

where $R=R(x,y)$ and $\tau_{k}=\tau_{k}(x,y)$ are some homogeneous functions
([18]). If in (1) there holds $R_{.i}=2\tau_{i}$, then ${\bf G}$ is said to be
of isotropic curvature ([12]). A spray ${\bf G}$ is said to be of constant
curvature, a new notion we introduce in this paper, if ${\bf G}$ satisfies (1)
with

$\tau_{i;k}=0\ (\ \Leftrightarrow\ R=\tau_{k}=0,\ or\ R_{;i}=0).$

In the above, we use $T_{i;j}$ and $T_{i.j}$ to denote respectively the
horizontal and vertical covariant derivatives of the tensor $T$ with respect
to Berwald connection of a given spray. Some basic properties for a spray of
constant curvature are given in Theorem 5.2 below, in which, it especially
shows that an $n(\geq 3)$-dimensional spray of isotropic curvature is not
necessarily of constant curvature, which is different from the Finslerian
case.

Now we show the following Theorems 1.1 and 1.2 on the metrizability for sprays
of isotropic and constant curvature respectively.

###### Theorem 1.1

Let G be an $n$-dimensional spray of isotropic curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\frac{1}{2}R_{.k}$.

  1. (i)

If $R=0$, then G is locally Finsler-metrizable.

  2. (ii)

If $R\neq 0$ and $R$ is not a Finsler metric, then G is not Finsler-
metrizable.

  3. (iii)

If $R$ is a Finsler metric, then G is (locally) Finsler-metrizable if and only
if $R_{;i}=R\omega_{i}$ for some closed 1-form $\omega=\omega_{i}(x)dx^{i}$.
In this case, we have

    1. (iiia)

if $n\geq 3$, then $R_{;i}=0$ (or $\omega=0$).

    2. (iiib)

if $\omega=0$, then G is induced by the Finlser metric $R$ with the flag
curvature ${\bf K}=1$.

    3. (iiib)

if $\omega\neq 0$, then G is induced by the Finlser metric $R/\lambda$ with
the flag curvature ${\bf K}=\lambda$, where $\lambda\neq 0$ is given by
$\omega_{i}=(\ln|\lambda|)_{;i}$. This case happens only in dimension $n=2$.

###### Theorem 1.2

Let G be an $n$-dimensional spray of constant curvature and $Ric$ be the Ricci
curvature of G. Then G is (locally) Finsler-metrizable iff. $Ric=0$ or $Ric$
is a Finsler metric. In this case, the Finsler metric $L$ inducing G has
vanishing flag curvature or $L$ is given by $L=Ric/(n-1)$ with the flag
curvature $1$.

By Theorems 1.1 and 1.2, for a given spray G of isotropic or constant
curvature, it is easy to check directly whether G is (locally) Finsler-
metrizable or not. Meanwhile, if it is metrizable, the corresponding Finsler
metric is easily obtained if $Ric\neq 0$. The metric in Theorem 1.1 can be
multiplied by a suitable non-zero constant if we need the metric to be
positive. Theorem 1.1(i) shows that a spray with vanishing Riemann curvature
(not necessary to be locally projectively flat) is locally metrizable (cf.
[17, 12]).

As applications of Theorems 1.1 and 1.2, we generalize a result in [19]
(Theorem 4.10 below), and make a check on some spays whether they are Finsler-
metrizable or not (see Sec.7 below). On the other hand, we can use Theorem 1.1
to obtain the local structure for locally projectively flat Berwald sprays of
isotropic curvature when they are Finsler-metrizable.

###### Theorem 1.3

Let G be a projectively flat Berwald spray of isotropic curvature on an open
set $U\subset R^{n}$ with $Ric\neq 0$ (on a conical region $\mathcal{C}(U)$).
Then G is Finsler metrizable (on $\mathcal{C}(U)$) if and only if G can be
expressed as

$G^{i}=Py^{i},\ \ \ \ \ \ P:=-\frac{1}{2}\Big{[}\ln|x^{\prime}Ax+\langle
B,x\rangle+C|\Big{]}_{x^{k}}y^{k},$ (2)

where $A\neq 0$ is a constant symmetric matrix, $B$ is a constant vector and
$C$ is a constant number satisfying certain condition such that the following
function $L$ is a metric (defined on $\mathcal{C}(U)$),

$L:=\frac{4(x^{\prime}Ax+\langle
B,x\rangle+C)y^{\prime}Ay-(2x^{\prime}Ay+\langle
B,y\rangle)^{2}}{4(x^{\prime}Ax+\langle B,x\rangle+C)^{2}}.$ (3)

In this case, G is induced by the metric $L=Ric/(n-1)$ of constant sectional
curvature 1.

In [17], there is a general description of the construction for locally
projectively flat Finsler metrics with constant flag curvature 1. As a special
case, putting $2A=(\delta_{ij}),B=0,C=1/2$ in Theorem 1.3, we obtain

$L=\frac{(1+|x|^{2})|y|^{2}-\langle x,y\rangle^{2}}{(1+|x|^{2})^{2}}.$ (4)

Theorem 1.3 also implies that if we take in (2),
$P=-\big{[}\ln\sqrt{|f(x)|}\big{]}_{x^{k}}y^{k}$ for an arbitrary non-constant
function $f(x)$ which is not a polynomial of degree two, then the spray G in
(2) is not Finsler-metrizable. If G in Theorem 1.3 has zero Riemann curvature
($Ric=0$), then G can be locally induced by a Minkowski metric (a trivial
case, Remark 6.7 below).

For a spray of scalar curvature $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ on
a manifold $M$, the two quantities $R$ and $\tau_{k}$ are closely related (see
Proposition 3.1 below). Now we consider the following condition for $R$ and
$\tau_{k}$,

$R_{.i}-2\tau_{i}=\omega_{i0},\ \ \ (\omega_{i0}:=\omega_{ir}y^{r}),$ (5)

where $\omega=\omega_{ij}dx^{i}\wedge dx^{j}$ is a 2-form on the manifold $M$.
A spray satisfying (5) is considered to be some kind of weakly isotropic
curvature. For a spray of scalar curvature, the condition (5) is a special
case of $\chi_{i}=\omega_{i0}$ (see [11]), where $\chi_{i}$ is called the
$\chi$-curvature originally defined in [15]. For a spray satisfying (5), we
have the following theorem.

###### Theorem 1.4

Let G be an $n$-dimensional spray of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$.

  1. (i)

If G is induced by a Finsler metric $L$ and satisfies (5). Then

    1. (ia)

([8]) ($n\geq 3$) $L$ is of constant flag curvature with $\omega=0$.

    2. (ib)

($n=2$) For $L=F^{2}$, the flag curvature $\lambda$ of $L$ satisfies
$\lambda^{\prime\prime}(\theta)+\epsilon I(\theta)\lambda^{\prime}(\theta)=0$
on each tangent space, where $\epsilon=\pm 1$ is the sign of $L$, and $\theta$
is the Landsberg angle. Further, if $L$ is a Riemann metric, or regular
Finsler metric, then $L$ is of isotropic flag curvature ($\lambda=\lambda(x)$)
with $\omega=0$.

  2. (ii)

($n=2$) If G is induced by a Finsler metric $L>0$ with constant main scalar,
then G satisfies (5) with $\omega$ not necessarily zero.

Theorem 1.4(i) has essentially been proved in [8]. Starting from (5), we are
also going to give a little different version of proof from that in [8]. If a
spray G satisfying (5) with $\omega\neq 0$ and $n\geq 3$, then the spray G is
not Finsler-metrizabl by Theorem 1.4(ia). If a two-dimensionalspray ${\bf G}$
satisfying (5) with $\omega\neq 0$, then ${\bf G}$ cannot be induced by a
Riemann metric or a regular Finsler metric by Theorem 1.1(ib), but Theorem
1.4(ii) shows that possibly there are such sprays which can be induced by a
singular Finsler metric.

The problem still remains open to look for complete conditions for a spray of
scalar curvature to be Finsler-metrizable. Only some special conditions are
concerned in Theorem 3.5 below.

##  2 Preliminaries

Let $M$ be an $n$-dimensional manifold. A conical region
$\mathcal{C}=\mathcal{C}(M)$ of $TM\setminus\\{0\\}$ means
$\mathcal{C}_{x}:=\mathcal{C}\cap T_{x}M\setminus\\{0\\}$ is conical region
for $x\in M$ ($\lambda y\in\mathcal{C}_{x}$ if
$\lambda>0,y\in\mathcal{C}_{x}$). A spray on $M$ is a smooth vector field
${\bf G}$ on a conical region $\mathcal{C}$ of $TM\setminus\\{0\\}$ (an
important case is $\mathcal{C}=TM\setminus\\{0\\}$) expressed in a local
coordinate system $(x^{i},y^{i})$ in $TM$ as follows

${\bf G}=y^{i}\frac{\partial}{\partial x^{i}}-2G^{i}\frac{\partial}{\partial
y^{i}},$

where $G^{i}=G^{i}(x,y)$ are local functions satisfying $G^{i}(x,\lambda
y)=\lambda^{2}G^{i}(x,y)$ for any constant $\lambda>0$. The integral curves of
G projected onto $M$ are the geodesics of G.

The Riemann curvature tensor $R^{i}_{\ k}$ of a given spray $G^{i}$ is defined
by

$R^{i}_{\
k}:=2\partial_{k}G^{i}-y^{j}(\partial_{j}\dot{\partial_{k}}G^{i})+2G^{j}(\dot{\partial_{j}}\dot{\partial_{k}}G^{i})-(\dot{\partial_{j}}G^{i})(\dot{\partial_{k}}G^{j}),$
(6)

where we define $\partial_{k}:=\partial/\partial
x^{k},\dot{\partial}_{k}:=\partial/\partial y^{k}$. The trace of $R^{i}_{\ k}$
is called the Ricci curvature, $Ric:=R^{i}_{\ i}$. A spray ${\bf G}$ is said
to be R-flat if $R^{i}_{\ k}=0$. In [15], Z. Shen defines a non-Riemannian
quantity called $\chi$-curvature $\chi=\chi_{i}dx^{i}$ expressed as follows

$\chi_{i}:=2R^{m}_{\ i.m}+R^{m}_{\ m.i}.$ (7)

Plugging (1) into (7) yields

$\chi_{i}=(n+1)(R_{.i}-2\tau_{i}).$ (8)

So a spray of scalar curvature is of isotropic curvature iff. it has vanishing
$\chi$-curvature.

A spray G is called a Berwald spray if its Berwald curvature vanishes
$G^{i}_{hjk}:=\dot{\partial}_{h}\dot{\partial}_{j}\dot{\partial}_{k}G^{i}=0$.
A spray G is said to be locally projectively flat if locally $G^{i}$ can be
expressed as $G^{i}=Py^{i}$, where $P$ is a positively homogeneous local
function of degree one.

In the calculation of some geometric quantities of a spray, it is very
convenient to use Berwald connection as a tool. For a spray manifold $({\bf
G},M)$, Berwald connection is usually defined as a linear connection on the
pull-back $\pi^{*}\mathcal{C}$ ($\pi:TM\rightarrow M$ the natural projection)
over the base manifold $\mathcal{C}$. The Berwald connection is defined by

$D(\partial_{i})=(G^{k}_{ir}dx^{r})\partial_{k},\ \ \ \ \
(G^{k}_{ir}:=\dot{\partial}_{r}\dot{\partial}_{i}G^{k}),$

For a spray tensor $T=T_{i}dx^{i}$ as an example, the horizontal and vertical
derivatives of $T$ with respect to Berwald connection are given by

$T_{i;j}=\delta_{j}T_{i}-T_{r}G^{r}_{ij},\ \ \ \ \ \ \
T_{i.j}=\dot{\partial}_{j}T_{i},\ \ \ \
(\delta_{i}:=\partial_{i}-G^{r}_{i}\dot{\partial}_{r}).$

The $hh$-curvature tensor $H^{\ i}_{j\ kl}$ of Berwald connection is defined
by

$H^{\ i}_{j\ kl}:=\frac{1}{3}\big{\\{}R^{i}_{\ l.j.k}-(k/l)\big{\\}},\ \
H_{ij}:=H^{\ m}_{i\ jm},\ \ H_{i}:=\frac{1}{n-1}(nH_{0i}+H_{i0}),$

where $T_{ij}-(i/j)$ means $T_{ij}-T_{ji}$, and $T_{0}$ is defined by
$T_{0}:=T_{r}y^{r}$, as an example. For the Ricci identities and Bianchi
identities of Berwald connection, readers can refer to [1].

In this paper, we define a Finsler metric $L(\neq 0)$ on a manifold $M$ as
follows (cf. [18]): (i) for any $x\in M$, $L_{x}$ is defined on a conical
region of $T_{x}M\setminus\\{0\\}$ and $L$ is $C^{\infty}$; (ii) $L$ is
positively homogeneous of degree two; (iii) the fundamental metric tensor
$g_{ij}:=(\frac{1}{2}L)_{y^{i}y^{j}}$ is non-degenerate. A Finsler metric $L$
is said to be regular if additionally $L$ is defined on the whole
$TM\setminus\\{0\\}$ and $(g_{ij})$ is positively definite. Otherwise, $L$ is
called singular. In general case, we don’t require that $L$ be regular. If a
Finsler metric $L>0$, we put $L=F^{2}$, and in this case, $F$ is also called a
Finsler metric being positively homogeneous of degree one.

Any Finsler metric $L$ induces a natural spray whose coefficients $G^{i}$ are
given by

$G^{i}:=\frac{1}{4}g^{il}\big{\\{}L_{x^{k}y^{l}}y^{k}-L_{x^{l}}\big{\\}},$

where $(g^{ij})$ is the inverse of $(g_{ij})$. $L$ is said to be of scalar
flag curvature $K=K(x,y)$ if

$R^{i}_{\ k}=K(L\delta^{i}_{k}-y^{i}y_{k}),$

where $y_{k}:=(L/2)_{.k}=g_{km}y^{m}$. If $K_{.i}=0$, then $L$ is said to be
of isotropic flag curvature, and in this case, $K$ is a constant if the
dimension $n\geq 3$.

A spray G is (globally) Finsler-metrizable on $M$ (or on $\mathcal{C}(M)$) if
there is a Finlser metric $L$ defined on a conical region $\mathcal{C}(M)$ and
$L$ induces G. A spray G is locally Finsler-metrizable on $M$ if for each
$x\in M$, there is a neighborhood $U$ of $x$ such that G is Finsler-metrizable
on $U$.

Let $(M,F)$ be a two-dimensional Finsler space with the Finsler metric $F$. We
use $\epsilon(=\pm 1)$ to denote the sign of the determinant of the metric
matrix. We have

$\displaystyle g_{ij}=\ell_{i}\ell_{j}+\epsilon
m_{i}m_{j}=\ell_{i}\ell_{j}+h_{ij},$ $\displaystyle\ell^{i}=y^{i}/F,\ \ \ \
(m^{1},m^{2})=\big{(}\sqrt{\epsilon g}\big{)}^{-1}(-\ell_{2},\ell_{1}),\ \ \
(g:=det(g_{ij})),$ $\displaystyle Fm_{i.j}=-(\ell_{i}-\epsilon Im_{i})m_{j},\
\ \ \ FC_{ijk}=Im_{i}m_{j}m_{k},$ (9)

where $\ell=(\ell^{1},\ell^{2}),m=(m^{1},m^{2})$ is called the Berwald frame,
$C_{ijk}$ is the Cartan tensor and $I$ is the main scalar. The system
$F_{.i}=\ell_{i},F\theta_{.i}=m_{i}$ is integrable. It defines the so-called
Landsberg angle $\theta$, which is the arc-length parameter of the indicatrix
$S_{x}M:=\\{y\in T_{x}M|F(x,y)=1\\}$ with respect to the Riemann metric
$ds^{2}=g_{ij}dy^{i}dy^{j}$ on the Minkowski plane $(M_{x},F_{x})$.

###### Lemma 2.1

For a positively homogeneous function $\lambda=\lambda(x,y)$ of degree zero,
it satisfies on each tangent space,

$F\lambda_{.i}=\lambda^{\prime}(\theta)m_{i}.$

##  3 Spays of scalar curvature

In this section, we will introduce some basic properties of sprays with scalar
curvature, and the metrizability of such sprays under certain conditions.

###  3.1 Some basic formulas

For a spray of scalar curvature, $R$ and $\tau_{k}$ in (1) are related in the
following formula (10).

###### Proposition 3.1

Let G be an $n(\geq 3)$-dimensional spray of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$. Then there holds

$R_{.i;0}-3R_{;i}+\tau_{i;0}=0.$ (10)

In particular, if G is of isotropic curvature, then (10) becomes
$\tau_{i;0}=R_{;i}$, or $\tau_{i;0}=\tau_{0;i}$.

Proof : By a Bianchy identity of Berwald connection

$R^{i}_{\ jk;l}+R^{i}_{\ kl;j}+R^{i}_{\ lj;k}=0,$

we have

$R^{m}_{\ k;m}+R^{m}_{\ km;0}-R^{m}_{\ m;k}=0.$ (11)

Since $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$, a direct computation gives

$R^{m}_{\ k;m}=R_{;k}-\tau_{k;0},\ \ \ R^{m}_{\
km;0}=\frac{1}{3}\big{[}(n-1)(R_{.k;0}+\tau_{k;0})+\tau_{k;0}-\tau_{m.k;0}y^{m}\big{]}.$
(12)

By $\tau_{0}=R$, we obtain

$\tau_{m.k;0}y^{m}=R_{.k;0}-\tau_{k;0}.$ (13)

Now plugging (12) and (13) into (11) we have

$(n-2)(R_{.i;0}-3R_{;i}+\tau_{i;0})=0,$

which gives the proof. Q.E.D.

The isotropic case of Proposition 3.1 is given by Z. Shen in [16]. In
Proposition 3.1, if G is Finsler-metrizable induced by a Finsler metric $L$
with the flag curvature $\lambda$, we have $R=\lambda L,\tau_{i}=\lambda
y_{i}$ and then putting them into (10) gives a known formula

$L\lambda_{.i;0}+3\lambda_{;0}y_{i}-3L\lambda_{;i}=0.$ (14)

If $L=F^{2}$ is of weakly isotropic flag curvature $\lambda=3\theta/F+\sigma$,
then (14) becomes

$\theta_{;i}-\theta_{.i;0}+(F\sigma_{;r}+2\theta_{;r})h^{r}_{i}=0,\ \ \
(h^{i}_{j}:=\delta^{i}_{j}-\ell^{i}\ell_{j}).$

For a two-dimensional spray G in Proposition 3.1, (10) generally does not
hold. For example, if G is a two-dimensional spray induced by a Riemann metric
$L$ of isotropic Gauss curvature $\lambda=\lambda(x)$, then (14) reduces to
$\lambda_{;i}L=\lambda_{;0}y_{i}$, which is impossible if $\lambda$ is not
constant.

In Proposition 3.1, if G is induced by a Finsler metric $L$ of isotropic flag
curvature $\lambda$, then we have $\lambda=constant$ by (14), which is just
the Schur’s Theorem. But for a general spray, we cannot conclude from
Proposition 3.1 that a spray of isotropic curvature in dimension $n\geq 3$
must be of constant curvature (see Examples 7.3 and 7.4 below).

The following proposition gives a useful formula on a spray manifold of scalar
curvature.

###### Proposition 3.2

Let G be a spray of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$, and $T$ be a homogeneous scalar function of
degree $p$ satisfying $T_{;i}=0$. Then we have

$RT_{.k}=pT\tau_{k}.$ (15)

  1. (i)

If G is of isotropic curvature with $R\neq 0$, then there holds

$T=c|R|^{\frac{1}{2}p},\ \ \ (c=c(x)).$ (16)

  2. (ii)

If G is induced by a Finsler metric $L$ of non-zero flag curvature, then there
holds

$T=c|L|^{\frac{1}{2}p},\ \ \ (c=constant).$ (17)

Proof : By a Ricci identity and $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$,
we have

$0=y^{j}(T_{;i;j}-T_{;j;i})=T_{.r}R^{r}_{\ i}=RT_{.k}-y^{r}T_{.r}\tau_{k}.$

Then we obtain (15) since $T$ is a homogeneous scalar function of degree $p$.

(i) Since G is of isotropic curvature, we have $\tau_{k}=\frac{1}{2}R_{.i}$.
Putting it into (15) gives (16) since

$\frac{T_{.i}}{T}=\frac{p}{2}\frac{R_{.i}}{R}\ \
\Big{(}\Leftrightarrow\big{(}\frac{T}{|R|^{\frac{1}{2}p}}\big{)}_{.i}=0\Big{)}.$

(ii) The original version is given in [1]. Let $\lambda\neq 0$ be the flag
curvature of $L$. Similarly as the proof in (i), plugging $R=\lambda L$ and
$\tau_{i}=\frac{1}{2}L_{.i}$ into (15) gives

$T=c|L|^{\frac{1}{2}p},\ \ \ c=c(x).$

Then by $T_{;i}=0$ and $L_{;i}=0$ we obtain $c_{;i}=0$, which means
$c=constant$. Q.E.D.

###  3.2 Some basic properties

Let ${\bf G}$ be a spray of scalar curvature satisfying (1). Then we have

$H^{\ i}_{h\
jk}=\frac{1}{3}\big{[}R_{.j.h}\delta^{i}_{k}-\tau_{k.j.h}y^{i}-\tau_{k.j}\delta^{i}_{h}-\tau_{k.h}\delta^{i}_{j}-(j/k)\big{]}.$
(18)

By (18) we obtain

$H_{ij}-H_{ji}=\frac{1}{3}(n+1)(\tau_{j.i}-\tau_{i.j}),$ (19)
$H_{0i}=\frac{1}{3}\big{[}(n-2)R_{.i}+(n+1)\tau_{i}\big{]},\ \ \ \
H_{0i}=\frac{1}{3}\big{[}(2n-1)R_{.i}-(n+1)\tau_{i}\big{]}$ (20)
$H_{0i}-H_{i0}=-\frac{1}{3}(n+1)(R_{.i}-2\tau_{i}),$ (21)
$H_{i}=\frac{1}{3}(n+1)(R_{.i}+\tau_{i}).$ (22)

###### Proposition 3.3

Let G be a spray of scalar curvature satisfying (1).

  1. (i)

If G is of isotropic curvature, then $H_{ij}=H_{ji}$, $H_{i0}=H_{0i}$ and
$H_{i}$ is proportional to $\tau_{i}$.

  2. (ii)

If $H_{ij}=H_{ji}$, or $R_{i0}=H_{0i}$, or $H_{i}$ is proportional to
$\tau_{i}$ with $R\neq 0$, then G is of isotropic curvature.

Proof : Firstly, we prove $R_{.i}=2\tau_{i}$ is equivalent to $H_{ij}=H_{ji}$.
If $R_{.i}=2\tau_{i}$, then it is easy to see that $\tau_{i.j}=\tau_{j.i}$ and
then by (19), we have $H_{ij}=H_{ji}$. Conversely, if $H_{ij}=H_{ji}$, then by
(19) we have $\tau_{i.j}=\tau_{j.i}$. Differentiating $R=\tau_{0}$ by $y^{i}$
yields

$R_{.i}=\tau_{i}+\tau_{m.i}y^{m}=\tau_{i}+\tau_{i.m}y^{m}=2\tau_{i}.$

Secondly, it is clear from (21) that $R_{.i}=2\tau_{i}$ is equivalent to
$H_{i0}=H_{0i}$.

Finally, if $R_{.i}=2\tau_{i}$, it is an obvious result that $H_{i}$ is
proportional to $\tau_{i}$ by (22). Conversely, if $H_{i}$ is proportional to
$\tau_{i}$ and $R\neq 0$, then by (22) we get $R_{.i}=\lambda\tau_{i}$ for
some scalar function $\lambda=\lambda(x,y)$. Contracting this by $y^{i}$ gives
$2R=\lambda\tau_{0}=\lambda R$. Since $R\neq 0$ by assumption, we have
$\lambda=2$ and thus $R_{.i}=2\tau_{i}$. Q.E.D.

###### Proposition 3.4

A spray G of scalar curvature is R-flat if and only if $H_{ij}=0$, or
$H_{i0}=0$, or $H_{0i}=0$ or $H_{i}=0$.

Proof : The Riemann curvature tensor of G satisfies (1). If $H_{ij}=0$, then
$H_{i}=0$. So by (22) we have $R_{.i}+\tau_{i}=0$. Contracting this by $y^{i}$
gives $3R=0$ and so $R=0,\tau_{i}=0$. This shows that G is R-flat. If
$H_{0i}=0$ or $H_{i0}=0$, we have $(n-2)R_{.i}+(n+1)\tau_{i}=0$ or
$(2n-1)R_{.i}-(n+1)\tau_{i}=0$ by (20). Contracting either one by $y^{i}$
gives $R=0$. Similarly we see that G is R-flat. Q.E.D.

###  3.3 Metrizability under certain conditions

For a spray of scalar curvature in general case, we have the following result
and Theorem 1.4 on metrizability.

###### Theorem 3.5

Let G be a spray of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$.

  1. (i)

If $R=\tau_{k}=0$, then G is locally Finsler-metrizable.

  2. (ii)

If $R=0,\tau_{k}\neq 0$, then G is not Finsler-metrizable.

  3. (iii)

If $R\neq 0$ and

$\big{(}\frac{\tau_{i}}{R}\big{)}_{;j}\neq 0\ \ or\ \
\big{(}\frac{\tau_{i}}{R}\big{)}_{.j}\neq\big{(}\frac{\tau_{j}}{R}\big{)}_{.i},$
(23)

then G is not Finsler-metrizable.

Proof : (i) is just Theorem 1.1(i). For (ii), if G is induced by a Finsler
metric $L$, then $L$ is of scalar flag curvature with $R=\lambda
L,\tau_{k}=\lambda y_{k}$ for some scalar function $\lambda=\lambda(x,y)$. So
if $R=0$, then $\tau_{k}=0$. Thus G is not Finsler-metrizable.

For (iii), if G is induced by a Finsler metric $L$, then similarly we have
$R=\lambda L,\tau_{k}=\lambda y_{k}$. Thus we have

$\big{(}\frac{\tau_{i}}{R}\big{)}_{;j}=\big{(}\frac{y_{i}}{L}\big{)}_{;j}=0,\
\ \
\big{(}\frac{\tau_{i}}{R}\big{)}_{.j}=\big{(}\frac{y_{i}}{L}\big{)}_{.j}=\big{(}\frac{y_{j}}{L}\big{)}_{.i}=\big{(}\frac{\tau_{j}}{R}\big{)}_{.i}.$

So G is not Finsler-metrizable. Q.E.D.

In [18], there are some two-dimensional sprays (of scalar curvature)
satisfying the condition of Theorem 3.5(ii) ($R=0,\tau_{k}\neq 0$) (also see
Example 7.6 below).

###### Remark 3.6

Denote by $\mathcal{D}_{hol}(G)$ the holonomy algebra of a spray G generated
by the horizontal vector fields and their successive Lie brackets. Define

$e_{i}:=(\delta^{r}_{i}-\frac{\tau_{i}}{R}y^{r})\dot{\partial}_{r},\ \
(i=1,\cdots,n),$

where G has the dimension $n$ and is of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with $R\neq 0$. Then
$\mathcal{V}:=span\\{e_{1},\cdots,e_{n}\\}$ has the dimension $n-1$ and
$\mathcal{V}\subset\mathcal{D}_{hol}(G)$ since
$Re_{i}=y^{j}[\delta_{j},\delta_{i}]\in\mathcal{D}_{hol}(G)$. So
$\mathcal{D}_{hol}(G)$ has the dimension at least $2n-1$. Further, we have

$[\delta_{i},e_{j}]=G^{r}_{ij}e_{r}-\big{(}\frac{\tau_{j}}{R}\big{)}_{;i}Y,\ \
\ \ \
[e_{i},e_{j}]=\frac{\tau_{i}}{R}e_{j}-\frac{\tau_{j}}{R}e_{i}+\big{[}\big{(}\frac{\tau_{i}}{R}\big{)}_{.j}-\big{(}\frac{\tau_{j}}{R}\big{)}_{.i}\big{]}Y,\
\ \ \ (Y:=y^{i}\dot{\partial}_{i}).$

Note that $Y$ is not in $\mathcal{V}$. Here we can give a different proof of
Theorem 3.5(iii): if (23) holds, then we have $Y\in\mathcal{D}_{hol}(G)$ and
thus $0=YL=2L$ (a contradiction). On the other hand, even if

$\big{(}\frac{\tau_{i}}{R}\big{)}_{;j}=0\ \ and\ \
\big{(}\frac{\tau_{i}}{R}\big{)}_{.j}=\big{(}\frac{\tau_{j}}{R}\big{)}_{.i},$
(24)

we still do not make sure whether $Y$ is in $\mathcal{D}_{hol}(G)$ or not.

It is easy to check that a spray of cosntant curvature ($R\neq 0$) always
satisfies (24) (see (41) below). The condition (24) is a necessary but not
sufficient for G to be Finsler-metrizable in Theorem 3.5(iii). See Example 7.4
below based on Theorem 1.3.

Proof of Theorem 1.4 : Let G be induced by the Finsler metric $L$. Here we
provide a version of proof of Theorem 1.4(ia) for $L>0$. Put $R=\lambda
L,\tau_{i}=\lambda y_{i}=\lambda F\ell_{i}$, where $F$ is given by $F^{2}=L$
and $\lambda$ is the flag curvature. By (5) we have
$(R_{.i}-2\tau_{i})_{.k}+(i/k)=0$, which just is

$F\lambda_{.i.k}+\lambda_{.k}\ell_{i}+\lambda_{.i}\ell_{k}=0.$ (25)

Differentiating (25) by $y^{j}$ gives

$F\lambda_{.i.j.k}+\lambda_{.i.k}\ell_{j}+\lambda_{.j.k}\ell_{i}+F^{-1}\lambda_{.k}h_{ij}+\lambda_{.i.j}\ell_{k}+F^{-1}\lambda_{.i}h_{jk}=0.$
(26)

Interchanging $i,j$ in (26) and making a subtraction, we obtain
$h_{jk}\lambda_{.i}-h_{ik}\lambda_{.j}=0.$ Contracting this by $g^{jk}$ gives
$(n-2)\lambda_{.i}=0$.

(i)(ia) If $n>2$, then we have $\lambda_{.i}=0$. So $\lambda$ is constant by
Schur’s Theorem.

(i)(ib) If $n=2$, let $(\ell,m)$ be the Berwald frame and $\theta$ be the
Landsberg angle. By Lemma 2.1, we put

$F\lambda_{.i}=\eta m_{i},\ \ \ \ \eta(\theta)=\lambda^{\prime}(\theta).$ (27)

Differentiating (27) by $y^{j}$ and using (9) we obtain

$\displaystyle\ell_{j}\lambda_{.i}+F\lambda_{.i.j}$
$\displaystyle=\eta_{.j}m_{i}-\eta F^{-1}(\ell_{i}-\epsilon Im_{i})m_{j},$
$\displaystyle\ell_{i}\lambda_{.j}+\ell_{j}\lambda_{.i}+F\lambda_{.i.j}$
$\displaystyle=\eta_{.j}m_{i}+\epsilon\eta F^{-1}Im_{i}m_{j}.$ (28)

Then by (25) we have

$F\eta_{.j}+\epsilon\eta Im_{j}=0,\ \ \ {\rm or}\ \ \
\eta^{\prime}+\epsilon\eta I=0,\ \ \ {\rm or}\ \ \
\lambda^{\prime\prime}+\epsilon\lambda^{\prime}I=0.$

If $F$ is regular ($\epsilon=1$), then $T_{x}M$ is compact. There is a
$\theta_{0}$ such that $\lambda^{\prime}(\theta_{0})=0$. So we get
$\eta(\theta)=\lambda^{\prime}(\theta)\equiv 0$. Thus we have $\lambda_{.i}=0$
from (27).

(ii) It is known that all Finsler metrics with isotropic main scalar $I=I(x)$
on a two-dimensional manifold are divided into the following three classes
([1]):

$\displaystyle L=\beta^{2s}\gamma^{2(1-s)},\quad(s=s(x)\neq 0,\quad s(x)\neq
1),$ (29) $\displaystyle L=\beta^{2}e^{\frac{2\gamma}{\beta}},$ (30)
$\displaystyle L=(\beta^{2}+\gamma^{2})e^{2r\cdot
arctan(\frac{\beta}{\gamma})},\quad r=r(x),$ (31)

where $\beta=p_{i}(x)y^{i}$ and $\gamma=q_{i}(x)y^{i}$ are two independent
1-forms. The main scalar $I=I(x)$ is given respectively by

$\epsilon I^{2}=\frac{(2s-1)^{2}}{s(s-1)},\ \ \ I^{2}=4,\ \ \ \
I^{2}=\frac{4r^{2}}{1+r^{2}}.$

In the following we assume that the Finsler metric $L$ is of constant main
scalar, that is, $s$ in (29) and $r$ in (31) are constant. For the convenience
of computation, we may put $\beta=py^{1}$ and $\gamma=qy^{2}$ under certain
local coordinate system, where $p=p(x^{1},x^{2})$ and $q=q(x^{1},x^{2})$ are
scalar functions. By a direct computation, we can obtain the flag curvature
$\lambda$ of the Finsler metric given by (29)-(31) respectively (see [20]).
Now we can obtain (5) from $R=\lambda L$ and $\tau_{k}=\lambda y_{k}$. If $L$
is given by (29), then (5) holds with

$\omega_{12}=\frac{1-2s}{s(1-s)}\frac{(p^{2}qq_{12}-pq^{2}p_{12}+q^{2}p_{1}p_{2}-p^{2}q_{1}q_{2})s-p^{2}qq_{12}+p^{2}q_{1}q_{2}}{p^{2}q^{2}}.$

If $L$ is given by (30), then (5) holds with

$\omega_{12}=-\frac{2(p^{2}qp_{22}+pq^{2}p_{12}-p^{2}qq_{12}+p^{2}q_{1}q_{2}-q^{2}p_{1}p_{2}-p^{2}p_{2}q_{2})}{p^{2}q^{2}}.$

If $L$ is given by (31), then (5) holds with

$\omega_{12}=\frac{2r}{1+r^{2}}\frac{(p^{2}qq_{12}-pq^{2}p_{12}-p^{2}q_{1}q_{2}+q^{2}p_{1}p_{2})r+p^{2}qp_{22}+pq^{2}q_{11}-q^{2}p_{1}q_{1}-p^{2}p_{2}q_{2}}{p^{2}q^{2}}.$

So generally, $\omega_{12}$ in the above is not zero. Q.E.D.

##  4 Sprays of isotropic curvature

In this section, we are going to prove Theorem 1.1 and give a metrizibility
result as an application of Theorem 1.1. For this, we first introduce a simple
necessary and sufficient condition for a general spray to be metrizable (see
Lemma 4.3).

In the following lemmas, the horizontal and vertical covariant derivative is
taken with respect to the given spray G.

###### Lemma 4.1

Let G be a spray and $L$ be a Finsler function. Then we have

$L_{.i;0}-L_{;i}=0\Longleftrightarrow
G^{i}=\frac{1}{4}g^{il}\big{\\{}L_{x^{k}y^{l}}y^{k}-L_{x^{l}}\big{\\}}.$

Proof : It follows from
$L_{.i;0}-L_{;i}=-2G^{r}L_{.r.i}+L_{x^{r}y^{i}}y^{r}-L_{x^{i}}.$ Q.E.D.

###### Lemma 4.2

Let G be a spray and $L$ be a Finsler function. Then we have

$L_{.0}=2L,\ \ \ L_{;i}=0\ \Longleftrightarrow\ L_{.0}=2L,\ \ \
L_{.i;0}-L_{;i}=0.$

Proof : We only need to prove ”$\Longleftarrow$”. By $L_{.i;0}=L_{;i}$ we have
$L_{;0.i}-L_{;i}=L_{;i}$, or $L_{;0.i}=2L_{;i}$. Further, $L_{.i;0}=L_{;i}$
implies $L_{.0;0}=L_{;0}$. So by $L_{.0}=2L$ we obtain $L_{;0}=0$. Thus it
follows from $L_{;0.i}=2L_{;i}$ that $L_{;i}=0$. Q.E.D.

###### Lemma 4.3

A spray G is Finsler-metrizable if and only if there is a Finsler function $L$
satisfies $L_{;i}=0$. In this case, G is induced by $L$.

###  4.1 Formal integrability

To prove Theorem 1.1(i), we need a theory on Spencer’s technique of formal
integrability for linear partial differential systems. Here we only give some
basic notions for this theory and more details are refereed to [9, 3].

Let $B$ be a vector bundle over an $n$-dimensional manifold $M$, and denote by
$J_{k}B$ the bundles of $k$th order jets of the sections of $B$. For two
vector bundles $B_{1},B_{2}$ over $M$, consider $P:Sec(B_{1})\rightarrow
Sec(B_{2})$, which is a linear partial differential operator of order $k$. $P$
can be identified with a map $p_{0}(P):J_{k}B_{1}\rightarrow B_{2}$, a
morphism of vector bundles over $M$. We also denote by
$p_{l}(P):J_{k+l}B_{1}\rightarrow J_{l}B_{2}$ the morphisms of vector bundles
over $M$, which is called the $l$th order jet prolongation of $P$. Let
$R_{k+l,x}(P):=Kerp_{l}(P)_{x}$ be the space of $(k+l)$th order formal
solutions of $P$ at a point $x\in M$. The operator $P$ is said to be formally
integrable at $x\in M$, if $R_{k+l}(P)$ is a vector bundle for all $l\geq 0$
and the projection $\pi_{k+l,x}(P):R_{k+l,x}(P)\rightarrow R_{k+l-1,x}(P)$ is
onto for all $l\geq 1$.

Let $\sigma_{k}(P):S^{k}(T^{*}M)\otimes B_{1}\rightarrow B_{2}$ be the symbol
of $P$, which is defined by the highest order term of $P$, and let
$\sigma_{k+l}(P):S^{k+l}(T^{*}M)\otimes B_{1}\rightarrow S^{l}(T^{*}M)\otimes
B_{2}$ be the symbol of the $l$th order prolongation of $P$. Define

$\displaystyle g_{k,x}(P):$ $\displaystyle=Ker\ \sigma_{k,x}(P),$
$\displaystyle g_{k,x}(P)_{e_{1}\ldots e_{j}}:$ $\displaystyle=\\{A\in
g_{k,x}(P)|i_{e_{1}}A=\ldots=i_{e_{j}}A=0\\},\ \ \ 1\leq j\leq n,$

where $\\{e_{1},\cdots,e_{n}\\}$ is a basis of $T_{x}M$. Such a basis is said
to be quasi-regular if it satisfies

$dim\ g_{k+1,x}(P)=dim\ g_{k,x}(P)+\sum^{n}_{j=1}dim\ g_{k,x}(P)_{e_{1}\cdots
e_{j}}.$

The symbol $\sigma_{k}(P)$ is said to be involutive at $x\in M$ if there
exists a quasi-regular basis of $T_{x}M$. For the proof of Theorem 1.1(i), we
need the following theorem and lemma ([9]).

###### Theorem 4.4

(Cartan-Kahler) Let $P$ be a regular linear partial differential operator of
order $k$. If $\pi_{k+1,x}(P):R_{k+1,x}(P)\rightarrow R_{k,x}(P)$ is onto and
the symbol $\sigma_{k}(P)$ is involutive, then $P$ is formally integrable.

###### Lemma 4.5

For two vector bundles $B_{1},B_{2}$ over $M$, let $P:Sec(B_{1})\rightarrow
Sec(B_{2})$ be a regular linear partial differential operator of order $k$.
Then $\pi_{k+1,x}(P):R_{k+1,x}(P)\rightarrow R_{k,x}(P)$ is onto iff.

$P(s)_{x}=0\Longrightarrow(DP(s))_{x}=\sigma_{k+1}(P)(A):$

for some $A\in S^{k+1}(T^{*}_{x}M)\otimes B_{1}$, where $D$ is an arbitrary
linear connection of the bundle $B_{2}$ over the base manifold $M$.

###  4.2 Proof of Theorem 1.1(i)

Let $T^{*}_{v}TM$ denote the subbundle of $T^{*}TM$, in which, if $\omega\in
T^{*}_{v}TM$, then $\omega$ can be written locally as
$\omega=\omega_{i}(x,y)dx^{i}$, and $\wedge^{2}T^{*}_{v}TM$ the subbundle of
$T^{*}TM$, every element $\omega$ of which is locally in the form
$\omega=\omega_{ij}(x,y)dx^{i}\wedge dx^{j}$ with $\omega_{ij}=-\omega_{ji}$.
Besides, we designate $S^{k}(T^{*}TM)$ as the bundle of symmetric $k$-forms
over $TM$. For an $n$-dimensional manifold $M$, let $B_{1},B_{2}$ be two
vector bundles over $TM$ with

$B_{1}:=T^{*}_{v}TM,\ \ \
B_{2}:=T^{*}_{v}TM\oplus\wedge^{2}T^{*}_{v}TM\oplus(T^{*}_{v}TM\otimes
T^{*}_{v}TM).$

We define a linear partial differential operator $P:Sec(B_{1})\rightarrow
Sec(B_{2})$ in component form as follows

$P(\theta_{i})=(\theta_{i.0},\theta_{i.j}-\theta_{j.i},\theta_{i;j}).$ (32)

###### Lemma 4.6

For the operator $P$ in (32), the symbol $\sigma_{1}(P)$ is involutive.

Proof : We are going to prove that
$\\{\dot{\partial}_{1},\cdots,\dot{\partial}_{n},\delta_{1},\cdots,\delta_{n}\\}$
is a quasi-regular basis of $P$.

By definition, for $A=(A_{ji},A_{\underline{j}i})\ (=A_{ji}dx^{j}\otimes
dx^{i}+A_{\underline{j}i}\delta y^{j}\otimes dx^{i})\in T^{*}TM\otimes B_{1}$,
we have

$\sigma_{1}(P)A=(A_{\underline{0}i},A_{\underline{j}i}-A_{\underline{i}j},A_{ji})\in
B_{2}.$

Assume $\sigma_{1}(P)(A)=0$. Then for the computation of $dim(g_{1}(P))$, we
see that $A_{\underline{0}i}=0$ and $A_{\underline{i}j}=A_{\underline{j}i}$
together contribute the number $(n-1)n/2$, and $A_{ij}=0$ gives $0$.
Therefore, we obtain

$dim(g_{1}(P))=\frac{(n-1)n}{2}.$ (33)

Now with respect to the basis $\\{dx^{i},\delta y^{i}\\}$, an element $B\in
S^{2}(T^{*}TM)\otimes B_{1}$ can be expressed as

$\displaystyle\hskip 22.76228ptB=(B_{ijk},\ B_{\underline{i}jk},\
B_{i\underline{j}k},\ B_{\underline{i}\underline{j}k})$
$\displaystyle\big{(}B_{ijk}=B_{jik},B_{\underline{i}jk}=B_{j\underline{i}k},B_{\underline{i}\underline{j}k}=B_{\underline{j}\underline{i}k}\big{)}.$

By definition, we have

$\sigma_{2}(P)B=(B_{i\underline{0}k},B_{\underline{i}\underline{0}k};B_{i\underline{j}k}-B_{i\underline{k}j},B_{\underline{i}\underline{j}k}-B_{\underline{i}\underline{k}j};B_{ijk},B_{\underline{i}jk}).$

Assume $\sigma_{2}(P)(B)=0$. Then $B_{ijk}$, $B_{i\underline{j}k}$ and
$B_{\underline{i}jk}$ gives 0 to $dim(g_{2}(P))$. By the fact that
$B_{\underline{i}\underline{j}k}$ is symmetric in $i,j,k$ satisfying
additional condition $B_{\underline{i}\underline{0}k}=0$, we obtain the number

$\sum^{n-2}_{k=0}\frac{(n-k-1)(n-k)}{2}.$

to $dim(g_{2}(P))$. So altogether, we have

$dim(g_{2}(P))=\sum^{n-2}_{k=0}\frac{(n-k-1)(n-k)}{2}.$ (34)

Next we verify under the basis
$\\{\dot{\partial}_{1},\cdots,\dot{\partial}_{n},\delta_{1},\cdots,\delta_{n}\\}$
at a point $(x,y)\in TM$,

$dim(g_{2}(P))=dim\ g_{1}(P)+\sum^{n}_{j=1}dim\
g_{1}(P)_{\dot{\partial}_{1}\cdots\dot{\partial}_{j}}+\sum^{n}_{j=1}dim\
g_{1}(P)_{\dot{\partial}_{1}\cdots\dot{\partial}_{n}\delta_{1}\cdots\delta_{j}}.$
(35)

By a direct computation, we see

$\displaystyle dim\
g_{1}(P)_{\dot{\partial}_{1}\cdots\dot{\partial}_{j}}=\frac{(n-j-1)(n-j)}{2},\
\ \ \ dim\
g_{1}(P)_{\dot{\partial}_{1}\cdots\dot{\partial}_{n}\delta_{1}\cdots\delta_{j}}=0,\
\ (1\leq j\leq n).$

Plugging them into (35) and using (33), (34), we see that both sides of (35)
are equal. Therefore, $\sigma_{1}(P)$ is involutive. Q.E.D.

###### Lemma 4.7

For the operator $P$ in (32), a first-order solution of $P(\theta)=0$ can be
lifted into a second-order solution iff.

$\theta_{r}H^{\ r}_{i\ jk}+\theta_{i.r}R^{r}_{\ jk}=(\theta_{r}R^{r}_{\
jk})_{.i}=0.$ (36)

Proof : If $P(\theta)=0$, then we have $\theta_{i;j}=\theta_{j;i}$. Then by a
Ricci identity and $\theta_{i.j}=\theta_{j.i}$, it gives (36):

$0=\theta_{i;j;k}-\theta_{i;k;j}=-\theta_{r}H^{\ r}_{i\
jk}-\theta_{i.r}R^{r}_{jk}=-(\theta_{r}R^{r}_{\ jk})_{.i}.$

Conversely, suppose that (36) holds for a $\theta$ satisfying $P(\theta)=0$
with $\theta_{0}\neq 0$. Let $D$ be the Berwald connection of the bundle
$\pi^{*}TM$ over the base manifold $TTM$. Then $D$ can be naturally extended
to the bundle $B_{2}$. We use Lemma 4.5. In component form, we have

$DP(\theta_{i})=(\theta_{i.0;j},\ \theta_{i.0.j},\ \
\theta_{i.j;k}-\theta_{j.i;k},\ \theta_{i.j.k}-\theta_{j.i.k},\ \
\theta_{i;j;k},\ \theta_{i;j.k}),$ (37)

where $\theta$ ($\theta_{0}\neq 0$) satisfies $P(\theta)=0$ at a point
$w=(x,y)\in TM$, that is, at the point $w$ there holds

$\theta_{i.0}=0,\ \ \ \ \theta_{i.j}=\theta_{j.i},\ \ \ \ \theta_{i;j}=0.$
(38)

Next we are going to prove that there is an $A_{\alpha\beta i}\in
S^{2}(T^{*}_{w}TM)\otimes B_{1}$ satisfying

$DP(\theta_{i})=\sigma_{2}(A_{\alpha\beta i})=(A_{\alpha\underline{r}i}y^{r},\
\ A_{\alpha\underline{j}i}-A_{\alpha\underline{i}j},\ \ A_{\alpha ji}).$ (39)

By (37), we see that (39) is equivalent to

$\displaystyle\text{\char 172}\
A_{j\underline{r}i}y^{r}=\theta_{i.r;j}y^{r},\hskip 88.2037pt\text{\char 173}\
A_{\underline{j}\underline{r}i}y^{r}=\theta_{i.0.j}=\theta_{i.r.j}y^{r}+\theta_{i.j},$
$\displaystyle\text{\char 174}\
A_{k\underline{j}i}-A_{k\underline{i}j}=\theta_{i.j;k}-\theta_{j.i;k},\hskip
42.67912pt\text{\char 175}\
A_{\underline{k}\underline{j}i}-A_{\underline{k}\underline{i}j}=\theta_{i.j.k}-\theta_{j.i.k},$
$\displaystyle\text{\char 176}\ A_{kji}=\theta_{i;j;k},\hskip
106.69783pt\text{\char 177}\ A_{\underline{k}ji}=\theta_{i;j.k}.$

In the following, we will construct $A_{\alpha\beta i}$ which satisfies the
above six relations. Put

$\displaystyle
A_{\underline{j}\underline{k}i}=\theta_{i.k.j}+\theta_{0}^{-1}\big{[}\theta_{i.j}\theta_{k}+(i,j,k)\big{]},$
$\displaystyle A_{j\underline{k}i}=\theta_{i.k;j}-\theta_{r}G^{r}_{ijk},\ \ \
\ \ A_{\underline{k}ji}=\theta_{i;j.k},$ $\displaystyle
A_{kji}=\theta_{i;j;k}.$

Firstly, $A_{\underline{j}\underline{k}i}$ is symmetric in $j,k$ by
$\theta_{i.j}=\theta_{j.i}$ in (38) and it also satisfies ②④ by
$\theta_{i.j}=\theta_{j.i},\theta_{i.0}=0$ in (38). Next, by a Ricci identity
of Berwald connection, we see that

$A_{\underline{k}ji}-A_{j\underline{k}i}=\theta_{i;j.k}-\theta_{i.k;j}+\theta_{r}G^{r}_{ijk}=0,$

which gives $A_{\underline{k}ji}=A_{j\underline{k}i}$. It is also clear that
$A_{\underline{k}ji}$ and $A_{j\underline{k}i}$ satisfy ①③⑥. Finally,
$A_{kji}$ is symmetric in $j,k$, and satisfies ⑤ by a Ricci identity of
Berwald connection and (36). So (39) holds. This finishes the proof of the
lemma. Q.E.D.

Now in Theorem 1.1(i), we have $R=0$. So (36) automatically holds. It follows
from Lemmas 4.6, 4.7 and then Theorem 4.4, the operator $P$ is formally
integrable, that is, for each point $u_{0}:=(x^{i}_{0},y^{i}_{0})$, there
exist a neighborhood $U$ of $u_{0}$ and a analytic $\theta$ defined on $U$
such that $P(\theta)=0$.

Under the basis $(\delta_{i},\dot{\partial}_{i})$, the local coordinate of
$J_{1}B_{1}$ is expressed as
$(x^{i},y^{i},\theta_{i},\theta_{i\underline{j}},\theta_{ij})$. An initial
data
$(x^{i}_{0},y^{i}_{0},\theta_{i}^{0},\theta^{0}_{i\underline{j}},\theta^{0}_{ij})$
satisfied by the operator $P$ means
$\theta_{i\underline{0}}^{0}=0,\theta^{0}_{i\underline{j}}=\theta^{0}_{j\underline{i}},\theta^{0}_{ij}-\theta_{r}^{0}G^{r}_{i}=0$.
Further, we let the initial data satisfy $\theta_{i}^{0}y^{i}_{0}>0$ and
$Rank(\theta^{0}_{i\underline{j}})=n-1$.

Now for the above analytic solution $\theta$ of $P$ which is defined on a
neighborhood $U$ of $u_{0}$ and satisfies the above initial data, we obtain a
local metric $F:=\theta_{0}$ defined on $U$. To prove that $F$ is a Finsler
metric, we need the following lemma which can be proved by an elementary
discussion in linear algebra.

###### Lemma 4.8

Let $F$ be positively homogeneous of degree one with $F(y)\neq 0$ at a point
$y$. Then $g_{ij}:=\frac{1}{2}(F^{2})_{.i.j}$ is non-degenerate at $y$ iff.
$Rank(F_{.i.j})=n-1$ at $y$.

Now for $F=\theta_{0}$, we have $F_{.i.j}=\theta_{i.j}$. Then at $u_{0}$, we
have $Rank(F_{.i.j})=Rank(\theta^{0}_{i\underline{j}})=n-1$. So by the above
lemma, $g_{ij}$ is non-degenerate at $u_{0}$. By continuity, $g_{ij}$ is non-
degenerate in $U$ (when it is small enough). Thus we obtain a Finsler metric
$F$ with each $F_{x}$ defined on the conical region formed by $y=0$ and
$\\{y|(x,y)\in U\\}$. Further, $F$ satisfies $F_{;i}=0$, which means that the
spray G in Theorem 1.1 is induced by $F$ by Lemma 4.3. This completes the
proof of Theorem 1.1(i). Q.E.D.

###### Remark 4.9

The idea of the proof of Theorem 1.1(i) can be referred to that in [4] for the
formal integrability of the operator
$P_{1}(\theta):=(\theta_{i.0},\theta_{i.j}-\theta_{j.i},\delta_{i}\theta_{j}-\delta_{j}\theta_{i})$.
On the other hand, a suitable change of the proof of Theorem 1.1(i) can give
the proof for the formal integrability of the operator $P_{1}$ in [4] (where
actually we can redefine $P_{1}$ as
$\bar{P}_{1}:=(\theta_{i.0},\theta_{i.j}-\theta_{j.i},\theta_{j;i}-\theta_{i;j})$).
Besides, we may also consider the system $F_{.0}=F,\ F_{;i}=0$ for a possible
proof (cf. [14] for a more general discussion).

###  4.3 Proof of Theorem 1.1(ii) and (iii)

(ii) Assume that $R\neq 0$ is not a Finsler metric. If G is Finsler-metrizable
induced by a Finsler metric $F$, then $F$ is of isotropic curvature
$\lambda\neq 0$. By $R=\lambda F^{2}$, we see that $R$ is a Finsler metric,
which gives a contradiction.

(iii) Assume that $R$ is a Finsler metric. If G is Finsler-metrizable induced
by a Finsler metric $L$, then $L$ is of isotropic curvature
$\lambda=\lambda(x)\neq 0$. By $R=\lambda L$, we have $R_{;i}=\lambda_{;i}L$.
Thus we obtain

$R_{;i}=\frac{\lambda_{;i}}{\lambda}R=R(\ln|\lambda|)_{;i}.$

Let $\omega_{i}:=(\ln|\lambda|)_{;i}$. Then $\omega$ is closed and
$R_{;i}=R\omega_{i}$. Conversely, if $R_{;i}=R\omega_{i}$ for some closed
1-form $\omega=\omega_{i}(x)dx^{i}$, then locally there is a scalar function
$\lambda=\lambda(x)\neq 0$ such that $\omega_{i}=(\ln|\lambda|)_{;i}$. It is
easy to check that $R_{;i}=R(\ln|\lambda|)_{;i}$ is equivalent to
$(R/\lambda)_{;i}=0$. Therefore, G is Finsler-metrizable induced by the
Finsler metric $L:=R/\lambda$ by Lemma 4.3. By $R=\lambda L$, we see that $L$
is of isotropic flag curvature $\lambda$. If $\omega=0$, we may choose
$\lambda=1$, and then the Finlser metric $L=R$ is of constant flag curvature
$\lambda=1$. If $n\geq 3$, then the Finsler metric $L$ is of constant flag
curvature $\lambda$ by Schur’s theorem, which gives $R_{;i}=0$. Q.E.D.

###  4.4 A metrizability result

As an application of Theorem 1.1, we show the following theorem.

###### Theorem 4.10

Let $G^{i}$ be the spray of a Finsler metric $F$ of constant flag curvature
$\lambda$ and $\bar{G}^{i}$ be a spray defined by $\bar{G}^{i}=G^{i}+cFy^{i}$
for a constant $c$. Then ${\bf\bar{G}}$ is (locally) Finsler-metrizable iff.
$\lambda=-c^{2}$ or $c=0$. When $\lambda=-c^{2}$, $\bar{G}^{i}$ is locally
induced by a Finsler metric of zero flag curvature.

Proof : The Riemann curvature $R^{i}_{\ k}$ of G is given by

$R^{i}_{\ k}=\lambda(F^{2}\delta^{i}_{k}-FF_{.k}y^{i}).$

Then by a direct computation, the Riemann curvature $\bar{R}^{i}_{\ k}$ of
${\bf\bar{G}}$ is given by

$\bar{R}^{i}_{\ k}=\bar{R}\delta^{i}_{k}-\bar{\tau}_{k}y^{i},\ \ \ \
\big{(}\bar{R}:=(\lambda+c^{2})F^{2},\ \ \
\bar{\tau}_{k}:=(\lambda+c^{2})FF_{.k}\big{)}.$

So ${\bf\bar{G}}$ is of isotropic curvature since
$\bar{R}_{.i}=2\bar{\tau}_{i}$. Further, we have

$\bar{R}_{\bar{;}i}=\bar{R}_{;i}+\bar{R}_{.r}(cF_{.i}y^{r}+cF\delta^{r}_{i})=4c(\lambda+c^{2})F^{2}F_{.i}.$

Assume that ${\bf\bar{G}}$ is Finsler-metrizable. If $c(\lambda+c^{2})\neq 0$,
then $\bar{R}$ is a Finsler metric. By Theorem 1.1(iii), we have
$\bar{R}_{\bar{;}i}=\bar{R}\omega_{i}$ for some closed 1-form
$\omega=\omega_{i}(x)dx^{i}$. But clearly this does not hold. Therefore, we
have $c=0$ or $\lambda=-c^{2}$. Conversely, if $c=0$, then ${\bf\bar{G}}$ is
induced by $F$. If $\lambda+c^{2}=0$, then ${\bf\bar{G}}$ has zero Riemann
curvature. So ${\bf\bar{G}}$ is (locally) Finsler-metrizable by Theorem
1.1(i).Q.E.D.

Theorem 4.10 is a generalization of a result in [19], where we have an
additonal condition that $F$ is projectively flat.

##  5 Sprays of constant curvature

In this part, we introduce a new notion: a spray of constant curvature, which
is a generalization of a Finsler metric of constant flag curvature. For this
new notion, some basic properties for Finsler metrics still remain unchanged
for sprays (see Theorem 5.2 below).

###### Definition 5.1

A spray G of scalar curvature $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ is
said to be of constant curvature if $\tau_{i;j}=0$.

The following theorem gives some basic properties for sprays of constant
curvature.

###### Theorem 5.2

A spray has the following properties on constant curvature:

  1. (i)

A spray of scalar curvature is of constant curvature iff. its Riemann
curvature is zero or its Ricci curvature $Ric$ satisfies $Ric_{;i}=0(Ric\neq
0)$.

  2. (ii)

A spray of constant curvature must be of isotropic curvature.

  3. (iii)

A Finsler metric is of constant flag curvature iff. its spray is of constant
curvature.

  4. (iv)

An $n$-dimensional spray of isotropic curvature is not necessarily of constant
curvature even for $n\geq 3$.

A Finsler metric has the same conclusions as shown for sprays in Theorem
5.2(i)(ii). Meanwhile, Theorem 5.2(iv) shows a different property of sprays
from that of Finsler metrics.

To prove Theorem 5.2, we first show the following lemma.

###### Lemma 5.3

Let G be a spray of scalar curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$. Then we have

$\displaystyle R_{;i}=0\ (R\neq 0)\ \Longrightarrow\ R_{.i}=2\tau_{i},$ (40)
$\displaystyle\tau_{i;k}=0\ \Longrightarrow\ R=\tau_{k}=0\ \ or\ \ R_{;i}=0\
(R\neq 0),$ (41) $\displaystyle R_{;i}=0\ (R\neq 0)\ \Longrightarrow\
\tau_{i;k}=0.$ (42)

Proof : Assume $R_{;i}=0(R\neq 0)$. We have $RR_{.i}=2R\tau_{i}$ from (15),
where we have put $T=R$ with $p=2$ in (15). This gives $R_{.i}=2\tau_{i}$
since $R\neq 0$, which gives the proof of (40).

Assume $\tau_{i;k}=0$. We have $R_{;i}=0$ since $\tau_{0}=R$ and then
$R_{;i}=\tau_{0;i}=\tau_{m;i}y^{m}=0$. Now we prove that if $R=0$, then
$\tau_{i}=0$. By a Ricci identity we obtain

$0=y^{j}(\tau_{i;j;k}-\tau_{i;k;j})=y^{j}(-\tau_{r}H^{\ r}_{i\
jk}-\tau_{i.r}R^{r}_{\ jk})=-\tau_{r}H^{\ r}_{i\ 0k}-\tau_{i.r}R^{r}_{\ k}.$
(43)

Now by $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ we have

$H^{\ r}_{i\
jk}=\frac{1}{3}\big{[}R_{.j.i}\delta^{r}_{k}-\tau_{k.j.i}y^{r}-\tau_{k.j}\delta^{r}_{i}-\tau_{k.i}\delta^{r}_{j}-(j/k)\big{]}.$
(44)

Plugging (44), $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$, $\tau_{i;j}=0$ and
$R=0$ into (43) we obtain $\tau_{i}\tau_{k}=0$, which gives $\tau_{i}=0$. This
gives the proof of (41).

Assume $R_{;i}=0(R\neq 0)$. By (40) we have $R_{.i}=2\tau_{i}$. So
$\tau_{i;k}=\frac{1}{2}R_{.i;k}=\frac{1}{2}R_{;k.i}=0$. This gives the proof
of (42). Q.E.D.

Proof of Theorem 5.2 : (i) Assume that G is of constant curvature. Then we
have $R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with $\tau_{i;k}=0$ by
definition. By (41) we immediately obtain the desired conclusion since
$Ric=(n-1)R$. Conversely, let the Riemann curvature be zero or the Ricci
curvature $Ric$ satisfy $Ric_{;i}=0(Ric\neq 0)$. If $R^{i}_{\ k}=0$, then
$\tau_{i}=0$ and so $\tau_{i;k}=0$. If $Ric_{;i}=0(Ric\neq 0)$, then we have
$R_{;i}=0\ (R\neq 0)$. So by (40) we have $R_{.i}=2\tau_{i}$. Thus we obtain
$\tau_{i;k}=\frac{1}{2}R_{.i;k}=\frac{1}{2}R_{;k.i}=0$. By definition, G is of
constant curvature.

(ii) It follows directly from (41) and (40), and the definitions for a spray
of isotropic curvature and constant curvature.

(iii) If a Finsler metric $L$ is of constant flag curvature $\lambda$, then
its spray has the Riemann curvature $R^{i}_{\
k}=\lambda(L\delta^{i}_{k}-y_{k}y^{i})$. So its spray is of scalar curvature
$R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with $R=\lambda L$ and
$\tau_{k}=\lambda y_{k}$. Thus we have $\tau_{i;k}=0$, which implies that the
spray is of constant curvature. Conversely, if the spray G of a Finsler metric
$L$ is of constant curvature, then the Riemann curvature of G has the form
$R^{i}_{\ k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with $\tau_{i;k}=0$. So $L$ is of
scalar flag curvature (put the flag curvature as $\lambda$). We have
$R=\lambda L$ and $\tau_{k}=\lambda y_{k}$. Thus we get $\lambda_{;k}=0$ by
$\tau_{i;k}=0$. If $\lambda=0$, then $L$ has constant flag curvature $0$. If
$\lambda\neq 0$, then since $\lambda$ is a homogeneous function of degree $0$
satisfying $\lambda_{;k}=0$, we immediately obtain $\lambda=constant$ by
putting $T=\lambda,p=0$ in (17) of Proposition 3.2.

(iv) See Examples 7.3 and 7.4 below. Q.E.D.

Proof of Theorem 1.2 : The spray G has the Riemann curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with $\tau_{i;k}=0$ (note that
$Ric=(n-1)R$). By Theorem 5.2(ii), the spray G is of isotropic curvature since
G is of constant curvature.

If G is (locally) Finsler-metrizable, then by Theorem 1.1, it is clear that
there has $Ric=0$ or $Ric$ is a Finsler metric when $Ric\neq 0$.

Conversely, if $Ric=0$, then by Theorem 1.1(i), G is (locally) Finsler-
metrizable. If $Ric\neq 0$, then by assumption $Ric$ is a Finsler metric. It
follows from Theorem 5.2(i) that $Ric_{;i}=0$. By Theorem 1.1(iii), G is
(locally) Finsler-metrizable. Q.E.D.

##  6 Locally projectively flat sprays

In this section, we consider the properties and metrizability of locally
projectively flat sprays on a manifold. A locally projectively flat spray is
always of scalar curvature. Locally, we let G be a projectively flat spray
with $G^{i}=Py^{i}$ defined on an open set of $R^{n}$. Then by (6), the
Riemann curvature tensor $R^{i}_{\ k}$ is in the form $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with

$R=P^{2}-P_{x^{r}}y^{r},\ \
\tau_{k}=PP_{y^{k}}+P_{x^{r}y^{k}}y^{r}-2P_{x^{k}}.$ (45)

###  6.1 Some basic results

###### Lemma 6.1

Let $G^{i}=Py^{i}$ be a spray defined on an open set of $R^{n}$. Then (45)
becomes

$R=-P^{2}-P_{;0},\ \ \ \ \tau_{k}=P_{k;0}-2P_{;k}-PP_{k},\ \ \
(P_{k}:=P_{.k}).$ (46)

###### Lemma 6.2

In (46), $R=0$ and $\tau_{k}=0$ $\Longleftrightarrow$ $P_{;k}+PP_{k}=0$. So
the spray $G^{i}=Py^{i}$ has vanishing Riemann curvature iff.
$P_{;k}+PP_{k}=0$.

Proof : If $R=0$ and $\tau_{k}=0$, then $P_{;0}=-P^{2}$ implies
$P_{;k}+P_{k;0}=-2PP_{k}$. Further by $P_{k;0}=2P_{;k}+PP_{k}$ we obtain
$P_{;k}+PP_{k}=0$. Conversely, let $P_{;k}+PP_{k}=0$. Then we have
$P_{;0}=-P^{2}$ $(R=0)$. Now again $P_{;0}=-P^{2}$ implies
$P_{;k}+P_{k;0}=-2PP_{k}$, which implies $\tau_{k}=0$ by $P_{;k}+PP_{k}=0$.
Q.E.D.

By Theorem 1.1(i) and Lemma 6.2, a spray $G^{i}=Py^{i}$ satisfying
$P_{;k}+PP_{k}=0$ is locally Finsler-metrizable (also see [17, 12]). By (46)
we can easily obtain the following lemma.

###### Lemma 6.3

Let $G^{i}=Py^{i}$ be a spray with $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$. Then we have

$R_{.i}-2\tau_{i}=3(P_{;i}-P_{i;0}).$ (47)

So G is of isotropic curvature iff. $P_{;i}=P_{i;0}$.

By Lemma 6.3, we can easily obtain the following Proposition 6.4.

###### Proposition 6.4

Let $G^{i}=Py^{i}$ be a Berwald spray. Then G is of isotropic curvature iff.
$P$ is a (local) exact 1-form given by $P=\sigma_{x^{i}}y^{i}$ for a scalar
function $\sigma=\sigma(x)$.

The following Proposition 6.5 directly follows from Lemma 6.3 and Theorem 1.1.

###### Proposition 6.5

Let $G^{i}=Py^{i}$ ($P\neq 0$) be a spray defined on an open set of $R^{n}$
with $P_{;i}=0$. Then G is of isotropic curvature. Further G is Finsler
metrizable iff. $P$ is a Finsler metric, and in this case, $P^{2}$ is a
Finsler metric of constant flag curvature $-1$.

By Theorem 1.4(ia) and (47) we have the following Proposition 6.6 (cf. [10]).

###### Proposition 6.6

For an $n(\geq 3)$-dimensional Berwald spray G with $G^{i}=Py^{i}$, if G is
not of isotropic curvature (or equivalently, $P$ is not a closed 1-form), then
G is not Finsler-metrizable.

###  6.2 Proof of Theorem 1.3

Now we give the proof of Theorem 1.3 as follows.

If the spray G is given by (2) (defined on $\mathcal{C}(U)$), then G is
Finsler-metrizable (on $\mathcal{C}(U)$) since it is easy to check that the
metric $L$ given by (3) induces the spray G. Further, a direct computation
shows that the Ricci curvature $Ric$ of G is equal to $(n-1)L^{*}$ (on
$\mathcal{C}(U)$) since G is given by (2). So $L:=Ric/(n-1)$ is a Riemann
metric and $L$ induces G.

Conversely, suppose that the spray G is Finsler-metrizable (on
$\mathcal{C}(U)$). Since G is a projectively flat Berwald spray of isotropic
curvature, it follows from Proposition 6.4 that G has the form
$G^{i}=\sigma_{0}y^{i}$, where $\sigma_{i}=\sigma_{x^{i}}$ is the differential
of some scalar function $\sigma=\sigma(x)$. Since $Ric\neq 0$ and G is
Finsler-metrizable, $Ric$ is a Finsler metric and $Ric_{;i}=0$ by theorem 1.1
and the fact that a locally projectively flat Finsler metric of isotropic flag
curvature is of constant flag curvature ([2]). Since
$Ric=(n-1)[-(\sigma_{0})^{2}-\sigma_{0;0}]$ (see (46)) and $Ric_{;i}=0$, we
have

$\big{[}(\sigma_{0})^{2}+\sigma_{0;0}\big{]}_{;i}=0.$ (48)

Now we only need to solve the scalar function $\sigma$ from (48). It is easy
to see that (48) is equivalent to

$(\sigma_{j}\sigma_{k}+\sigma_{j;k})_{;i}=0.$ (49)

Now by $G^{i}=\sigma_{0}y^{i}$ we have

$G^{i}_{j}=\sigma_{j}y^{i}+\sigma_{0}\delta^{i}_{j},\ \ \ \
G^{i}_{jk}=\sigma_{j}\delta^{i}_{k}+\sigma_{k}\delta^{i}_{j}.$

For convenience, we put
$\sigma_{ij}:=\sigma_{x^{i}x^{j}},\sigma_{ijk}:=\sigma_{x^{i}x^{j}x^{k}}$. By
a direct computation we obtain

$\displaystyle\sigma_{i;j}$ $\displaystyle=\sigma_{ij}-2\sigma_{i}\sigma_{j},\
\ \ \ \ \sigma_{i}\sigma_{j}+\sigma_{i;j}=\sigma_{ij}-\sigma_{i}\sigma_{j},$
$\displaystyle(\sigma_{j}\sigma_{k}+\sigma_{j;k})_{;i}$
$\displaystyle=\sigma_{ijk}-2(\sigma_{i}\sigma_{jk}+\sigma_{j}\sigma_{ik}+\sigma_{k}\sigma_{ij})+4\sigma_{i}\sigma_{j}\sigma_{k}.$
(50)

Putting $u=e^{-2\sigma}$, it follows from (50) that (49) is equivalent to
$u_{ijk}=0$. So $u$ is a polynomial in ($x^{i})$ of degree two. Thus the
scalar function $\sigma$ satisfying (49) is given by

$\sigma(x)=-\frac{1}{2}\ln|x^{\prime}Ax+\langle B,x\rangle+C|,$ (51)

where ′ denotes the transpose of a matrix, $A$ is a constant symmetric matrix,
$B$ is a constant vector and $C$ is a constant number. By
$G^{i}=\sigma_{0}y^{i}$ and (51), we see that G is locally given by (2).
Meanwhile, by (51), we have

$-(\sigma_{0})^{2}-\sigma_{0;0}=\frac{4(x^{\prime}Ax+\langle
B,x\rangle+C)y^{\prime}Ay-(2x^{\prime}Ay+\langle
B,y\rangle)^{2}}{4(x^{\prime}Ax+\langle B,x\rangle+C)^{2}}.$ (52)

Since $R(=-(\sigma_{0})^{2}-\sigma_{0;0})\neq 0$ by assumption, by Theorem
1.1, $R$ is a metric, which means that $A,B,C$ should satisfy certain
condition such that $L$ given by (3) is a metric.

A simple computation shows that if $det(A)\neq 0$, then $L$ given by (3) is a
metric iff. the constant quantities $A,B,C$ satisfy $4C\neq
B^{\prime}A^{-1}B$. Q.E.D.

###### Remark 6.7

If $-(\sigma_{0})^{2}-\sigma_{0;0}=0$, we can obtain $\sigma=-\ln|\langle
B,x\rangle+C|$. In this case, G has vanishing Riemann curvature, and it can be
locally induced by a Finsler metric $L$. It is clear that $L$ is a Minkowskin
metric.

The above proof and Remark 6.7 have actually given the proof of the following
theorem for the local structure of a locally projectively flat Berwald spray
with constant curvature.

###### Theorem 6.8

Let G be a locally projectively flat Berwald spray on a manifold $M$. Then G
is of constant curvature if and only if G can be locally expressed as (2).

##  7 Examples

As an application of Theorem 1.1, Theorem 1.3 gives the necessary and
sufficient condition for the Berwald spray $G^{i}=(\sigma_{x^{r}}y^{r})y^{i}$
to be Finsler-metrizable. In this section, we are going to give more examples
to support some of the basic theories we introduce in the above sections.

###### Example 7.1

Let ${\bf G}$ be a two-dimensional Berwald spray given by

$G^{1}=f(x^{1})[(y^{1})^{2}-(y^{2})^{2}],\ \ \ \ \ G^{2}=2f(x^{1})y^{1}y^{2},$

where $f(x^{1})$ is a non-constant function. It can be directly verified that
${\bf G}$ is of isotropic curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\frac{1}{2}R_{.k}y^{i}$ satisfying

$R=-2f^{\prime}(x^{1})[(y^{1})^{2}+(y^{2})^{2}],\ \ \ \ R_{;i}=R\omega_{i}$
$\omega_{1}:=\frac{f^{\prime\prime}(x^{1})}{f^{\prime}(x^{1})}-4f(x^{1})=\big{[}\ln|f^{\prime}(x^{1})e^{-4\int
f(x^{1})dx^{1}}|\big{]}_{x^{1}}=[\ln|\lambda|]_{x^{1}},\ \ \ \ \
\omega_{2}:=0.$

Then by Theorem 1.1, ${\bf G}$ is Finsler-metrizable induced by the Riemann
metric

$L:=R/\lambda=-2e^{4\int f(x^{1})dx^{1}}[(y^{1})^{2}+(y^{2})^{2}]$

of isotropic sectional curvature

$\lambda=f^{\prime}(x^{1})e^{-4\int f(x^{1})dx^{1}}.$

###### Example 7.2

Let ${\bf G}$ be a two-dimensional spray given by

$G^{1}=\frac{1}{2r}y^{2}\sqrt{(y^{1})^{2}+(y^{2})^{2}},\ \ \ \ \
G^{2}=-\frac{1}{2r}y^{1}\sqrt{(y^{1})^{2}+(y^{2})^{2}},$

where $r$ is a constant. This spray appears in Example 4.1.3 of [18]. From the
completeness of geodesics of ${\bf G}$ and Hopf-Rinow theorem, it is concluded
that ${\bf G}$ is not globally Finsler-metrizable. But actually, it is even
not locally Finsler-metrizable by Theorem 1.1 (Example 12.4.1 in [18] shows it
is locally projectively Finsler-metrizable).

By a direct computation, ${\bf G}$ is of isotropic curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\frac{1}{2}R_{.k}y^{i}$ satisfying

$R=r^{-2}\big{[}(y^{1})^{2}+(y^{2})^{2}\big{]},\ \ \ \
R_{;1}=r^{-3}y^{2}\sqrt{(y^{1})^{2}+(y^{2})^{2}},\ \ \ \
R_{;2}=-r^{-3}y^{1}\sqrt{(y^{1})^{2}+(y^{2})^{2}}.$

Therefore, by Theorem 1.1, ${\bf G}$ is not locally Finsler-metrizable
anywhere since there exists no closed 1-form $\omega=\omega_{i}(x)dx^{i}$ such
that $R_{;i}=R\omega_{i}$.

###### Example 7.3

Let ${\bf G}_{c}$ be a Berwald spray given by

$G^{i}_{c}=\frac{-|y|^{2}x^{i}+c\langle x,y\rangle y^{i}}{1-|x|^{2}},$

where $c$ is a constant. The spray ${\bf G}_{2}$ is induced by the Riemann
metric $L=4|y|^{2}/(1-|x|^{2})^{2}$ of constant sectional curvature $-1$, and
by computation, we can see that ${\bf G}_{1}$ is just the spray given by
Example 4.1.4 in [18]. ${\bf G}_{c}$ actually is projectively flat although at
present coordinate it is not in the form $Py^{i}$.

By a direct computation, ${\bf G}_{c}$ is of isotropic curvature $R^{i}_{\
k}=R\delta^{i}_{k}-\frac{1}{2}R_{.k}y^{i}$ with

$R=-\frac{\big{[}(c-2)|x|^{2}+c+2\big{]}|y|^{2}-c(c-2)\langle
x,y\rangle^{2}}{(1-|x|^{2})^{2}}.$

Further, we have

$R_{;i}=\frac{2(c-2)}{(1-|x|^{2})^{3}}\Big{\\{}\big{[}(c-1)|x|^{2}+c+1\big{]}(|y|^{2}x^{i}+2\langle
x,y\rangle y^{i})-2c(c-1)\langle x,y\rangle^{2}x^{i}\Big{\\}}.$

(i) By Theorem 1.1, it is easy to see that ${\bf G}_{c}$ is Finsler-metrizable
iff. $c=2$. When $c=2$, we obtain the Riemann metric
$L:=R=-4|y|^{2}/(1-|x|^{2})^{2}$ of constant curvature 1.

(ii) ${\bf G}_{c}$ is of constant curvature iff. $c=2$.

(iii) When $c\neq 2$, ${\bf G}_{c}$ is of isotropic curvature but not of
constant curvature in any dimension. This fact is different from the
Finslerian case.

###### Example 7.4

Let ${\bf G}$ be a spray given by $G^{i}=\sigma_{0}y^{i}$ with
$\sigma_{i}:=\sigma_{x^{i}},\sigma=\sigma(x)$.

(i) ${\bf G}$ is of isotropic curvature for any $\sigma$ but not of constant
curvature by Theorem 6.8 if

$\sigma(x)\neq-\frac{1}{2}\ln|x^{\prime}Ax+\langle B,x\rangle+C|.$

(ii) Taking $A=(\delta_{ij}),B=0,C=0$ in (2) and (3), we have

$G^{i}=-\frac{\langle x,y\rangle}{|x|^{2}}y^{i},\ \ \ \
L=\frac{|x|^{2}|y|^{2}-\langle x,y\rangle^{2}}{|x|^{4}}.$ (53)

In (53), $L$ is not a metric since $(L_{.i.j})$ is degenerate. By Theorem 1.3,
or Theorem 1.1(ii) ($R=L$ is not a Finsler metric), the spray G given by (53)
is not Finsler-metrizable, although this spray G satisfies (24).

###### Example 7.5

Let G be a spray on $R^{3}$ given by ([18], Example 4.1.2)

$G^{1}=0,\ \ G^{2}=x^{1}(y^{1})^{2}+x^{3}(y^{3})^{2},\ \ G^{3}=0.$

G has zero Riemann curvature. Then by Theorem 1.1(i), G is locally Finsler-
metrizable. Actually, if the following function is a Finsler metric

$L(x,y)=\Big{[}f\Big{(}\frac{y^{3}}{y^{1}},(x^{1})^{2}+\frac{y^{2}}{y^{1}}+(x^{3})^{2}\frac{y^{3}}{y^{1}}\Big{)}y^{1}\Big{]}^{2},$

then $L$ induces G on some open set of $R^{3}$, and $F$ should be a locally
Minkowski metric since G is a Berwald spray (with zero Riemann curvature).
Consider a special case with $f(r,s)=\sqrt{r^{2}+s}$ and we obtain

$L(x,y)=(y^{3})^{2}+(x^{1})^{2}(y^{1})^{2}+y^{1}y^{2}+(x^{3})^{2}y^{1}y^{3},$

which is a singular Riemann metric globally defined on $R^{3}$. So the spray G
is globally Finsler-metrizable.

###### Example 7.6

Let G be a two-dimensional spray given by

$G^{1}=-f(x^{1})g^{\prime}(t)(y^{1})^{2},\ \ \
G^{2}=f(x^{1})\big{[}g(t)y^{1}-g^{\prime}(t)y^{2}\big{]}y^{1},\ \ \ \
(t:=y^{2}/y^{1}).$

where $f,g$ are two smooth functions. A direct computation gives $R^{i}_{\
k}=R\delta^{i}_{k}-\tau_{k}y^{i}$ with

$R=0,\ \ \ \tau_{1}=Ay^{2},\ \ \ \tau_{2}=-Ay^{1},\ \ \
A:=2f^{2}(x^{1})g(t)g^{\prime\prime\prime}(t)-f^{\prime}(x^{1})g^{\prime\prime}(t).$

We can choose the two functions $f,g$ satisfying $A\neq 0$. So in this case, G
is not Finsler metrizable by Theorem 3.5(ii).

## References

  * [1] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, Kluwer Academic Publishers, 1993.
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Guojun Yang
Department of Mathematics
Sichuan University
Chengdu 610064, P. R. China
<EMAIL_ADDRESS>