On Sprays of Scalar Curvature and Metrizability

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 $n\ge 3$. Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler metrizable. Based on this 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.

1 Introduction

Spray geometry studies the properties of sprays on a manifold, and it 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 [8, 13, 20]. A spray $\textbf{G}$ 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 $\textbf{G}$ 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 $\textbf{G}$ seeks for a Finsler metric whose spray is just G, or whose geodesics coincide with that of $\textbf{G}$. 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 [14], 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, 7]. In [15], 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 [20], the present author constructs a class of sprays whose metrizable and non-metrizable conditions are completely determined respectively. Inspired by the sprays constructed in [20], Elgendi and 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 [8]. In [5], Bucataru and Muzsnay characterize metrizable sprays with non-zero Ricci constant in dimension greater than 2, and further they give necessary and sufficient conditions for sprays of scalar curvature and non-zero Ricci curvature to be metrizable in [6]. In [12], Li and 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 [13], Li, Mo, and 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 $\textbf{G}$ is said to be of scalar curvature if its Riemann curvature $R^i_{\ k}$ satisfies

$$\begin{aligned} R^i_{\ k}=R\delta ^i_k-\tau _ky^i, \end{aligned}$$

(1)

where $R=R(x,y)$ and $\tau _k=\tau _k(x,y)$ are some homogeneous functions [16]. In [5], the condition $R_{.i}=2\tau _i$ is considered for a spray satisfying (1). If in (1) there holds $R_{.i}=2\tau _i$, then $\textbf{G}$ is said to be of isotropic curvature [12]. A spray $\textbf{G}$ is said to be of constant curvature, a new notion we introduce in this paper, if $\textbf{G}$ satisfies (1) with

$$\begin{aligned} \tau _{i;k}=0 \ (\ \Leftrightarrow \ R=\tau _k=0,\ or \ R_{;i}=0). \end{aligned}$$

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 $n(\ge 3)$-dimensional spray of isotropic curvature is not necessarily of constant curvature, which is different from the Finslerian case.

Theorem 1.1

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

• (i) If the spray manifold $(\textbf{G},M)$ is analytical with $R=0$, then $\textbf{G}$ is locally Finsler metrizable.

• (ii) If $R\ne 0$ and R is not a Finsler metric, then $\textbf{G}$ is not Finsler metrizable.

• (iii) If R is a Finsler metric, then $\textbf{G}$ 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

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

  • (iiib) if $\omega =0$, then $\textbf{G}$ is induced by the Finsler metric R with the flag curvature $\textbf{K}=1$.

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

Theorem 1.2

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

Note that Theorem 1.1 for $R\ne 0$ can be concluded from Theorem 3.1 in [6], and Theorem 1.1 (iiia) has been proved in [5]. In Theorem 1.1, if G is metrizable, the corresponding Finsler metric is easily obtained if $Ric\ne 0$. 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. [12, 17]). As applications of Theorems 1.1 and 1.2, we generalize a result in [20] (Theorem 4.10 below), and make a check on some spays whether they are Finsler metrizable or not (see Sect. 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 $Ric\ne 0$ (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

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

(2)

where $A\ne 0$ 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 $\mathcal {C}(U)$),

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

(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

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

(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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$, the quantities R and $\tau _k$ are closely related (Proposition 3.1 below). Now we consider the following condition for R and $\tau _k$,

$$\begin{aligned} R_{.i}-2\tau _i=\omega _{i0},\quad (\omega _{i0}:=\omega _{ir}y^r), \end{aligned}$$

(5)

where $\omega =\omega _{ij}(x)dx^i\wedge dx^j$ is a 2-form. 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 [18]. For a spray satisfying (5), we have the following theorem.

Theorem 1.4

Let G be an n-dimensional spray of scalar curvature $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ satisfying (5). Suppose that G is induced by a Finsler metric L.

• (i) For $n\ge 3$, L is of constant flag curvature with $\omega =0$ [9].

• (ii) For $n=2$, the flag curvature $\lambda$ of L satisfies $\lambda ''(\theta )+\epsilon I(\theta )\lambda '(\theta )=0$ on each tangent space, where $\epsilon =\pm 1$ is the sign of L, and $\theta$ is the Landsberg angle.

  • (iia) If L is a Riemann metric, or regular Finsler metric, then L is of isotropic flag curvature ($\lambda =\lambda (x)$) with $\omega =0$.

  • (iib) If L has constant main scalar, then (5) is satisfied with $\omega$ not necessarily zero.

Theorem 1.4(i) has essentially been proved in [9]. Starting from (5), we are also going to give a little different version of proof from that in [9]. If a spray G satisfies (5) with $\omega \ne 0$ and $n\ge 3$, then the spray G is not Finsler metrizable by Theorem 1.4(i). If a two-dimensional spray $\textbf{G}$ satisfies (5) with $\omega \ne 0$, then $\textbf{G}$ cannot be induced by a Riemann metric or a regular Finsler metric by Theorem 1.4(iia), but Theorem 1.4(iib) shows that there are such sprays which can be induced by a singular Finsler metric.

2 Preliminaries

Let M be an n-dimensional manifold. A conical region $\mathcal {C}=\mathcal {C}(M)$ of $TM\setminus \{0\}$ means $\mathcal {C}_x:=\mathcal {C}\cap T_xM\setminus \{0\}$ 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 $\textbf{G}$ 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:

$$\begin{aligned} \textbf{G}=y^i\frac{\partial }{\partial x^i}-2G^i\frac{\partial }{\partial y^i}, \end{aligned}$$

where $G^i=G^i(x,y)$ are local functions satisfying $G^i(x,\lambda y)=\lambda ^2G^i(x,y)$ 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

$$\begin{aligned} R^i_{\ k}:=2\partial _k G^i-y^j(\partial _j\dot{\partial _k}G^i)+2G^j(\dot{\partial _j}\dot{\partial _k}G^i)-(\dot{\partial _j}G^i)(\dot{\partial _k}G^j), \end{aligned}$$

(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 $\textbf{G}$ is said to be R-flat if $R^i_{\ k}=0$. In [18], Shen defines a non-Riemannian quantity called $\chi$-curvature $\chi =\chi _idx^i$ expressed as follows:

$$\begin{aligned} \chi _i:=2R^m_{\ i.m}+R^m_{\ m.i}. \end{aligned}$$

(7)

Plugging (1) into (7) yields

$$\begin{aligned} \chi _i=(n+1)(R_{.i}-2\tau _i). \end{aligned}$$

(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 $G^i_{hjk}:={\dot{\partial }}_h{\dot{\partial }}_j{\dot{\partial }}_kG^i=0$. 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 $(\textbf{G}, M)$, Berwald connection is usually defined as a linear connection on the pull-back $\pi ^*TM$ ($\pi : TM\rightarrow M$ the natural projection) over the base manifold M. The Berwald connection is defined by

$$\begin{aligned} D(\partial _i)=(G^k_{ir}dx^r)\partial _k,\quad (G^k_{ir}:={\dot{\partial }}_r{\dot{\partial }}_iG^k), \end{aligned}$$

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

$$\begin{aligned} T_{i;j}=\delta _jT_i-T_rG^r_{ij},\quad T_{i.j}={\dot{\partial }}_jT_i,\quad (\delta _i:=\partial _i-G^r_i{\dot{\partial }}_r). \end{aligned}$$

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

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

where $T_{ij}-(i/j)$ means $T_{ij}-T_{ji}$, and $T_0$ is defined by $T_0:=T_ry^r$, 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 $L(\ne 0)$ on a manifold M as follows (cf. [16]): (i) for any $x\in M$, $L_x$ is defined on a conical region of $T_xM\setminus \{0\}$ and L is $C^{\infty }$; (ii) L is positively homogeneous of degree two; (iii) the fundamental metric tensor $g_{ij}:=(\frac{1}{2}L)_{y^iy^j}$ 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 called a Finsler norm function which is positively homogeneous of degree one.

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

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

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

$$\begin{aligned} R^i_{\ k}=K(L\delta ^i_k-y^iy_k), \end{aligned}$$

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 $n\ge 3$.

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

$$\begin{aligned}{} & {} Lg_{ij}=y_iy_j+\epsilon Y_iY_j,\nonumber \\{} & {} \quad (Y^1,Y^2)=\big (\sqrt{\epsilon g}\big )^{-1}(-y_2,y_1),\quad (g:=det(g_{ij})),\nonumber \\{} & {} LY_{i.j}=y_jY_i-y_iY_j+\epsilon IY_iY_j,\quad L^2C_{ijk}=IY_iY_jY_k, \end{aligned}$$

(9)

where (y, Y) with $y=(y^1,y^2),Y=(Y^1,Y^2)$ is called the Berwald frame, $C_{ijk}$ is the Cartan tensor and I is the main scalar. The system $L_{.i}=2y_i,L\theta _{.i}=Y_i$ is integrable. It defines the so-called Landsberg angle $\theta$, which is the arc-length parameter of the indicatrix $S_xM:=\{y\in T_xM|F(x,y)=1\}$ with respect to the Riemann metric $ds^2=g_{ij}dy^i\otimes dy^j$ on the Minkowski plane $(M_x,F_x)$ if L is positively definite. When $L=F^2$, the Berwald frame is denoted by $(\ell ,m)$ with $\ell :=y/F,m:=Y/F$ (see [1]).

Lemma 2.1

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

$$\begin{aligned} L\lambda _{.i}=\lambda '(\theta )Y_i. \end{aligned}$$

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 $n(\ge 3)$-dimensional spray of scalar curvature $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$. Then there holds

$$\begin{aligned} R_{.i;0}-3R_{;i}+\tau _{i;0}=0. \end{aligned}$$

(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

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

we have

$$\begin{aligned} R^{m}_{\ k;m}+R^{m}_{\ km;0}-R^{m}_{\ m;k}=0. \end{aligned}$$

(11)

Since $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$, a direct computation gives

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

(12)

By $\tau _0=R$, we obtain

$$\begin{aligned} \tau _{m.k;0}y^m=R_{.k;0}-\tau _{k;0}. \end{aligned}$$

(13)

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

$$\begin{aligned} (n-2)(R_{.i;0}-3R_{;i}+\tau _{i;0})=0, \end{aligned}$$

which gives the proof. $\square$

The isotropic case of Proposition 3.1 is given by Z. Shen in [19]. 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:

$$\begin{aligned} L\lambda _{.i;0}+3\lambda _{;0}y_i-3L\lambda _{;i}=0. \end{aligned}$$

(14)

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

$$\begin{aligned} \theta _{;i}-\theta _{.i;0}+(F\sigma _{;r}+2\theta _{;r})h^r_i=0,\ \ \ (h^i_j:=\delta ^i_j-\ell ^i\ell _j). \end{aligned}$$

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 $n\ge 3$ 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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$, and T be a homogeneous scalar function of degree p satisfying $T_{;i}=0$. Then we have

$$\begin{aligned} RT_{.k}=pT\tau _k. \end{aligned}$$

(15)

• (i) If $p=0$ and $R\ne 0$, then T is a constant.

• (ii) If G is of isotropic curvature with $R\ne 0$, then there holds

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

  (16)

• (iii) If G is induced by a Finsler metric L of non-zero flag curvature, then there holds

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

  (17)

Proof

By a Ricci identity and $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$, we have

$$\begin{aligned} 0=y^j(T_{;i;j}-T_{;j;i})=T_{.r}R^r_{\ i} =RT_{.k}-y^rT_{.r}\tau _k. \end{aligned}$$

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

• (i) If $p=0$ and $R\ne 0$, then $T_{.i}=0$ by (15). So T is independent of y. Using $T_{;i}=0$ again, we obtain $T=constant$.

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

  $$\begin{aligned} \frac{T_{.i}}{T}=\frac{p}{2}\frac{R_{.i}}{R}\ \ \Bigg (\Leftrightarrow \bigg (\frac{T}{|R|^{\frac{1}{2}p}}\bigg )_{.i}=0\Bigg ). \end{aligned}$$

• (iii) The original version is given in [1]. Let $\lambda \ne 0$ 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

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

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

$\square$

3.2 Some Basic Properties

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

$$\begin{aligned} H^{\ i}_{h\ jk}=\frac{1}{3}\big [R_{.j.h}\delta ^i_k-\tau _{k.j.h}y^i-\tau _{k.j}\delta ^i_h-\tau _{k.h}\delta ^i_j -(j/k)\big ]. \end{aligned}$$

(18)

By (18) we obtain

$$\begin{aligned}{} & {} H_{ij}-H_{ji}=\frac{1}{3}(n+1)(\tau _{j.i}-\tau _{i.j}), \end{aligned}$$

(19)

$$\begin{aligned}{} & {} H_{0i}=\frac{1}{3}\big [(n-2)R_{.i}+(n+1)\tau _i\big ],\quad H_{0i}=\frac{1}{3}\big [(2n-1)R_{.i}-(n+1)\tau _i\big ] \end{aligned}$$

(20)

$$\begin{aligned}{} & {} H_{0i}-H_{i0}=-\frac{1}{3}(n+1)(R_{.i}-2\tau _i), \end{aligned}$$

(21)

$$\begin{aligned}{} & {} H_i=\frac{1}{3}(n+1)(R_{.i}+\tau _i). \end{aligned}$$

(22)

Proposition 3.3

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

• (i) If G is of isotropic curvature, then $H_{ij}=H_{ji}$, $H_{i0}=H_{0i}$ and $H_i$ is proportional to $\tau _i$.

• (ii) If $H_{ij}=H_{ji}$, or $R_{i0}=H_{0i}$, or $H_i$ is proportional to $\tau _i$ with $R\ne 0$, 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

$$\begin{aligned} R_{.i}=\tau _i+\tau _{m.i}y^m=\tau _i+\tau _{i.m}y^m=2\tau _i. \end{aligned}$$

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 $R\ne 0$, 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 $R\ne 0$ by assumption, we have $\lambda =2$, and thus, $R_{.i}=2\tau _i$. $\square$

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. $\square$

3.3 Metrizability

Let G be a spray of scalar curvature $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$. 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 $\lambda =\lambda (x,y)$. Then if $R=0$, then $\tau _k=0$; if $R\ne 0$, then we can easily obtain

$$\begin{aligned} \bigg (\frac{\tau _i}{R}\bigg )_{;j}= 0, \quad \bigg (\frac{\tau _i}{R}\bigg )_{.j}= \bigg (\frac{\tau _j}{R}\bigg )_{.i}. \end{aligned}$$

(23)

In [16], there are some two-dimensional sprays (of scalar curvature) satisfying the condition $R=0,\tau _k\ne 0$ (also see Example 7.6 below). It is easy to check that a spray of constant curvature ($R\ne 0$) always satisfies (23). Example 7.4 below shows that the condition (23) is not sufficient for a spray of scalar curvature to be Finsler metrizable.

The condition (23) is almost sufficient for a spray of scalar curvature with $R\ne 0$ to be Finsler metrizable. In [6], Bucataru and Muzsnay show that (23) and the following non-degenerate condition:

$$\begin{aligned} \det \Big (\big (\frac{\tau _i}{R}\big )_{.j}+\frac{2\tau _i\tau _j}{R^2}\Big )\ne 0, \end{aligned}$$

are necessary and sufficient for a spray of scalar curvature ($R\ne 0$) to be Finsler metrizable.

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 $L=F^2$. Put $R=\lambda L, \tau _i=\lambda y_i=\lambda F\ell _i$, where $\lambda$ is the flag curvature of F. By (5) we have $(R_{.i}-2\tau _i)_{.k}+(i/k)=0$, which just is

$$\begin{aligned} F\lambda _{.i.k}+\lambda _{.k}\ell _i+\lambda _{.i}\ell _k=0,\quad (or \ L\lambda _{.i.k}+\lambda _{.k}y_i+\lambda _{.i}y_k=0). \end{aligned}$$

(24)

Differentiating (24) by $y^j$ gives

$$\begin{aligned} F\lambda _{.i.j.k}+\lambda _{.i.k}\ell _j+\lambda _{.j.k}\ell _i+F^{-1}\lambda _{.k}h_{ij}+ \lambda _{.i.j}\ell _k+F^{-1}\lambda _{.i}h_{jk}=0, \end{aligned}$$

(25)

where $h_{ij}:=g_{ij}-\ell _i\ell _j$. Interchanging i, j in (25) 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) If $n>2$, then we have $\lambda _{.i}=0$. So $\lambda$ is constant by Schur’s Theorem.

• (ii) If $n=2$, let (y, Y) be the Berwald frame and $\theta$ be the Landsberg angle. By Lemma 2.1, we put

  $$\begin{aligned} L\lambda _{.i}=\eta Y_i, \quad \eta (\theta )=\lambda '(\theta ). \end{aligned}$$

  (26)

  Differentiating (26) by $y^j$ and using (9) we obtain

  $$\begin{aligned} 2y_j\lambda _{.i}+L\lambda _{.i.j}=\eta _{.j}Y_i-\eta L^{-1}(y_jY_i-y_iY_j+\epsilon IY_iY_j), \end{aligned}$$

  which, by (26) again, is reduced to

  $$\begin{aligned} y_i\lambda _{.j}+y_j\lambda _{.i}+L\lambda _{.i.j} =\eta _{.j}Y_i+\epsilon \eta L^{-1}IY_iY_j. \end{aligned}$$

  (27)

  Then by (24) we have

  $$\begin{aligned} L\eta _{.j}+\epsilon \eta IY_j=0,\ \ \ \textrm{or}\ \ \ \eta '+\epsilon \eta I=0,\ \ \ \textrm{or}\ \ \ \lambda ''+\epsilon \lambda ' I=0. \end{aligned}$$

  • (iia) If L is regular, then $\epsilon =1$ and $T_xM$ is compact. There is a $\theta _0$ such that $\lambda '(\theta _0)=0$. So we get $\eta (\theta )=\lambda '(\theta )\equiv 0$. Thus, we have $\lambda _{.i}=0$ from (26).

  • (iib) 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]):

    $$\begin{aligned}{} & {} L=c\beta ^{2s}\gamma ^{2(1-s)},\quad (s=s(x)\ne 0,\quad s(x)\ne 1), \end{aligned}$$

    (28)
    $$\begin{aligned}{} & {} L=c\beta ^2e^{\frac{2\gamma }{\beta }},\end{aligned}$$

    (29)
    $$\begin{aligned}{} & {} L=c(\beta ^2+\gamma ^2)e^{2r\cdot arctan(\frac{\beta }{\gamma })},\quad r=r(x), \end{aligned}$$

    (30)

    where $\beta =p_i(x)y^i$ and $\gamma =q_i(x)y^i$ are two independent 1-forms, and $c\ne 0$ is constant. The main scalar $I=I(x)$ is given respectively by

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

    In the following, we assume that the Finsler metric L is of constant main scalar, that is, s in (28) and r in (30) is 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 (28)–(30) respectively taking $c=1$ there for convenience). Then we can verify (5) from $R=\lambda L$ and $\tau _k=\lambda y_k$. If L is given by (28), then (5) holds with

    $$\begin{aligned} \omega _{12}=\frac{1-2s}{s(1-s)}\frac{(p^2qq_{12}-pq^2p_{12}+q^2p_1p_2-p^2q_1q_2)s-p^2qq_{12}+p^2q_1q_2}{p^2q^2}. \end{aligned}$$

    If L is given by (29), then (5) holds with

    $$\begin{aligned} \omega _{12}=-\frac{2(p^2qp_{22}+pq^2p_{12}-p^2qq_{12}+p^2q_1q_2-q^2p_1p_2-p^2p_2q_2)}{p^2q^2}. \end{aligned}$$

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

    $$\begin{aligned}{} & {} \omega _{12}\\ {}{} & {} =\frac{2r}{1+r^2}\frac{(p^2qq_{12}-pq^2p_{12}-p^2q_1q_2+q^2p_1p_2)r+p^2qp_{22}+pq^2q_{11}-q^2p_1q_1-p^2p_2q_2}{p^2q^2}. \end{aligned}$$

    So generally, $\omega _{12}$ in the above is not zero.

$\square$

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

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

Proof

It follows from $L_{.i;0}-L_{;i}=-2G^rL_{.r.i}+L_{x^ry^i}y^r-L_{x^i}.$ $\square$

Lemma 4.2

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

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

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 $L_{;i}=0$. $\square$

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 [3, 10].

Let B be a vector bundle over an n-dimensional manifold M, and denote by $J_kB$ the bundles of kth-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 $p_0(P): J_kB_1\rightarrow B_2$, a morphism of vector bundles over M. We also denote by $p_l(P): J_{k+l}B_1\rightarrow J_lB_2$ the morphisms of vector bundles over M, which is called the lth-order jet prolongation of P. Let $R_{k+l,x}(P):=Ker p_l(P)_x$ 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 $l\ge 0$ and the projection $\pi _{k+l,x}(P):R_{k+l,x}(P)\rightarrow R_{k+l-1,x}(P)$ is onto for all $l\ge 1$.

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 lth-order prolongation of P. Define

$$\begin{aligned} g_{k,x}(P):{} & {} \quad =Ker\ \sigma _{k,x}(P),\\ g_{k,x}(P)_{e_1\ldots e_j}:{} & {} \quad =\{A\in g_{k,x}(P)|i_{e_1}A=\ldots =i_{e_j}A=0\},\ \ \ 1\le j\le n, \end{aligned}$$

where $\{e_1,\ldots , e_n\}$ is a basis of $T_xM$. Such a basis is said to be quasi-regular if it satisfies

$$\begin{aligned} dim\ g_{k+1,x}(P)= dim\ g_{k,x}(P)+\sum ^n_{j=1}dim\ g_{k,x}(P)_{e_1\cdots e_j}. \end{aligned}$$

The symbol $\sigma _k(P)$ is said to be involutive at $x\in M$ if there exists a quasi-regular basis of $T_xM$. For the proof of Theorem 1.1(i), we need the following theorem and lemma ([10]).

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.

$$\begin{aligned} P(s)_x=0\Longrightarrow (DP(s))_x=\sigma _{k+1}(P)(A): \end{aligned}$$

for some $A\in S^{k+1}(T^*_xM)\otimes B_1$, 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 $T^*_vTM$ denote the subbundle of $T^*TM$, in which, if $\omega \in T^*_vTM$, then $\omega$ can be written locally as $\omega =\omega _i(x,y)dx^i$, and $\wedge ^2T^*_vTM$ 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

$$\begin{aligned} B_1:= T^*_vTM,\ \ \ B_2:=T^*_vTM\oplus \wedge ^2 T^*_vTM\oplus (T^*_vTM\otimes T^*_vTM). \end{aligned}$$

We define a linear partial differential operator $P:Sec(B_1)\rightarrow Sec(B_2)$ in component form as follows

$$\begin{aligned} P(\theta _i)=(\theta _{i.0},\theta _{i.j}-\theta _{j.i}, \theta _{i;j}). \end{aligned}$$

(31)

Lemma 4.6

For the operator P in (31), the symbol $\sigma _1(P)$ is involutive.

Proof

We are going to prove that $\{{\dot{\partial }}_1,\ldots ,{\dot{\partial }}_n,\delta _1,\ldots ,\delta _n\}$ 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

$$\begin{aligned} \sigma _1(P)A=(A_{{\underline{0}}i},A_{{\underline{j}}i}-A_{{\underline{i}}j},A_{ji})\in B_2. \end{aligned}$$

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

$$\begin{aligned} dim(g_1(P))=\frac{(n-1)n}{2}. \end{aligned}$$

(32)

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 follows:

$$\begin{aligned}{} & {} \quad B=(B_{ijk},\ B_{{\underline{i}}jk},\ B_{i{\underline{j}}k},\ B_{{\underline{i}}{\underline{j}}k}) \\{} & {} \big (B_{ijk}=B_{jik},B_{{\underline{i}}jk}=B_{j{\underline{i}}k},B_{{\underline{i}}{\underline{j}}k}=B_{{\underline{j}}{\underline{i}}k}\big ). \end{aligned}$$

By definition, we have

$$\begin{aligned} \sigma _2(P)B=(B_{i{\underline{0}}k},B_{{\underline{i}}{\underline{0}}k};B_{i{\underline{j}}k}-B_{i{\underline{k}}j}, B_{{\underline{i}}{\underline{j}}k}-B_{{\underline{i}}{\underline{k}}j};B_{ijk},B_{{\underline{i}}jk}). \end{aligned}$$

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

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

to $dim(g_2(P))$. So altogether, we have

$$\begin{aligned} dim(g_2(P))=\sum ^{n-2}_{k=0}\frac{(n-k-1)(n-k)}{2}. \end{aligned}$$

(33)

Next we verify under the basis $\{{\dot{\partial }}_1,\ldots ,{\dot{\partial }}_n,\delta _1,\ldots ,\delta _n\}$ at a point $(x,y)\in TM$,

$$\begin{aligned} dim(g_2(P))=dim\ g_{1}(P)+\sum ^n_{j=1}dim\ g_{1}(P)_{{\dot{\partial }}_1\cdots {\dot{\partial }}_j}+\sum ^n_{j=1}dim\ g_{1}(P)_{{\dot{\partial }}_1\cdots {\dot{\partial }}_n\delta _1\cdots \delta _j}.\nonumber \\ \end{aligned}$$

(34)

By a direct computation, we see

$$\begin{aligned} dim\ g_{1}(P)_{{\dot{\partial }}_1\ldots {\dot{\partial }}_j}=\frac{(n-j-1)(n-j)}{2},\ \ \ \ dim\ g_{1}(P)_{{\dot{\partial }}_1\ldots {\dot{\partial }}_n\delta _1\ldots \delta _j} =0,\ \ (1\le j\le n). \end{aligned}$$

Plugging them into (34) and using (32), (33), we see that both sides of (34) are equal. Therefore, $\sigma _1(P)$ is involutive. $\square$

Lemma 4.7

For the operator P in (31), a first-order solution of $P(\theta )=0$ can be lifted into a second-order solution iff.

$$\begin{aligned} \theta _rH^{\ r}_{i\ jk}+\theta _{i.r}R^r_{\ jk}=(\theta _{r}R^r_{\ jk})_{.i}=0. \end{aligned}$$

(35)

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 (35):

$$\begin{aligned} 0=\theta _{i;j;k}-\theta _{i;k;j}=-\theta _rH^{\ r}_{i\ jk}-\theta _{i.r}R^r_{jk}=-(\theta _{r}R^r_{\ jk})_{.i}. \end{aligned}$$

Conversely, suppose that (35) holds for a $\theta$ satisfying $P(\theta )=0$ with $\theta _0\ne 0$. 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

$$\begin{aligned} DP(\theta _i)=(\theta _{i.0;j},\ \theta _{i.0.j},\ \ \theta _{i.j;k}-\theta _{j.i;k},\ \theta _{i.j.k}-\theta _{j.i.k},\ \ \theta _{i;j;k},\ \theta _{i;j.k}), \end{aligned}$$

(36)

where $\theta$ ($\theta _0\ne 0$) satisfies $P(\theta )=0$ at a point $w=(x,y)\in TM$, that is, at the point w there holds

$$\begin{aligned} \theta _{i.0}=0,\ \ \ \ \theta _{i.j}=\theta _{j.i},\ \ \ \ \theta _{i;j}=0. \end{aligned}$$

(37)

Next we are going to prove that there is an $A_{\alpha \beta i}\in S^2(T^*_wTM)\otimes B_1$ satisfying

$$\begin{aligned} DP(\theta _i)=\sigma _2(A_{\alpha \beta i})=(A_{\alpha {\underline{r}}i}y^r,\ \ A_{\alpha {\underline{j}}i}-A_{\alpha {\underline{i}}j},\ \ A_{\alpha ji}). \end{aligned}$$

(38)

By (36), we see that (38) is equivalent to

$$\begin{aligned} \begin{aligned}&{\textcircled {1}}\ A_{j{\underline{r}}i}y^r=\theta _{i.r;j}y^r,\qquad \qquad \qquad \qquad \qquad {\textcircled {2}}\ A_{{\underline{j}}{\underline{r}}i}y^r=\theta _{i.0.j}=\theta _{i.r.j}y^r+\theta _{i.j},\\&{\textcircled {3}}\ A_{k{\underline{j}}i}-A_{k{\underline{i}}j}=\theta _{i.j;k}-\theta _{j.i;k},\qquad \quad \,\,\,\, \,\,{\textcircled {4}}\ A_{{\underline{k}}{\underline{j}}i}-A_{{\underline{k}}{\underline{i}}j}=\theta _{i.j.k}-\theta _{j.i.k},\\&{\textcircled {5}}\ A_{kji}=\theta _{i;j;k},\qquad \qquad \qquad \qquad \qquad \qquad \,\,\, {\textcircled {6}}\ A_{{\underline{k}}ji}=\theta _{i;j.k}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned}{} & {} A_{{\underline{j}}{\underline{k}}i}=\theta _{i.k.j}+\theta _0^{-1}\big [\theta _{i.j}\theta _k+(i,j,k)\big ],\\{} & {} A_{j{\underline{k}}i}=\theta _{i.k;j}-\theta _rG^r_{ijk},\ \ \ \ \ A_{{\underline{k}}ji}=\theta _{i;j.k},\\{} & {} A_{kji}=\theta _{i;j;k}. \end{aligned}$$

Firstly, $A_{{\underline{j}}{\underline{k}}i}$ is symmetric in j, k by $\theta _{i.j}=\theta _{j.i}$ in (37) and it also satisfies $\textcircled {2}\textcircled {4}$ by $\theta _{i.j}=\theta _{j.i}, \theta _{i.0}=0$ in (37). Next, by a Ricci identity of Berwald connection, we see that

$$\begin{aligned} A_{{\underline{k}}ji}-A_{j{\underline{k}}i}=\theta _{i;j.k}-\theta _{i.k;j}+\theta _rG^r_{ijk}=0, \end{aligned}$$

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 $\textcircled {1}\textcircled {3}\textcircled {6}$. Finally, $A_{kji}$ is symmetric in j, k, and satisfies $\textcircled {5}$ by a Ricci identity of Berwald connection and (35). So (38) holds. This finishes the proof of the lemma. $\square$

Now in Theorem 1.1(i), we have $R=0$. So (35) 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 $J_1B_1$ 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 $\theta _{i{\underline{0}}}^0=0,\theta ^0_{i{\underline{j}}}=\theta ^0_{j{\underline{i}}},\theta ^0_{ij}-\theta _r^0G^r_i=0$. Further, we let the initial data satisfy $\theta _i^0y^i_0>0$ 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 $F(y)\ne 0$ 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). $\square$

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. [15] for a more general discussion).

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

(ii) Assume that $R\ne 0$ is not a Finsler metric. If G is Finsler metrizable induced by a Finsler metric F, then F is of isotropic curvature $\lambda \ne 0$. 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 $\lambda =\lambda (x)\ne 0$. By $R=\lambda L$, we have $R_{;i}=\lambda _{;i}L$. Thus, we obtain

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

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 $\lambda =\lambda (x)\ne 0$ 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 Finsler metric $L=R$ is of constant flag curvature $\lambda =1$. If $n\ge 3$, then the Finsler metric L is of constant flag curvature $\lambda$ by Schur’s theorem, which gives $R_{;i}=0$. $\square$

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 ${\bar{\varvec{G}}}$ 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

$$\begin{aligned} R^i_{\ k}=\lambda (F^2\delta ^i_k-FF_{.k}y^i). \end{aligned}$$

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

$$\begin{aligned} \bar{R}^i_{\ k}=\bar{R}\delta ^i_k-{\bar{\tau }}_ky^i,\ \ \ \ \big (\bar{R}:=(\lambda +c^2)F^2,\ \ \ {\bar{\tau }}_k:=(\lambda +c^2)FF_{.k}\big ). \end{aligned}$$

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

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

Assume that ${\varvec{\bar{G}}}$ is Finsler metrizable. If $c(\lambda +c^2)\ne 0$, 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 ${\varvec{\bar{G}}}$ is induced by F. If $\lambda +c^2= 0$, then ${\varvec{\bar{G}}}$ has zero Riemann curvature. So ${\varvec{\bar{G}}}$ is (locally) Finsler metrizable by Theorem 1.1(i). $\square$

Theorem 4.10 is a generalization of a result in [20], where we have an additional 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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ 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:

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

• (ii) A spray of constant curvature must be of isotropic curvature.

• (iii) A Finsler metric is of constant flag curvature iff. its spray is of constant curvature.

• (iv) An n-dimensional spray of isotropic curvature is not necessarily of constant curvature even for $n\ge 3$.

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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$. Then we have

$$\begin{aligned}{} & {} R_{;i}=0\ (R\ne 0)\ \Longrightarrow \ R_{.i}=2\tau _i,\end{aligned}$$

(39)

$$\begin{aligned}{} & {} \tau _{i;k}=0\ \Longrightarrow \ R=\tau _k=0\ \ or \ \ R_{;i}=0\ (R\ne 0), \end{aligned}$$

(40)

$$\begin{aligned}{} & {} R_{;i}=0\ (R\ne 0)\ \Longrightarrow \ \tau _{i;k}=0. \end{aligned}$$

(41)

Proof

Assume $R_{;i}=0 (R\ne 0)$. 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 $R\ne 0$, which gives the proof of (39).

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

$$\begin{aligned} 0=y^j(\tau _{i;j;k}-\tau _{i;k;j})=y^j(-\tau _rH^{\ r}_{i\ jk}-\tau _{i.r}R^r_{\ jk})=-\tau _rH^{\ r}_{i\ 0k}-\tau _{i.r}R^r_{\ k}. \end{aligned}$$

(42)

Now by $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ we have

$$\begin{aligned} H^{\ r}_{i\ jk}=\frac{1}{3}\big [R_{.j.i}\delta ^r_k-\tau _{k.j.i}y^r-\tau _{k.j}\delta ^r_i-\tau _{k.i}\delta ^r_j -(j/k)\big ]. \end{aligned}$$

(43)

Plugging (43), $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$, $\tau _{i;j}=0$ and $R=0$ into (42) we obtain $\tau _i\tau _k=0$, which gives $\tau _i=0$. This gives the proof of (40).

Assume $R_{;i}=0 (R\ne 0)$. By (39) 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 (41). $\square$

Proof of Theorem 5.2

• (i) Assume that G is of constant curvature. Then we have $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ with $\tau _{i;k}=0$ by definition. By (40) we immediately obtain the desired conclusion since $Ric=(n-1)R$. Conversely, let the Riemann curvature be zero or the Ricci curvature Ric satisfy $Ric_{;i}=0 (Ric\ne 0)$. If $R^i_{\ k}=0$, then $\tau _i=0$ and so $\tau _{i;k}=0$. If $Ric_{;i}=0 (Ric\ne 0)$, then we have $R_{;i}=0\ (R\ne 0)$. So by (39) 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 (40) and (39), 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 $R^i_{\ k}=\lambda (L\delta ^i_k-y_ky^i)$. So its spray is of scalar curvature $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ 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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ 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 $\lambda \ne 0$, 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.

$\square$

Proof of Theorem 1.2

The spray G has the Riemann curvature $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ 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 $Ric\ne 0$.

Conversely, if $Ric=0$, then by Theorem 1.1(i), G is (locally) Finsler metrizable. If $Ric\ne 0$, 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. $\square$

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 $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ with

$$\begin{aligned} R=P^2-P_{x^r}y^r,\ \ \tau _k=PP_{y^k}+P_{x^ry^k}y^r-2P_{x^k}. \end{aligned}$$

(44)

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 (44) becomes

$$\begin{aligned} R=-P^2-P_{;0},\ \ \ \ \tau _k=P_{k;0}-2P_{;k}-PP_k, \ \ \ (P_k:=P_{.k}). \end{aligned}$$

(45)

Lemma 6.2

In (45), $R=0$ 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 $P_{;k}+PP_k=0$. $\square$

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 [12, 17]). By (45) we can easily obtain the following lemma.

Lemma 6.3

Let $G^i=Py^i$ be a spray with $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$. Then we have

$$\begin{aligned} R_{.i}-2\tau _i=3(P_{;i}-P_{i;0}). \end{aligned}$$

(46)

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 $G^i=Py^i$ ($P\ne 0$) 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(i) and (46) we have the following Proposition 6.6 (cf. [13]).

Proposition 6.6

For an $n(\ge 3)$-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 $G^i=\sigma _0y^i$, where $\sigma _i=\sigma _{x^i}$ is the differential of some scalar function $\sigma =\sigma (x)$. Since $Ric\ne 0$ 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 (45)) and $Ric_{;i}=0$, we have

$$\begin{aligned} \big [(\sigma _0)^2+\sigma _{0;0}\big ]_{;i}=0. \end{aligned}$$

(47)

Now we only need to solve the scalar function $\sigma$ from (47). It is easy to see that (47) is equivalent to

$$\begin{aligned} (\sigma _j\sigma _k+\sigma _{j;k})_{;i}=0. \end{aligned}$$

(48)

Now by $G^i=\sigma _0y^i$ we have

$$\begin{aligned} G^i_j=\sigma _jy^i+\sigma _0\delta ^i_j,\ \ \ \ G^i_{jk}=\sigma _j\delta ^i_k+\sigma _k\delta ^i_j. \end{aligned}$$

For convenience, we put $\sigma _{ij}:=\sigma _{x^ix^j},\sigma _{ijk}:=\sigma _{x^ix^jx^k}$. By a direct computation we obtain

$$\begin{aligned} \sigma _{i;j}{} & {} \quad =\sigma _{ij}-2\sigma _i\sigma _j,\ \ \ \ \ \sigma _i\sigma _j+\sigma _{i;j} =\sigma _{ij}-\sigma _i\sigma _j,\nonumber \\ (\sigma _j\sigma _k+\sigma _{j;k})_{;i}{} & {} \quad =\sigma _{ijk}-2(\sigma _i\sigma _{jk}+\sigma _j\sigma _{ik}+\sigma _k\sigma _{ij}) +4\sigma _i\sigma _j\sigma _k. \end{aligned}$$

(49)

Putting $u=e^{-2\sigma }$, it follows from (49) that (48) is equivalent to $u_{ijk}=0$. So u is a polynomial in ($x^i)$ of degree two. Thus, the scalar function $\sigma$ satisfying (48) is given by

$$\begin{aligned} \sigma (x)=-\frac{1}{2}\ln |\langle Ax,x\rangle +\langle B,x\rangle +C|, \end{aligned}$$

(50)

where A is a constant symmetric matrix, B is a constant vector and C is a constant number. By $G^i=\sigma _0y^i$ and (50), we see that G is locally given by (2). Meanwhile, by (50), we have

$$\begin{aligned} -(\sigma _0)^2-\sigma _{0;0}=\frac{4(\langle Ax,x\rangle +\langle B,x\rangle +C)\langle Ay,y\rangle -(2\langle Ax,y\rangle +\langle B,y\rangle )^2}{4(\langle Ax,x\rangle +\langle B,x\rangle +C)^2}. \end{aligned}$$

(51)

Since $R(=-(\sigma _0)^2-\sigma _{0;0})\ne 0$ 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 $det(A)\ne 0$, then L given by (3) is a metric iff. the constant quantities A, B, C satisfy $4C\ne B'A^{-1}B$. $\square$

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

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

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

$$\begin{aligned}{} & {} R=-2f'(x^1)[(y^1)^2+(y^2)^2],\ \ \ \ R_{;i}=R\omega _i\\{} & {} \omega _1:=\frac{f''(x^1)}{f'(x^1)}-4f(x^1)=\big [\ln |f'(x^1)e^{-4\int f(x^1)dx^1}|\big ]_{x^1}=[\ln |\lambda |]_{x^1},\ \ \ \ \ \omega _2:=0. \end{aligned}$$

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

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

of isotropic sectional curvature

$$\begin{aligned} \lambda =f'(x^1)e^{-4\int f(x^1)dx^1}. \end{aligned}$$

Example 7.2

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

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

where r is a constant. This spray appears in Example 4.1.3 of [16]. From the completeness of geodesics of $\textbf{G}$ and Hopf-Rinow theorem, it is concluded that $\textbf{G}$ is not globally Finsler metrizable. But actually, it is even not locally Finsler metrizable by Theorem 1.1 (Example 12.4.1 in [16] shows it is locally projectively Finsler metrizable).

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

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

Therefore, by Theorem 1.1, $\textbf{G}$ 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 $\textbf{G}_c$ be a Berwald spray given by

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

where c is a constant. The spray $\textbf{G}_2$ 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 $\textbf{G}_1$ is just the spray given by Example 4.1.4 in [16]. $\textbf{G}_c$ actually is projectively flat although at present coordinate it is not in the form $Py^i$.

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

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

Further, we have

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

• (i) By Theorem 1.1, it is easy to see that $\textbf{G}_c$ 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) $\textbf{G}_c$ is of constant curvature iff. $c=2$.

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

Example 7.4

Let $\textbf{G}$ be a spray given by $G^i=\sigma _0y^i$ with $\sigma _i:=\sigma _{x^i}, \sigma =\sigma (x)$.

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

  $$\begin{aligned} \sigma (x)\ne -\frac{1}{2}\ln |\langle Ax,x\rangle +\langle B,x\rangle +C|. \end{aligned}$$

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

  $$\begin{aligned} G^i=-\frac{\langle x,y\rangle }{|x|^2}y^i,\ \ \ \ L=\frac{|x|^2|y|^2-\langle x,y\rangle ^2}{|x|^4}. \end{aligned}$$

  (52)

  In (52), 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 (52) is not Finsler metrizable, although this spray G satisfies (23).

Example 7.5

Let G be a spray on $R^3$ given by ([16], Example 4.1.2)

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

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

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

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

$$\begin{aligned} L(x,y)=(y^3)^2+(x^1)^2(y^1)^2+y^1y^2+(x^3)^2y^1y^3, \end{aligned}$$

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

$$\begin{aligned} G^1=-f(x^1)g'(t)(y^1)^2,\ \ \ G^2=f(x^1)\big [g(t)y^1-g'(t)y^2\big ]y^1,\ \ \ \ (t:=y^2/y^1). \end{aligned}$$

where f, g are two smooth functions. A direct computation gives $R^i_{\ k}=R\delta ^i_k-\tau _ky^i$ with

$$\begin{aligned} R=0,\ \ \ \tau _1=Ay^2,\ \ \ \tau _2=-Ay^1,\ \ \ A:=2f^2(x^1)g(t)g'''(t)-f'(x^1)g''(t). \end{aligned}$$

We can choose the two functions f, g satisfying $A\ne 0$. So in this case, G is not Finsler metrizable.