It is an important problem in Finsler geometry to study and characterize projectively flat Finsler metrics on an open domain in $\mathbb{R}^{m}$. Hilbert's 4th problem is to characterize the distance functions on an open subset in $\mathbb{R}^{m}$ such that straight lines are geodesics [5]. Regular distance functions with straight geodesics are projectively flat Finsler metrics. A Finsler metric $F=F(x, y)$ on an open subset $U\subset \mathbb{R}^{m}$ is projectively flat if and only if it satisfies the following equation

$F_{x^{i}y^{j}}y^{i}=F_{x^{j}}.$

(1.1)

In Finsler geometry, the flag curvature $\textbf{K}(P, y)$ is an analogue of the sectional curvature in Riemannian geometry. It is known that every projective Finsler metric is of scalar curvature, namely, the flag curvature $\textbf{K}(P, y)=\textbf{K}(y)$ is a scalar function of tangent vectors $\textbf{y}$. Shen discussed the classification problem on projective Finsler metrics of constant flag curvature [14]. The second author provided the projective factor of a class of projectively flat general $(\alpha, \beta)$-metrics [12] and studied a necessary and sufficient condition for a class of Finsler metric to be projectively flat [13]. Li proved the locally projectively flat Finsler metrics with constant flag curvature $\textbf{K}$ are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when $\textbf{K}=0$, $\textbf{K}=-1$, $\textbf{K}=1$ are given in an algebraic way [15].

For a Finsler metric $F=F(x, y)$ on a manifold $M$, the geodesics $c=c(t)$ of $F$ in local coordinates $(x^i)$ are characterized by

$\frac{{\rm{d}}^2x^i}{{\rm{d}}t^2}+2G^i(x, \frac{{\rm{d}}x}{{\rm{d}}t})=0, $

where $(x^i(t))$ are the coordinates of $c(t)$ and $G^i=G^i(x, y)$ are defined by

$G^i=\frac{g^{il}}{4}\{[F^2]_{x^ky^l}y^k-[F^2]_{x^l}\}, $

where $g_{ij}=\frac{1}{2}[F^2]_{y^iy^i}$ and $(g^{ij})=(g_{ij})^{-1}$. $G^i$ are called the spray coefficients. The Riemann curvature is a family of linear maps

$\textbf{R}_{y}=R^{i}_{\ k}\frac{\partial}{\partial x^{i}}\bigotimes dx^{k} : T_{x}M\rightarrow T_{x}M$

defined by

$R^{i}_{\ k}=2\frac{\partial G^{i}}{\partial x^{k}}-y^{j}\frac{\partial^{2} G^{i}}{\partial x^{j}\partial y^{k}}+2G^{j}\frac{\partial^{2} G^{i}}{\partial y^{j}\partial y^{k}}-\frac{\partial G^{i}}{\partial y^{j}}\frac{\partial G^{j}}{\partial y^{k}}.$

For a tangent plane $P\subset T_{p}M$ and a non-zero vector $\textbf{y}\in T_{p}M$, the flag curvature $\textbf{K}(P, y)$ is defined by

$\textbf{K}(P, y):=\frac{g_{\textbf{y}}(\textbf{u}, \textbf{R}_{\textbf{y}}(\textbf{u}))}{g_{\textbf{y}}(\textbf{y}, \textbf{y})g_{\textbf{y}}(\textbf{u}, \textbf{u})-g_{\textbf{y}}(\textbf{y}, \textbf{u})^{2}}, $

where $P=\mbox{span}\{\textbf{y}, \textbf{u}\}$. It is known that if $F$ is projectively flat, the spray coefficients of $F$ are in the form $G^{i}=Py^{i}$ where

$P=\frac{F_{x^{k}}y^{k}}{2F}, $

then $F$ is of scalar curvature with flag curvature

$\textbf{K}=\frac{P^{2}-P_{x^{k}}y^{k}}{F^{2}}.$

On the other hand, the study of spherically symmetric Finsler metrics attracted a lot of attention. Many known Finsler metrics are spherically symmetric [1, 4, 7, 14, 15, 17]. A Finsler metric $F$ is said to be spherically symmetric (orthogonally invariant in an alternative terminology in [6]) if $F$ satisfies

$F(Ax, Ay)=F(x, y)$

(1.2)

for all $A\in O(m)$, equivalently, if the orthogonal group $O(m)$ acts as isometrics of $F$. Such metrics were first introduced by Rutz [16].

It was pointed out in [6] that a Finsler metric $F$ on $\mathbb{B}^{m}(\mu)$ is a spherically symmetric if and only if there is a function $\phi: [0, \mu)\times \mathbb{R}\rightarrow \mathbb{R}$ such that

$F(x, y)=\mid y\mid\phi(\mid x\mid, \frac{\langle x, y\rangle}{\mid y\mid}), $

(1.3)

where $(x, y)\in T\mathbb{R}^m(\mu)\backslash \{0\}$. The spherically symmetric Finsler metric of form (1.3) can be rewritten as the following form [8]

$F=\mid y\mid\phi(\frac{\mid x\mid^2}{2}, \frac{\langle x, y\rangle}{\mid y\mid}).$

Spherically symmetric Finsler metrics are the simplest and most important general $(\alpha, \beta)$-metrics [4]. Mo, Zhou and Zhu classified the projective spherically symmetric Finsler metrics with constant flag curvature in [2, 9, 10]. A lot of spherically symmetric Finsler metrics with nice curvature properties were investigated by Mo, Huang and et al. [3, 6-11].

An important example of projectively flat Finsler metric was given by Berwald. It can be written as

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

on the unit ball $\subset \mathbb{R}^{m}$, where $y\in T_{x}\mathbb{B}^{m}\subset \mathbb{R}^{m}$. It could also be expressed as

$F=\mid y\mid[g(t, s)(f_{0}(t)+f_{2}(t)s^{2})+h(t)s], $

where

$ g(t, s)=\frac{1}{\sqrt{1-2t+s^{2}}}, \ \ f_{0}(t)=\frac{1}{1-2t}, \ \ f_{2}(t)=\frac{2}{(1-2t)^{2}}, \\ h(t)=\frac{2s}{1-2t}, \ \ t=\frac{\mid x\mid^{2}}{2}, \ \ s=\frac{\langle x, y\rangle}{\mid y\mid}.$

Inspired by the Berwald metric, we try to find the solution of the projectively flat eq.(1.1) in the following forms

$F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})=\mid y\mid\phi(t, s), $

where

$\phi(t, s)=\sum \limits_{i=0}^{n}\phi_{i}(t)s^{i}+(1-2t+s^{2})^{-\frac{1}{r}}\sum \limits_{j=0}^{l}f_{j}(t)s^{j}.$

Through caculations, we have the following conclusions.

Theorem 1.1  Let $\phi(t, s)$ be a function defined by

$\begin{aligned} \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}\\ +b(1-2t+s^{2})^{-\frac{1}{2}}[f_{0}(t)+(\frac{f_{0}(t)}{1-2t}+\frac{1}{2}f_{0}'(t))s^{2}] \end{aligned}$

and $f_{0}(t)$ is a differentiable function which satisfies

$f_{0}(t)=C_{1}+\frac{C_{2}}{t-\frac{1}{2}}, $

where $b$, $C_{1}$, $C_{2}$ are constants and $\phi_{1}$ is an any continuous function, $\phi_{0}$ is a polynomial function of $N$ degree where $N\leqslant n$, $\phi_{0}^{(j)}$ denotes the $j$-order derivative for $\phi_{0}(t)$, $\phi(t, s)$ needs to satisfy $\phi-s\phi_{s}>0, $ when $m=2.$ Moreover, the additional equality holds

$\phi-s\phi_{s}+(t-s^{2})\phi_{ss}>0, $

when $m\geq3$. Then the following spherically symmetric Finsler metric on $\mathbb{B}^{m}(\mu)$

$F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$

is projectively flat.

Remark 1  Let us take a look at a special case $b=1$, $C_{1}=0$, $C_{2}=-\frac{1}{2}$, setting $\phi_{0}(t)=0$, $\phi_{1}(t)=\frac{2}{1-2t}, $ we obtain the Berwald metric.

Theorem 1.2  Let $\phi(t, s)$ be a function defined by

$ \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}\\ \quad \quad \quad \quad +b\frac{1}{1-2t+s^{2}}(f_{0}(t)-\frac{3}{5}f_{0}'(t)s^{2}) $

and $f_{0}(t)$ is a differentiable function which satisfies $f_{0}(t)=C_{1}(-1+2t), $ where $b$, $C_{1}$ are constants and $\phi_{1}$ is an any continuous function, $\phi_{0}$ is a polynomial function of $N$ degree where $N\leqslant n$, $\phi_{0}^{(j)}$ denotes the $j$-order derivative for $\phi_{0}(t)$, $\phi(t, s)$ needs to satisfy $\phi-s\phi_{s}>0, $ when $m=2.$ Moreover, the additional equality holds

$\phi-s\phi_{s}+(t-s^{2})\phi_{ss}>0, $

when $m\geq3$. Then the following spherically symmetric Finsler metric on $\mathbb{B}^{m}(\mu)$

$F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$

is projectively flat.

Theorem 1.3  Let $\phi(t, s)$ be a function defined by

$\begin{aligned} \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}+b\frac{1}{1-2t+s^{2}}\\ [f_{0}(t)+(\frac{2f_{0}(t)}{1-2t}+\frac{1}{2}f_{0}'(t))s^{2} +(\frac{2}{3}\frac{1}{(1-2t)^{2}}f_{0}(t)+\frac{1}{3}\frac{1}{1-2t}f_{0}'(t)-\frac{1}{2}f_{0}''(t))s^{4}] \end{aligned}$

and $f_{0}(t)$ is a differentiable function which satisfies $f_{0}(t)=C_{1}(t-\frac{1}{2})+C_{2}(t-\frac{1}{2})^{2}, $ where $b$, $C_{1}$, $C_{2}$ are constants and $\phi_{1}$ is an any continuous function, $\phi_{0}$ is a polynomial function of $N$ degree where $N\leqslant n$, $\phi_{0}^{(j)}$ denotes the $j$-order derivative for $\phi_{0}(t)$, $\phi(t, s)$ needs to satisfy $\phi-s\phi_{s}>0, $ when $m=2.$ Moreover, the additional equality holds

$\phi-s\phi_{s}+(t-s^{2})\phi_{ss}>0, $

when $m\geq3$. Then the following spherically symmetric Finsler metric on $\mathbb{B}^{m}(\mu)$

$F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$

is projectively flat.

Theorem 1.4  Let $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$ be a spherically symmetric Finsler metric on $\mathbb{B}^{m}(\mu)$. If $F$ is projectively flat, its projective factor $P$ is given by

$P=\mid y\mid\frac{\phi_{s}+s\phi_{t}}{2\phi}$

and its flag curvature $K$ is given by

$\textbf{K}=\frac{3}{4}\frac{(\phi_{s}+s\phi_{t})^{2}}{\phi^{4}}-\frac{\phi_{t}}{\phi^{3}}-\frac{\phi_{st}+s\phi_{tt}}{2\phi^{3}}s.$

In this section, we will construct a lot of projectively flat Finsler metrics which contains the Berwald metric. From [8], we know that

Lemma 2.1  $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$ is a solution of (1.1) if and only if $\phi$ satisfies

$s\phi_{ts}+\phi_{ss}-\phi_t=0,$

(2.1)

where $t=\frac{\mid x\mid^{2}}{2}$ and $s=\frac{\langle x, y\rangle}{\mid y\mid}$.

Consider the spherically symmetric Finsler metric $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$ on $\mathbb{B}^{m}(\mu)$, where $\phi=\phi(t, s)$ is given by $\phi(t, s)=\sum \limits_{j=0}^{l}\phi_{j}(t)s^{j}.$ By a direct caculation, we get

$ {\phi _t}(t,s) = \sum\limits_{j = 0}^l {{\phi _{j'}}} (t){s^j}, $

(2.2)

$ {\phi _{ts}}(t,s) = \sum\limits_{j = 0}^l j {\phi _{j'}}(t){s^{j - 1}}, $

(2.3)

$ {\phi _{ss}}(t,s) = \sum\limits_{j = 0}^l j (j - 1){\phi _j}(t){s^{j - 2}}. $

(2.4)

Plugging (2.2), (2.3), (2.4) into (2.1), the following equation is deduced,

$\sum \limits_{j=0}^{l}(j-1)\phi_{j}'(t)s^{j}+\sum \limits_{j=0}^{l-2}(j+2)(j+1)\phi_{j+2}s^{j}=0.$

(2.5)

It is equivalent to

$\sum \limits_{j=0}^{l-2}[(j-1)\phi_{j}'(t)+(j+2)(j+1)\phi_{j+2}(t)]s^{j}+\sum \limits_{j=l-1}^{l}(j-1)\phi_{j}'(t)s^{j}=0.$

(2.6)

By (2.6), $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$ is projectively flat if and only if

$ (j-1)\phi_{j}'(t)+(j+2)(j+1)\phi_{j+2}(t)=0, \ \ \ \ \ \ \ j=0, 1, 2\cdots l-2, \\ (j-1)\phi_{j}'(t)=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j=l-1, l.\\ $

(2.7)

When $j=0, $ from the first equation of (2.7), we get

$\phi_{0}'(t)=2\phi_{2}(t).$

(2.8)

Similarity, taking $j=1$ and $j=2$, we obtain

$\phi_{3}(t)=0, \ \ \phi_{2}'(t)+12\phi_{4}(t)=0.$

(2.9)

If $k=j+2, $ the first equation of (2.7) is equivalent to

$k(k-1)\phi_{k}(t)+(k-3)\phi_{k-2}'(t)=0.$

(2.10)

It is easy to see the recurrence fomula on $\phi_{k}(t)$ and $\phi_{k}'(t)$,

$\phi_{k}(t)=-\frac{k-3}{k(k-1)}\phi_{k-2}'(t).$

(2.11)

If $k=\mbox{odd}$, $k\geq3, $ then by (2.11),

$\phi_{k}(t)=(-1)^{\frac{k-3}{2}}\frac{(k-3)(k-5)\cdots2}{k(k-1)\cdots4}\phi_{3}^{(\frac{k-3}{2})}(t)=0.$

(2.12)

If $k=\mbox{even}$, $k\geq4, $ we have

$\phi_{k}(t)=(-1)^{\frac{k-4}{2}}\frac{24(k-3)!!}{k!}\phi_{4}^{(\frac{k-4}{2})}(t)=(-1)^{\frac{k-2}{2}}\frac{(k-3)!!}{k!}\phi_{0}^{(\frac{k}{2})}(t).$

(2.13)

Case 1  $k=\mbox{odd}\geq5, $ setting $l=2n+1, $ by the second equation of (2.7),

$\phi_{2n+1}(t)=0, \ \ \phi_{2n}(t)=\mbox{const.}, $

(2.14)

then it follows from (2.1), (2.12), (2.13), (2.14),

$ \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\phi_{2}(t)s^{2}+\cdots+\phi_{2n-1}(t)s^{2n-1}+\phi_{2n}(t)s^{2n}+\phi_{2n+1}(t)s^{2n+1}\\ \quad \quad =\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}. $

(2.15)

Case 2  $k=\mbox{even}\geq4, $ setting $l=2n+2, $ by the second equation of (2.7),

$\phi_{2n+2}(t)=\mbox{const.}, \ \ \phi_{2n+1}(t)=0, $

(2.16)

then it follows from (2.1), (2.12), (2.13), (2.16),

$ \phi(t, s) =\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=1}^{n}(-1)^{j}\frac{(2j-1)!!}{(2j+2)!}\phi_{0}^{(j+1)}(t)s^{2j+2}.$

(2.17)

The case $l\in\{1, 2, 3\}$ is similar. Through the above analysis, we obtain the following.

Proposition 2.1  $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2}, \frac{\langle x, y\rangle}{\mid y\mid})$ in the form $F=\mid y\mid\sum \limits_{j=0}^{l}\phi_{j}(t)s^{j}$ is a solution of the projectively flat eq.(2.1) if and only if $\phi(t, s)$ satisfies

$ \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\phi_{2}(t)s^{2}+\cdots+\phi_{l}(t)s^{l}\\ \quad \quad \quad =\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j} $

and $\phi_{0}^{(n)}=\mbox{const.}.$

Consider the solution of (2.1) where $\phi=\phi(t, s) $ is given by

$\phi(t, s)=(1-2t+s^{2})^{-\frac{1}{r}}\sum \limits_{j=0}^{l}f_{j}(t)s^{j}.$

(2.18)

Suppose that $g=(1-2t+s^{2})^{-\frac{1}{r}}, $ thus

$g_{t}=\frac{2}{r}g^{r+1}, \ \ g_{s}=-\frac{2}{r}g^{r+1}s.$

(2.19)

Differentiating (2.18), by using (2.19), we get

$ {\phi _t}(t,s) = \frac{2}{r}{g^{r + 1}}(\sum\limits_{j = 0}^l {{f_j}} (t){s^j}) + g(\sum\limits_{j = 0}^l {{f_{j'}}} (t){s^j}), $

(2.20)

$ {\phi _s}(t,s) = - \frac{2}{r}{g^{r + 1}}s(\sum\limits_{j = 0}^l {{f_j}} (t){s^j}) + g(\sum\limits_{j = 0}^l j {f_j}(t){s^{j - 1}}), $

(2.21)

$ \begin{array}{l} {\phi _{ts}}(t,s) = - \frac{2}{r}{g^{r + 1}}(\sum\limits_{j = 0}^l {{f_{j'}}} (t){s^{j + 1}}) + g(\sum\limits_{j = 0}^l j {f_{j'}}(t){s^{j - 1}})\\ \quad \quad \quad \quad - \frac{4}{{{r^2}}}(r + 1){g^{2r + 1}}(\sum\limits_{j = 0}^l {{f_j}} (t){s^{j + 1}}) + \frac{2}{r}{g^{r + 1}}(\sum\limits_{j = 0}^l j {f_j}(t){s^{j - 1}}), \end{array} $

(2.22)

$ \begin{array}{l} {\phi _{ss}}(t,s) = \frac{4}{{{r^2}}}(r + 1){g^{2r + 1}}(\sum\limits_{j = 0}^l {{f_j}} (t){s^{j + 2}}) - \frac{2}{r}{g^{r + 1}}[\sum\limits_{j = 0}^l {(j + 1)} {f_j}(t){s^j}]\\ \quad \quad \quad \quad - \frac{2}{r}{g^{r + 1}}(\sum\limits_{j = 0}^l j {f_j}(t){s^j}) + g[\sum\limits_{j = 0}^l j (j - 1){f_j}(t){s^{j - 2}}]. \end{array} $

(2.23)

Plugging (2.20), (2.22), (2.23) into (2.1), we get the following

$-\frac{2}{r}g^{r}(\sum \limits_{j=0}^{l}f_{j}'(t)s^{j+2})-\frac{2}{r}g^{r}[\sum \limits_{j=0}^{l}(j+2)f_{j}(t)s^{j}]+\sum \limits_{j=0}^{l}(j-1)f_{j}'(t)s^{j}+\sum \limits_{j=0}^{l}j(j-1)f_{j}(t)s^{j-2}=0.$

(2.24)

Multiplying $g^{-r}$ on the both sides of (2.24), then

$ 0=-\frac{2}{r}[\sum \limits_{j=0}^{l}(j+2)f_{j}(t)s^{j}]-\frac{2}{r}(\sum \limits_{j=0}^{l}f_{j}'(t)s^{j+2})+g^{-r}[\sum \limits_{j=0}^{l}(j-1)f_{j}'(t)s^{j}]\\ \quad \quad +g^{-r}[\sum \limits_{j=0}^{l}j(j-1)f_{j}(t)s^{j-2}]\\ \quad =-\frac{2}{r}[\sum \limits_{j=0}^{l}(j+2)f_{j}(t)s^{j}]+\sum \limits_{j=0}^{l}(j-1-\frac{2}{r})f_{j}'(t)s^{j+2}+\sum \limits_{j=0}^{l}(j-1)(1-2t)f_{j}'(t)s^{j}\\ \quad \quad +\sum \limits_{j=0}^{l}j(j-1)(1-2t)f_{j}(t)s^{j-2}+\sum \limits_{j=0}^{l}j(j-1)f_{j}(t)s^{j}\\ \quad =\sum \limits_{j=0}^{l}[j^{2}-(1+\frac{2}{r})j-\frac{4}{r}]f_{j}(t)s^{j}+\sum \limits_{j=2}^{l+2}(j-3-\frac{2}{r})f_{j-2}'(t)s^{j}+\sum \limits_{j=0}^{l}(j-1)(1-2t)f_{j}'(t)s^{j}\\ \quad \quad +\sum \limits_{j=0}^{l-2}(j+2)(j+1)(1-2t)f_{j+2}(t)s^{j}.$

(2.25)

From (2.25), we obtain the following equations

$\begin{array}{l} [{j^2} - (1 + \frac{2}{r})j - \frac{4}{r}]{f_j}(t) + (j - 3 - \frac{2}{r})f_{j - 2}^{_{^\prime }}(t) + (j - 1)(1 - 2t)f_j^{_{^\prime }}(t)\\ + (j + 2)(j + 1)(1 - 2t){f_{j + 2}}(t) = 0,\;\;j = 2, \cdots ,l - 2, \end{array}$

(2.26)

$\begin{array}{l} ({j^2} - j - \frac{{2j}}{r} - \frac{4}{r}){f_j}(t) + (j - 1)(1 - 2t)f_j^\prime (t) + (j + 2)(j + 1)(1 - 2t){f_{j + 2}}(t) = 0,\\ j = 0,1, \end{array}$

(2.27)

$\begin{array}{l} ({j^2} - j - \frac{{2j}}{r} - \frac{4}{r}){f_j}(t) + (j - 3 - \frac{2}{r})f_{j + 2}^\prime (t) + (j - 1)(1 - 2t)f_j^\prime (t) = 0,\\ j = l - 1,l, \end{array}$

(2.28)

$(j - 3 - \frac{2}{r})f_{j + 2}^\prime (t) = 0,\;\;j = l + 1,l + 2.$

(2.29)

Let us take a look at a special case $l=4, $ $f_{1}(t)=f_{3}(t)=0, $ then

$ - \frac{4}{r}{f_0}(t) - (1 - 2t)f_0^\prime (t) + 2(1 - 2t){f_2}(t) = 0,$

(2.30)

$(12 - \frac{{12}}{r}){f_4}(t) + (1 - \frac{2}{r})f_2^\prime (t) + 3(1 - 2t)f_4^\prime (t) = 0,$

(2.31)

$(2 - \frac{8}{r}){f_2}(t) - (1 + \frac{2}{r})f_0^\prime (t) + (1 - 2t)f_2^\prime (t) + 12(1 - 2t){f_4}(t) = 0,$

(2.32)

$f_4^\prime (t) = 0.$

(2.33)

Case 1  $r=2.$ In this case, by (2.30)-(2.33), we can get the following equations

$\left\{ {\begin{array}{*{20}{l}} { - 2{f_2}(t) - 2f_0^\prime (t) + (1 - 2t)f_2^\prime (t) = 0,}\\ { - 2{f_0}(t) + (2t - 1)f_0^\prime (t) + 2(1 - 2t){f_2}(t) = 0,}\\ {6{f_4}(t) + 3(1 - 2t)f_4^\prime (t) = 0,}\\ {f_4^\prime (t) = 0.} \end{array}} \right. $

(2.34)

Substituting the fourth equation of (2.34) into the third equation of it, we have $f_{4}(t)=0.$ From (2.30), we obtain

$f_{2}(t)=\frac{f_{0}(t)}{1-2t}+\frac{1}{2}f_{0}'(t).$

(2.35)

Differentiating (2.35), we get

$f_{2}'(t)=2\frac{f_{0}(t)}{(1-2t)^{2}}+\frac{f_{0}'(t)}{1-2t}+\frac{1}{2}f_{0}''(t).$

(2.36)

Substituting (2.35), (2.36) into the first equation of (2.34), we obtain that $f_{0}(t)$ satisfies

$-2f_{0}'(t)+\frac{1}{2}(1-2t)f_{0}''(t)=0.$

(2.37)

Solving (2.37), we have

$f_{0}(t)=C_{1}+\frac{C_{2}}{t-\frac{1}{2}}, $

(2.38)

where $C_{1}, C_{2}$ are constants. Thus we have the following proposition.

Proposition 2.2  $\phi(t, s)=\sum \limits_{i=0}^{n}\phi_{i}(t)s^{i}+(1-2t+s^{2})^{-\frac{1}{r}}\sum \limits_{j=0}^{l}f_{j}(t)s^{j}$ is a solution of the projectively flat eq.(2.2) if and only if

$\begin{aligned} \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}\\ +b(1-2t+s^{2})^{-\frac{1}{2}}[f_{0}(t)+(\frac{f_{0}(t)}{1-2t}+\frac{1}{2}f_{0}'(t))s^{2}] \end{aligned}$

and $f_{0}(t)=C_{1}+\frac{C_{2}}{t-\frac{1}{2}}, $ where $b, C_{1}, C_{2}$ are constants.

Case 2  $r\neq2$, $f_{4}(t)=0$. From (2.31), we know

$f_{2}'(t)=0.$

(2.39)

Plugging (2.39) into (2.32), we obtain

$(2-\frac{8}{r})f_{2}(t)=(1+\frac{2}{r})f_{0}'(t).$

(2.40)

If $r=4$, $f_{0}'(t)=0, $ from (2.30),

$f_{2}(t)=\frac{2}{r}\frac{1}{1-2t}f_{0}(t).$

(2.41)

Thus $f_{0}(t)$ and $f_{2}(t)$ can't be constants at the same time, so in this case, $r\neq4$, together with (2.39), (2.40), (2.41), we know that $f_{2}(t)$ needs to satisfy the following

$\left\{ {\begin{array}{*{20}{l}} {{f_2}(t) = \frac{2}{r}\frac{1}{{1 - 2t}}{f_0}(t) = \frac{{(1 + \frac{2}{r})f_0^\prime (t)}}{{2 - \frac{8}{r}}},}\\ {f_2^\prime (t) = 0.} \end{array}} \right.$

(2.42)

Through (2.42), we get that $f_{0}(t)$ needs to satisfy

$\left\{ {\begin{array}{*{20}{l}} {\frac{{(1 + \frac{2}{r})f_0^\prime (t)}}{{2 - \frac{8}{r}}} = \frac{2}{r}\frac{1}{{1 - 2t}}{f_0}(t),}\\ {f_0^{\prime \prime }(t) = 0.} \end{array}} \right.$

(2.43)

From the first equation of (2.43),

$f_{0}(t)=C_{1}(-1+2t)^{\frac{4}{3r}-\frac{1}{3}},$

(2.44)

where $C_{1}$ is a constant. But the $f_{0}(t)$ in (2.44) doesn't satisfy the second equation of (2.43) only if $r=1, $ thus we can get the following proposition.

Proposition 2.3  $\phi(t, s)=\sum \limits_{i=0}^{n}\phi_{i}(t)s^{i}+(1-2t+s^{2})^{-\frac{1}{r}}\sum \limits_{j=0}^{l}f_{j}(t)s^{j}$ is a solution of the projectively flat eq.(2.2) if and only if

$\phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}\\ \quad \quad \quad +b\frac{1}{1-2t+s^{2}}(f_{0}(t)-\frac{3}{5}f_{0}'(t)s^{2})$

and $f_{0}(t)=C_{1}(-1+2t), $ where $b, C_{1}$ are constants.

Case 3  $ r\neq2, $ $f_{4}(t)\neq0.$ In this case, from the first equation of (2.30),

$f_{2}(t)=\frac{2}{r}\frac{1}{1-2t}f_{0}(t)+\frac{1}{2}f_{0}'(t).$

(2.45)

Differentiating (2.45), we have

$f_2^\prime (t) = \frac{4}{r}\frac{1}{{{{(1 - 2t)}^2}}}{f_0}(t) + \frac{2}{r}\frac{1}{{1 - 2t}}f_0^\prime (t) + \frac{1}{2}f_0^{\prime \prime }(t),$

(2.46)

$f_2^{\prime \prime }(t) = \frac{{16}}{r}\frac{1}{{{{(1 - 2t)}^3}}}{f_0}(t) + \frac{8}{r}\frac{1}{{{{(1 - 2t)}^2}}}f_0^\prime (t) + \frac{2}{r}\frac{1}{{1 - 2t}}f_0^{\prime \prime }(t) + \frac{1}{2}f_0^{\prime \prime \prime }(t).$

(2.47)

From (2.32), we get

$f_{4}(t)=\frac{1}{6r}\frac{1}{(1-2t)^{2}}f_{0}(t)(\frac{8}{r}-4)+\frac{1}{3r}\frac{1}{1-2t}f_{0}'(t)-\frac{1}{2}f_{0}''(t).$

(2.48)

Differentiating (2.48), we obtain

$f_{4}'(t)=\frac{2}{3r}(\frac{8}{r}-4)\frac{f_{0}(t)}{(1-2t)^{3}}+\frac{4}{3r^{2}}\frac{1}{(1-2t)^{2}}f_{0}'(t)+\frac{1}{3r}\frac{1}{1-2t}f_{0}''(t)-\frac{1}{2}f_{0}'''(t).$

(2.49)

Plugging (2.33) into (2.31), we have

$(12-\frac{12}{r})f_{4}(t)+(1-\frac{2}{r})f_{2}'(t)=0.$

(2.50)

Thus from (2.50), no matter $r=1$ or not,

$f_{2}''(t)=0.$

(2.51)

Combining the fourth equation of (2.33) and (2.51), we obtain that $f_{0}(t)$ satisfies

$ \frac{16}{r}\frac{1}{(1-2t)^{3}}f_{0}(t)+\frac{8}{r}\frac{1}{(1-2t)^{2}}f_{0}'(t)+\frac{2}{r}\frac{1}{1-2t}f_{0}''(t)+\frac{1}{2}f_{0}'''(t)=0, \\ \frac{2}{3r}(\frac{8}{r}-4)\frac{f_{0}(t)}{(1-2t)^{3}}+\frac{4}{3r^{2}}\frac{1}{(1-2t)^{2}}f_{0}'(t)+\frac{1}{3r}\frac{1}{1-2t}f_{0}''(t)-\frac{1}{2}f_{0}'''(t)=0. $

(2.52)

Solving the first equation of (2.52), we get

$f_{0}(t)=C_{1}(t-\frac{1}{2})+C_{2}(t-\frac{1}{2})^{2}+C_{3}(t-\frac{1}{2})^{\frac{2}{r}}.$

(2.53)

Solving the second equation of (2.52), we know

$f_{0}(t)=C_{4}(t-\frac{1}{2})^{2}+C_{5}(t-\frac{1}{2})^{\frac{3r-1+\sqrt{9r^{2}-18r+25}}{6r}}+C_{6}(t-\frac{1}{2})^{\frac{3r-1-\sqrt{9r^{2}-18r+25}}{6r}}.$

(2.54)

If $r=1$, $C_{3}=C_{6}=0, $ two equations of (2.52) have the same solutions. Thus we have the following proposition.

Proposition 2.4  $\phi(t, s)=\sum \limits_{i=0}^{n}\phi_{i}(t)s^{i}+(1-2t+s^{2})^{-\frac{1}{r}}\sum \limits_{j=0}^{l}f_{j}(t)s^{j}$ is a solution of the projectively flat eq.(2.2) if and only if

$\begin{aligned} \phi(t, s)=\phi_{0}(t)+\phi_{1}(t)s+\frac{1}{2}\phi_{0}'(t)s^{2}+\sum \limits_{j=2}^{n}(-1)^{j-1}\frac{(2j-3)!!}{(2j)!}\phi_{0}^{(j)}(t)s^{2j}+b\frac{1}{1-2t+s^{2}}\\ [f_{0}(t)+(\frac{2f_{0}(t)}{1-2t}+\frac{1}{2}f_{0}'(t))s^{2} +(\frac{2}{3}\frac{1}{(1-2t)^{2}}f_{0}(t)+\frac{1}{3}\frac{1}{1-2t}f_{0}'(t)-\frac{1}{2}f_{0}''(t))s^{4}] \end{aligned}$

and $f_{0}(t)=C_{1}(t-\frac{1}{2})+C_{2}(t-\frac{1}{2})^{2}, $ where $b$, $C_{1}$, $C_{2}$ are constants.

$\phi(t,s)$ in Propositions 2.2, 2.3, 2.4 can't ensure that $F=\mid y\mid\phi(\frac{\mid x\mid^2}{2},\frac{\langle x,y\rangle}{\mid y\mid})$ is a Finsler metric. In order to obtain projectively flat Finsler metric, $\phi(t,s)$ in Propositions 2.2-2.4 needs to satisfy the necessary and sufficient condition for $F=\alpha\phi(\|\beta_{x}\|_{\alpha}, \frac{\beta}{\alpha})$ to be a Finsler metric for any $\alpha$ and $\beta$ with $\|\beta_{x}\|_{\alpha}<b_{0}$ given by Yu and Zhu [4]. In particular, considering $F=\mid y\mid\phi(\frac{\mid x\mid^{2}}{2},\frac{\langle x,y\rangle}{\mid y\mid})=\mid y\mid\phi(t,s)$, then $F$ is a Finsler metric if and only if the positive function $\phi$ satisfies

$\phi-s\phi_{s}>0, \ \ \phi-s\phi_{s}+(t-s^{2})\phi_{ss}>0,$

(3.1)

when $m\geq3$ or

$\phi-s\phi_{s}+(t-s^{2})\phi_{ss}>0,$

(3.2)

when $m=2$.

Proof of Theorem 1.1  Combine Proposition 2.2, (3.1), (3.2) and the fundamental property of the projectively flat equation (2.1).

Proof of Theorem 1.2  Combine Proposition 2.3, (3.1), (3.2) and the fundamental property of the projectively flat equation (2.1).

Proof of Theorem 1.3  Combine Proposition 2.4, (3.1), (3.2) and the fundamental property of the projectively flat equation (2.1).

Proof of Theorem 1.4  Suppose that

$r:=\mid y\mid, \ \ r^{i}:=r_{i}:=\frac{y^{i}}{\mid y\mid}, \ \ x_{i}:=x^{i}, \ \ s^{i}:=s_{i}:=x_{i}-sr_{i}.$

(3.3)

Direct computations yield that

$t_{x^{i}}=x^{i}=s_{i}+sr_{i}, \ \ s_{x^{i}}=r_{i}, \ \ r_{y^{i}}=r_{i}, \ \ s_{y^{i}}=\frac{s_{i}}{r},$

(3.4)

where we use of (3.3). By (3.3), (3.4), we get the following lemma.

Lemma 3.1  Let $f=f(r, t, s)$ be a function on a domain $\textrm{U}\subset \mathbb{R}^{3}.$ Then

$f_{x^{i}}=r_{i}(f_{s}+sf_{t})+s_{i}f_{t}, \ \ f_{y^{i}}=r_{i}f_{r}+s_{i}\frac{f_{s}}{r}.$

(3.5)

Let $F=\mid y\mid\phi(\frac{\mid x\mid^2}{2}, \frac{\langle x, y\rangle}{\mid y\mid})=r\phi(t, s)$ be a spherically symmetric Finsler metric on $\mathbb{B}^{m}(\mu)$. From (3.3), (3.4), (3.5), we have the following

$F_{r}=\phi, \ \ F_{t}=r\phi_{t}, \ \ F_{s}=s\phi_{s}.$

(3.6)

Note that $s_{i}$ and $r_{i}$ are positively homogeneous of degree 0 and 1. Hence

$s_{i}y^{i}=0, \ \ r_{i}y^{i}=0$

(3.7)

and we get

$F_{x^{i}}=r[\phi_{t}s_{i}+(\phi_{s}+s\phi_{t})r_{i}], \ \ F_{y^{i}}=\phi r_{i}+\phi_{s}s_{i}.$

(3.8)

Thus from (3.7), (3.8), we have $F_{0}=F_{x^{i}}y^{i}=r^{2}(\phi_{s}+s\phi_{t}), $

$P=\frac{F_{0}}{2F}=\frac{(\phi_{s}+s\phi_{t})r}{2\phi}.$

(3.9)

Differentiating (3.9), we know

$P_{x^{k}}=\frac{r}{2}[\frac{2\phi_{t}r_{k}+(\phi_{st}+s\phi_{tt})(s_{k}+sr_{k})}{\phi}-\frac{(\phi_{s}+s\phi_{t})(\phi_{s}r_{k}+\phi_{t}s_{k}+\phi_{t}sr_{k})}{\phi^{2}}].$

(3.10)

From (3.7), (3.10), we obtain

$P_{x^{k}}y^{k}=\frac{r^{2}}{2}[\frac{2\phi_{t}+(\phi_{st}+s\phi_{tt})s}{\phi}-\frac{(\phi_{s}+s\phi_{t})^{2}}{\phi^{2}}].$

(3.11)

Thus using (3.9), (3.11), we have

$\textbf{K}=\frac{P^{2}-P_{x^{k}}y^{k}}{F^{2}}=\frac{3}{4}\frac{(\phi_{s}+s\phi_{t})^{2}}{\phi^{4}}-\frac{\phi_{t}}{\phi^{3}}-\frac{\phi_{st}+s\phi_{tt}}{2\phi^{3}}s.$

Theorem 1.4 can be achieved.