The notion of dually flat metrics was first introduced by Amari and Nagaoka when they studied the information geometry on Riemannian space [1, 2]. Later on, the notion of locally dually flat Finsler metrics was introduced by Shen [3]. A Finsler metric $F=F( x,y)$ on an $n$-dimensional manifold $M$ is called the locally dually flat Finsler metric if at every point there is a coordinate system $({{x}^{i}})$ in which the geodesic coefficients are in the following form ${{G}^{i}}=-\frac{1}{2}{{g}^{ij}}{{H}_{{{y}^{j}}}},$ where $H=H(x,y)$ is a local scalar function on the tangent bundle $TM$ of $M$ and satisfies $H(x, \lambda y)=\lambda^3H(x,y)$ for all $\lambda>0$. Such a coordinate system is called an adapted coordinate system. It is shown that a Finsler metric on an open subset $U \subset {{\bf{R}}^n}$ is dually flat if and only if it satisfies the following PDE $ (F^2)_{x^ky^l}y^k-2(F^2)_{x^l}=0. $ In this case, $H=-\frac{1}{6}(F^{2})_{x^m}y^m$. Recently, Shen, Zhou and the second author studied locally dually flat Randers metrics $F=\alpha +\beta$ and classified locally dually flat Randers metrics $F=\alpha +\beta $ with isotropic $S$-curvature [4]. Later, Xia characterized locally dually flat $(\alpha ,\beta)$-metrics on an $n$-dimensional manifold $M( n\ge 3)$ [5].

The study on conformal properties has a long history. Two Finsler metrics $F$ and $\bar{F}$ on a manifold $M$ are said to be conformally related if there is a scalar function $\sigma(x)$ on $M$ such that $F=e^{\sigma(x)}\bar{F}$. A Finsler metric which is conformally related to a Minkowski metric is called conformally flat Finsler metric. In 1989, Ichijy$\overline {\rm{o}} $ and Hashiguchi defined a conformally invariant Finsler connection in a Finsler space with $(\alpha,\beta)$-metric and gave the condition for a Randers space to be conformally flat based on their connection (see [6]). Later, S. Kikuchi found a conformally invariant Finsler connection and gave a necessary and sufficient condition for a Finsler metric to be conformally flat by a system of partial differential equations under an extra condition (see [7]). By using Kikuchi's conformally invariant Finsler connection, Hojo, Matsumoto and Okubo studied conformally Berwald Finsler spaces and its applications to $(\alpha,\beta)$-metrics (see [8]). Recently, Kang proved that any conformally flat Randers metric of scalar flag curvature is projectively flat and classified completely conformally flat Randers metrics of scalar flag curvature (see [9]). On the other hand, Bacso and the second author studied the global conformal transformations on a Finsler space $(M,F)$. They obtain the relations between some important geometric quantities of $F$ and their correspondences respectively, including Riemann curvatures, Ricci curvatures and S-curvatures (see [10, 11]). The Weyl theorem states that the projective and conformal properties of a Finsler metric determine the metric properties uniquely. Thus the conformal properties of a Finsler metric deserve extra attention.

In this paper, we study and classify locally dually flat and conformally flat $(\alpha ,\beta)$-metrics. Firstly, we can prove the following theorem.

Theorem 1.1 Let $F=\alpha+\beta$ be a locally dually flat Randers metric on an n-dimensional manifold $M \ (n\ge3)$. Assume that $F$ is conformally flat. Then it must be Minkowskian.

Further, following Xia's main result on locally dually flat $(\alpha, \beta)$-metrics in [5], we study and characterize locally dually flat and conformally flat $(\alpha, \beta)$-metrics of non-Randers type. We get the following theorem.

Theorem 1.2 Let $F=\alpha \phi(s)$, $s=\frac{\beta }{\alpha }$, be an $(\alpha ,\beta)$-metric on an $n$-dimensional manifold $M \ ( n\ge 3)$. Suppose that $\phi$ satisfies one of the following conditions:

(i) $\phi(s)$ is a polynomial of $s$ with $\phi '(0)=0$;

(ii) $\phi(s)$ is an analytic function with $\phi '(0)=\phi ''(0)=0$;

(iii) $\phi'(0)\neq 0,\ s({{k}_{2}}-{{k}_{3}}{{s}^{2}})(\phi \phi '-s\phi {{'}^{2}}-s\phi \phi '')-( \phi {{'}^{2}}+\phi \phi '')+{{k}_{1}}\phi( \phi -s\phi ' )\ne 0$,

where $k_1, \ k_2$ and $k_3$ are constants. Then, if $F$ is locally dually flat with $\alpha$ conformally flat, $F$ must be Minkowskian.

Let $M$ be an $n$-dimensional $C^{\infty}$ mainfold and $TM$ denotes the tangent bundle of $M$. A Finsler metric on $M$ is a function $F: TM\rightarrow [{0,\infty})$ with the following properties:

(a) $F$ is $C^{\infty}$ on $TM \backslash\{0\}$;

(b) At any point $x\in M, F_x(y):=F(x,y)$ is a Minkowski norm on $T_xM$, we call the pair $(M,F)$ an $n$-dimensional Finsler manifold.

Let $(M,F)$ be a Finsler manifold and $ g_{ij}(x,y):=\frac{1}{2}[F^{2}(x,y)]_{y^{i}y^{j}}. $ For any non-zero vector $y=y^{i}{\frac{\partial}{\partial{x}^{i}}}\in T_{x}{M}$, $F$ induces an inner product $\textbf{g}_{y}$ on $T_{x}{M}$ as $ \textbf{g}_{y}(u,v):=g_{ij}(x,y)u^{i}v^{j}, $ where $u=u^{i}{\frac{\partial}{\partial{x}^{i}}}\in T_{x}{M}$, $v=v^{i}{\frac{\partial}{\partial{x}^{i}}}\in T_{x}{M}$.

The geodesic $\sigma=\sigma(t)$ of a Finsler metric $F$ is characterized by the following system of 2nd order ordinary differential equations

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

where $ G^{i}:=\frac{1}{4}g^{il}\{{[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}}\}, $ where $(g^{ij})=(g_{ij})^{-1}$. $G^{i}$ are called the geodesic coefficients of $F$.

By the definition, an $(\alpha,\beta)$-metric is a Finsler metric expressed in the following form

$F=\alpha\phi(s),\ s=\frac{\beta}{\alpha},$

where $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and $\beta= b_{i}(x)y^{i}$ is a 1-form with $\|\beta_{x}\|_{\alpha}<b_{0},\ x\in M$. It is proved (see [12]) that $F = \alpha \phi(\beta/\alpha)$ is a positive definite Finsler metric if and only if the function $\phi = \phi(s)$ is a $C^{\infty}$ positive function on an open interval $(-b_{0}, b_{0})$ satisfying

$\phi(s)-s\phi'(s)+(b^{2}-s^{2})\phi''(s)>0,\ \ |s|\leq b<b_{0}.$

In particular, when $\phi =1+s$, the metirc $F=\alpha \phi (\beta/\alpha)$ is just the Randers metric $F=\alpha+\beta$. Let $G^{i}$ and $G^{i}_{\alpha}$ denote the geodesic coefficients of $F$ and $\alpha$, respectively. Denote

$ r_{ij}:=(b_{i|j}+b_{j|i}),\ \ s_{ij}:=\frac{1}{2}(b_{i|j}-b_{j|i}), \\ s^{i}_{\ j}:=a^{il}s_{lj}, \ \ s_{i}:=b^{j}s_{ji},\ \ s_{0}:=s_{i}y^{i}, \ \ r_{00}:=r_{ij}y^{i}y^{j}, $

where $(a^{ij}):=(a_{ij})^{-1}$ and $b_{i|j}$ denote the covariant derivative of $\beta$ with respect to $\alpha$. Then we have

Lemma 2.1 (see [12]) The geodesic coefficients of $G^{i}$ are related to $G^{i}_{\alpha}$ by

$ G^{i}=G^{i}_{\alpha}+\alpha Qs^{i}_{\ 0}+\{-2Q\alpha s_{0}+r_{00}\}\{\Psi b^{i}+\Theta \alpha^{-1}y^{i}\},\label{GFGa} $

(2.1)

where $s^{i}_{\ 0}:=s^{i}_{\ j}y^{j}$ and

$ Q: = \frac{\phi'}{\phi-s\phi'}, \Theta: = \frac{\phi\phi'-s(\phi\phi''+\phi'\phi')}{2\phi\big[(\phi-s\phi')+(b^{2}-s^{2})\phi''\big]}, \Psi: = \frac{\phi''}{2\big[(\phi-s\phi')+(b^{2}-s^{2})\phi'')\big]}. $

In order to prove our theorems, we need some lemmas about locally dually flat $(\alpha, \beta)$-metrics. Shen, Zhou and the second author first characterized locally dually flat Randers metrics and obtained the following lemma.

Lemma 2.2 (see [4]) Let $F=\alpha +\beta $ be a Randers metric on an $n$-dimensional manifold $M.$ Then $F$ is locally dually flat if and only if in an adapted coordinate system, $ \beta $and $\alpha $ satisfy

${r}_{00}= \frac{2}{3}\theta \beta -\frac{5}{3}\tau {{\beta }^{2}}+[ \tau +\frac{2}{3}(\tau {{b}^{2}}-{{b}_{m}}{{\theta }^{m}})]{{\alpha }^{2}},$

(2.2)

$ {s}_{k0}= -\frac{1}{3}( \theta {{b}_{k}}-\beta {{\theta }_{k}}),$

(2.3)

$ G_{ \alpha }^{m}= \frac{1}{3}( 2\theta +\tau \beta){{y}^{m}}-\frac{1}{3}( \tau {{b}^{m}}-{{\theta }^{m}} ){{\alpha }^{2}},$

(2.4)

where $\tau =\tau(x)$ is a scalar function and $\theta ={{\theta }_{k}}{{y}^{k}}$ is a 1-form on $M$ and ${{\theta }^{m}}:={{a}^{mk}}{{\theta }_{k}}$.

Later, Xia characterized locally dually flat $(\alpha, \beta)$-metrics.

Lemma 2.3 (see [5]) Let $F=\alpha \phi (\beta/\alpha)$ be an $(\alpha ,\beta)$-metric on an $n$-dimensional manifold $M \ ( n\ge 3)$. Suppose $F$ is not Riemannian and $\phi$ satisfies one of the following:

(i) $\phi(s)$ is a polynomial of $s$ with $\phi '(0)=0$;

(ii) $\phi(s)$ is an analytic function with $\phi '(0)=\phi ''(0)=0$;

(iii) $\phi'(0)\neq 0,\ s({{k}_{2}}-{{k}_{3}}{{s}^{2}})(\phi \phi '-s\phi {{'}^{2}}-s\phi \phi '')-( \phi {{'}^{2}}+\phi \phi '')+{{k}_{1}}\phi( \phi -s\phi ' )\ne 0$, where $k_1, \ k_2$ and $k_3$ are constants. Then $F$ is locally dually flat on $M$ if and only if $\alpha $ and $\beta$ satisfy

$s_{l0}=\frac{1}{3}( \beta\theta_{l}-\theta b_{l}),$

(2.5)

$ r_{00}=\frac{2}{3}[\theta \beta -(\theta_{l}b^{l})\alpha^{2}],$

(2.6)

$G_{\alpha }^{l}=\frac{1}{3}(2\theta y^{l}+\theta^{l}\alpha^{2}),$

(2.7)

where $\theta:=\theta_{i}(x)y^{i}$ is a 1-form on $M$ and $\theta^{l}:=a^{lk}\theta_{k}$.

Now we are in the position to prove the theorems. First, we prove Theorem 1.1.

Proof of Theorem 1.1 Let $F=\alpha\phi(\beta/\alpha)$ and $\bar{F}=\bar{\alpha}\phi(\bar{\beta}/\bar{\alpha})$ be two $(\alpha, \beta)$-metrics. If $F$ and $\bar{F}$ are conformally related, that is $F=e^{\sigma(x)}\bar{F}$, then we have the following relations:

$\bar{\alpha}= e^{-\sigma(x)}\alpha, \ \ \ \ \ \bar{\beta}=e^{-\sigma(x)}\beta, \bar{a}_{ij}=e^{-2\sigma(x)}a_{ij}, \ \ \ \ \ {\bar{b}_{i}}=e^{-\sigma(x)}b_{i},\\ \bar{b}_{i\|j}={{e}^{-\sigma }}({b}_{i|j}+{b}_{j}\sigma_{i}-b_{r}\sigma^{r}a_{ij}),$

(3.1)

$\bar{r}_{ij}=e^{-\sigma(x)}(r_{ij}+\frac{1}{2}\sigma_ib_j+\frac{1}{2}\sigma_jb_i-{{b}_{r}}{{\sigma}^{r}}{{a}_{ij}}),$

(3.2)

$\bar{s}_{ij}=e^{-\sigma(x)}(s_{ij}+\frac{1}{2}\sigma_ib_j-\frac{1}{2}\sigma_jb_i),$

(3.3)

where ${\sigma _i}: = \frac{{\partial \sigma }}{{\partial {x^i}}}$, $\sigma^i:=a^{ij}\sigma_j$, and "$\|$" denotes the covariant derivative with respect to $\bar{\alpha}$.

Let $F=\alpha+\beta$ and $\bar{F}=\bar{\alpha}+\bar{\beta}$ be two Randers metrics and $F=e^{\sigma(x)}\bar{F}$. Then the above relations still hold. Assume $F$ is conformally flat, then $\bar{F}$ is Minkowskian. In this case, ${\bar{b}_{i\|j}}=0$ and (3.1), (3.2), (3.3) are reduced to:

$b_{i|j}=b_{r}\sigma^{r}a_{ij}-b_{j}\sigma_{i},$

(3.4)

$r_{ij}=b_{r}\sigma^{r}a_{ij}-\frac{1}{2}\sigma_ib_j-\frac{1}{2}\sigma_jb_i,$

(3.5)

$s_{ij}=\frac{1}{2}\sigma_jb_i-\frac{1}{2}\sigma_ib_j.$

(3.6)

For any Finsler metric $F$, the geodesic coefficients $G^i$ can be expressed as:

$G^i=\frac{1}{4}g^{il}\{(F^2)_{x^ky^l}y^k-(F^2)_{x^l}\}.$

(3.7)

In particular, for $\bar{\alpha}$ and $\alpha$, by (3.7), their geodesic coefficients $G^i_{\bar{\alpha}}$ and $G^i_{\alpha}$ have the relation

$G^i_{\bar{\alpha}}=G^i_{\alpha}-\sigma_0y^i+\frac{1}{2}\alpha^2\sigma^i,$

(3.8)

where $\sigma_{0}:=\sigma_{k}y^{k}$ and $\sigma^{i}:=a^{il}\sigma_{l}$.

If $F$ is locally dually flat, then Lemma 2.2 holds for $F$. Note that $\alpha$ is also conformally flat since $F$ is conformally flat, then $\bar{\alpha}$ is Euclidean and $G^i_{\bar{\alpha}}=0$. Combining (2.4) and (3.8) yields

$\{\frac{1}{3}(2\theta + \tau \beta )-\sigma_0\}y^i=\{\frac{1}{3}(\tau b^i-\theta^i)-\frac{1}{2}\sigma^i\}\alpha^2.$

For the dimension of manifold $M$ satisfies $n\geq3$ and $\alpha^2$ is not divisible in this circumstances, we immediately have $\sigma^i=\frac{2}{3}(\tau b^i-\theta^i), \sigma_0=\frac{1}{3}(2\theta+\tau\beta).$ Comparing the above two equations, one easily has

$\theta_i=\frac{1}{4}\tau b_i.$

(3.9)

Combining (2.2), (3.5) and (3.9) we get

$(\frac{3}{2}\tau \beta-\sigma_0)\beta=(t+\tau+\frac{1}{2}\tau b^2)\alpha^2,$

(3.10)

where $t:=b_i\sigma^i$.

When $n\geq3$, $\alpha^2$ is indivisible, then from (3.10) we have

$\sigma_i=\frac{3}{2}\tau b_i,$

(3.11)

$t+\tau+\frac{1}{2}\tau b^2=0.$

(3.12)

Plugging (3.11) into (3.12) yields $\tau(1+2b^2)=0$. Considering that $1+2b^{2}\neq0$, one has $\tau=0$. Then $\sigma_i=0$, i.e., $\sigma$ is a constant. In this case, $F$ is Minkowskian.

In the end, we are going to prove Theorem 1.2.

Proof of Theorem 1.2 Assume that $F=\alpha\phi(ta/\alpha) $ is an$(\alpha, \beta)$-metric satisfying the conditions in Theorem 1.2, $\alpha=e^{\sigma(x)}\bar{\alpha} $and $\alpha $is conformally flat. Then $\bar{\alpha}$ is Euclidean and (2.5), (2.6), (2.7) in Lemma 2.3 hold. By (2.7) and (3.8) we have

$(\frac{2}{3}\theta-\sigma_0)y^i=(-\frac{1}{2}\sigma^i-\frac{1}{3}\theta^i)\alpha^2. $

(3.13)

Then by (3.13) and the fact that $\alpha^2$ is indivisible when $n\geq3$ again, naturally we get

$ \theta_i=\frac{3}{2}\sigma_i,$

(3.14)

$ \theta^i=-\frac{3}{2}\sigma^i. $

(3.15)

We use $a_{ij}$ to lower the index of (3.15) and obtain

$ \theta_i=-\frac{3}{2}\sigma_i. $

(3.16)

Comparing (3.14) with (3.16), instantly we conclude $\sigma_i=0$ and $\theta_i=0$. Then $\sigma$ is a constant and obviously $\alpha$ is Euclidean. According to (2.5) and (2.6), we get $s_{ij}=0$ and $r_{ij}=0$, which implies that $\beta$ is parallel with respect to $\alpha$. Therefore, $F$ is Minkowskian.