The simplest form of Finsler metric is: $ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynomial that smoothly depends on $X$.

My question is: is it always true that $s^4(X)$, the fourth power of the geodesic distance from an origin point $O$ to $X$, constructed from the Finsler metric $ds$, is everywhere smooth?

My question arises from the fact that, when we take a Riemmanian metric $ds (X, dx)=\sqrt[2]{g_{\alpha, \beta}(X).dx^{\alpha}dx^{\beta}}$, then we know that the squarred geodesic distance $s^2(X)$ is everywhere smooth. Smoothness is easy to prove outside the origin $O$. At the origin, we need to use this argument: the exponential map is smooth and radially isometric.

Unfortunately, this argument seems not to be decisive in the Finslerian case, where the exponential map is not anymore smooth at the origin. I don't find any answer to my question in classical books on Finsler geometry.

Thanks.

The simplest form of Finsler metric is: $ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynomial that smoothly depends on $X$.

My question is: is it always true that $s^4(X)$, the fourth power of the geodesic distance from an origin point $O$ to $X$, constructed from the Finsler metric $ds$, is everywhere smooth on a neighboorhood of $O$?

My question arises from the fact that, when we take a Riemmanian metric $ds (X, dx)=\sqrt[2]{g_{\alpha, \beta}(X).dx^{\alpha}dx^{\beta}}$, then we know that the squarred geodesic distance $s^2(X)$ is everywhere smooth on a neighboord of $O$. Smoothness is easy to prove outside the origin $O$. At the origin, we need to use this argument: the exponential map is smooth and radially isometric.

Unfortunately, this argument seems not to be decisive in the Finslerian case, where the exponential map is not anymore smooth at the origin. I don't find any answer to my question in classical books on Finsler geometry.

Thanks.

The simplest form of Finsler metric is: $ds (X, dx)=^{4}\sqrt{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynom that smoothly depends on $X$.

My question is: is it always true that $s^4(X)$, the fourth power of the geodesic distance from an origin point $O$ to $X$, constructed from the Finsler metric $ds$, is everywhere smooth?

My question arises from the fact that, when we take a Riemmanian metric $ds (X, dx)=^{2}\sqrt{g_{\alpha, \beta}(X).dx^{\alpha}dx^{\beta}}$, then we know that the squarred geodesic distance $s^2(X)$ is everywhere smooth. Smoothness is easy to prove outside the origin $O$. At the origin, we need to use this argument: the exponential map is smooth and radially isometric.

Unfortunately, this argument seems not to be decisive in the Finslerian case, where the exponential map is not anymore smooth at the origin. I don't find any answer to my question in classical books on Finsler geometry.

Thanks.

The simplest form of Finsler metric is: $ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynomial that smoothly depends on $X$.

My question is: is it always true that $s^4(X)$, the fourth power of the geodesic distance from an origin point $O$ to $X$, constructed from the Finsler metric $ds$, is everywhere smooth?

My question arises from the fact that, when we take a Riemmanian metric $ds (X, dx)=\sqrt[2]{g_{\alpha, \beta}(X).dx^{\alpha}dx^{\beta}}$, then we know that the squarred geodesic distance $s^2(X)$ is everywhere smooth. Smoothness is easy to prove outside the origin $O$. At the origin, we need to use this argument: the exponential map is smooth and radially isometric.

Unfortunately, this argument seems not to be decisive in the Finslerian case, where the exponential map is not anymore smooth at the origin. I don't find any answer to my question in classical books on Finsler geometry.

Thanks.