**Problem.** *To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form.*

Some background may be useful since the problem is more of a problem in variational calculus than a problem in geometry. 

If $M$ is a smooth manifold and $L: TM \rightarrow \mathbb{R}$ is a (Lagrangian) fuction that is smooth outside the zero section and homogeneous of degree one in the velocities (i.e. $L(x,tv) = tL(x,v)$ for every $t > 0$), the variational problem 
$$
\gamma \mapsto \int_\gamma L
$$
is invariant under orientation-preserving reparametrization of the curve $\gamma$. The Lagrangian
$L$ is *geodesically reversible* if changing the orientation of any of its extremals yields another extremal. If the Lagrangian is *reversible* ($L(x,-v) = L(x,v)$), then it is geodesically reversible, but the converse is not true. For example, an asymmetric norm on $\mathbb{R}^n$ is geodesically reversible, but it is reversible if and only if the norm is symmetric. Another example can be constructed by taking a reversible Lagrangian and adding to it a closed $1$-form considered as a function on the tangent bundle that is linear in the velocities. This last example is somehow "trivial" and I would like to find many examples of geodesically reversible Lagrangians on compact manifolds that are **not** of this form. On the torus they are easy to construct because we can always compactify the example with the asymmetric norm, but what happens on the sphere? 

I can point to a non-solution: *If all the geodesics of a geodesically reversible Finsler metric on the $n$-sphere are closed and of the same length, then the metric is the sum of a reversible metric and an exact $1$-form.*  

**Added on 21/06/2021.**


**1.** The only examples of geodesically reversible $C^2$ Finsler metrics on closed manifolds that I know are still sums of flat metrics and closed $1$-forms or sums of reversible metrics and closed $1$-forms. 

**2.** *Let $M$ be a two- or three-dimensional manifold and let $F$ be
a $C^2$ geodesically reversible Finsler metric on M. If the geodesic flow does not admit any non-trivial continuous integral of motion, then the metric $F$ is necessarily the sum of a reversible metric and a closed 1-form.*

If the Nakajima-Schneider conjecture on bodies of constant breadth and constant brightness in relative geometry is true in dimensions greater than three, then the preceding result extends to all dimensions. 

**3.** If $M$ is a closed surface of genus greater than one and $F$ is a real-analytic, geodesically-reversible Finsler metric, then $F$ is the sum of a reversible metric and a closed $1$-form.

**4.** *Let $F$ be a $C^2$  geodesically-reversible Finsler metric on a manifold $M$. If every geodesic passes through some open set where $F$ is reversible, then $F$ is the sum of a reversible metric and a closed $1$-form.* 

See [Rigidity results for geodesically reversible Finsler metrics][1]


  [1]: https://arxiv.org/abs/2106.10095