In typical studies of Finsler geometry, the metric function $F: TM \to [0,\infty)$ is assumed to be smooth away from the zero section, and $F$ is assume to be sufficiently convex. Under these assumptions the geodesic spray is well-defined.
- Formally writing down the Euler-Lagrange equation for the length minimizer one sees that to apply Picard-Lindelof we would want $F$ to be at least $C^{2,1}$ (for simplicity let's work on a local coordinate patch).
- The convexity comes in because the ELE looks formally like $$ A(\gamma, \dot{\gamma}) \cdot \ddot{\gamma} = N(\gamma, \dot{\gamma}) $$ where $\gamma$ is vector valued and $A$ is a matrix of coefficients; the convexity would ensure that $A$ is invertible.
Are there any articles/surveys that describes (preferably with examples), what exactly goes wrong to the geodesic structure of the manifold if either $F$ is too rough, or if $F$ has flat points? Naively, I would expect to see something like the following:
- If $F$ is too rough, there may be some sort for "refraction" in the geodesic spray.
- If $F$ is insufficiently convex (so that $A$ is singular), locally the geometry picks up some sub-Riemannian flavor.
Has this been examined in the literature?