The geometry of geodesic invariant functions and applications to Landsberg surfaces

In this paper, for a given spray $S$ on an $n$-dimensional manifold $M$, we investigated the geometry of $S$-invariant functions. For an $S$-invariant function $\mathcal{P}$, we associated a vertical subdistribution ${\mathcal{V}}_{\mathcal{P}}$ and found the relation between the holonomy distribution and ${\mathcal{V}}_{\mathcal{P}}$ by showing that the vertical part of the holonomy distribution is the intersection of all spaces ${\mathcal{V}}_{{\mathcal{F}}_S}$ associated with ${\mathcal{F}}_S$ where ${\mathcal{F}}_S$ is the set of all Finsler functions that have the geodesic spray $S$. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $S$-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $S$-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $H$-invariant if and only if it is constant.