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 investigate the geometry of $S$-invariant functions. For an $S$-invariant function $\mathrm{\P }$, we associate a vertical subdistribution ${\text{V}}_{\mathrm{\P }}$ and find the relation between the holonomy distribution and ${\text{V}}_{\mathrm{\P }}$ by showing that the vertical part of the holonomy distribution is the intersection of \ok{all spaces ${\text{V}}_{{\text{F}}_S}$ associated to ${\text{F}}_S$ where ${\text{F}}_S$} is the set of all Finsler functions that have the geodesic spray $S$. As an application, we study the Landsberg Finsler surfaces. We prove that a Landsberg surface with $S$-invariant flag curvature is Riemannian or has a vanishing flag curvature. We show 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 prove that the flag curvature is $H$-invariant if and only if it is constant.