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, we associate a vertical subdistribution _ and find the relation between the holonomy distribution and _ by showing that the vertical part of the holonomy distribution is the intersection of all spaces ₒ ₀ₒₒ₎₂₈₀ₓ₄₃ ₓ₎ ₒ ₖ₇₄ₑ₄ ₒ 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.