The geometry of geodesic invariant functions and applications to Landsberg surfaces

Abstract

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.

Keywords

• spray
• holonomy distribution
• $ s $-invariant functions (first integrals)
• landsberg surfaces
• flag curvature