The geometry of geodesic invariant functions and applications to Landsberg surfaces

TL;DR

This paper explores the geometry of $S$-invariant functions on manifolds with a focus on Landsberg surfaces, revealing conditions under which these surfaces are Riemannian or have constant flag curvature, with implications for Finsler geometry.

Contribution

It establishes new relationships between $S$-invariant functions, holonomy, and flag curvature on Landsberg and Berwald surfaces, advancing understanding of their geometric properties.

Findings

Landsberg surfaces with $S$-invariant flag curvature are Riemannian or have zero flag curvature.

For non-vanishing flag curvature, invariance implies constancy, leading to Riemannian surfaces.

Flag curvature invariance on Berwald surfaces corresponds to constant curvature.

Abstract

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 $\P$, we associate a vertical subdistribution $\V_\P$ and find the relation between the holonomy distribution and $\V_\P$ by showing that the vertical part of the holonomy distribution is the intersection of \ok{all spaces $\V_{\F_S}$ associated to $\F_S$ where $\F_S$} 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.