First integrals for Finsler metrics with vanishing $\chi$-curvature

TL;DR

This paper demonstrates that in Finsler manifolds with zero chi-curvature, certain non-Riemannian structures are invariant along geodesics, leading to new first integrals expressed via mean curvature tensors.

Contribution

It introduces explicit formulas for non-Riemannian first integrals in Finsler geometry with vanishing chi-curvature, linking them to mean curvature tensors.

Findings

Non-Riemannian structures are geodesically invariant.

First integrals can be expressed via mean Berwald curvature.

Alternative expressions involve mean Cartan torsion and Landsberg curvature.

Abstract

We prove that in a Finsler manifold with vanishing $\chi$-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.