Two nonrelated Finsler structures on a manifold

TL;DR

This paper investigates the geometric relationships between two unrelated Finsler structures on the same manifold by analyzing their Cartan connections and intrinsic properties without using local coordinates.

Contribution

It introduces a $ pi$-tensor to compare the Cartan connections of two Finsler structures and derives conditions for their geometric objects to share properties.

Findings

Conditions for shared geodesic properties

Criteria for similar Jacobi fields

Characterization of when structures are Berwald or Landsberg

Abstract

In the present paper, we consider two different {\em Finsler} structures $L$ and $L^*$ on the same base manifold $M$, with no relation preassumed between them. \par Introducing the $\pi$-tensor field representing the difference between the Cartan connections associated with $L$ and $L^*$, we investigate the conditions, to be satisfied by this $\pi$-tensor field, for the geometric objects associated with $L$ and $L^*$ to have the same properties. Among the items investigated in the paper, we consider the properties of being a geodesic, a Jacobi field, a Berwald manifold, a locally Minkowskian manifold and a Landsberg manifold. \par It should be noticed that our approach is intrinsic, i.e., it does not make use of local coordinate techniques.