On a rigidity condition for Berwald Spaces

TL;DR

This paper establishes a characterization of Berwald spaces through Riemannian structures preserved by the Berwald connection and explores implications for Landsberg spaces, providing criteria for their existence.

Contribution

It introduces a new rigidity condition for Berwald spaces based on invariant Riemannian structures and offers a strategy to analyze pure Landsberg surfaces.

Findings

Riemannian structures preserved by the Berwald connection leave the indicatrix invariant.

Existence of an invariant Riemannian structure implies the space is Berwald.

Provides a necessary condition for pure Landsberg spaces.

Abstract

We show that which that for a Berwald structure, any Riemannian structure that is preserved by the Berwald connection leaves the indicatrix invariant under horizontal parallel transport. We also obtain the converse result: if $({\bf M},F)$ is a Finsler structure such that there exists a Riemannian structure that leaves invariant the indicatrix under parallel transport of the associated Levi-Civita connection, then the structure $({\bf M},F)$ is Berwald. As application, a necessary condition for pure Landsberg spaces is formulated. Using this criterion we provide an strategy to solve the existence or not of pure Landsberg surfaces