Some Rigidity Conditions on Berwald Structures

TL;DR

This paper explores rigidity conditions of Berwald Finsler structures using the method of average, establishing criteria for when a Finsler structure is Berwald or Landsberg, with implications for their geometric properties.

Contribution

It introduces new geodesic rigidity conditions for Berwald spaces and characterizes when a Finsler structure is Berwald based on Levi-Civita connection invariance.

Findings

Levi-Civita connection of Riemannian metrics affine equivalent to Berwald metrics leaves the indicatrix invariant.

A converse condition characterizes Berwald structures via Levi-Civita connection invariance.

Provides a necessary condition distinguishing Landsberg structures that are not Berwald.

Abstract

This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure such that there is a Riemannian connection whose Levi-Civita leaves invariant by parallel transport the indicatrix of the Finsler structure, then the structure (M,F) is Berwald. As an application we obtain a necessary condition for a structure to be Landsberg but not Berwald.