On generalized Berwald manifolds of dimension three

TL;DR

This paper characterizes compatible linear connections on three-dimensional Finsler manifolds, revealing conditions for their uniqueness and relating non-uniqueness to Euclidean symmetries of indicatrices.

Contribution

It introduces an intrinsic method to characterize compatible connections on 3D Finsler manifolds and links their non-uniqueness to Euclidean surface symmetries.

Findings

Compatible connections are unique unless indicatrices are Euclidean surfaces of revolution.

Non-uniqueness of connections relates to Euclidean symmetries.

Additional conditions can ensure the uniqueness of compatible connections.

Abstract

A linear connection on a Finsler manifold is called compatible to the Finsler function if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. In the paper we present a general and intrinsic method to characterize the compatible linear connections on a Finsler manifold of dimension three. We prove that if a compatible linear connection is not unique then the indicatrices must be Euclidean surfaces of revolution. The surplus freedom of choosing compatible linear connections is related to Euclidean symmetries. The unicity of the solution of the compatibility equations can be provided by some additional requirements. Following the idea in \cite{V14} we are also looking for the so-called extremal compatible linear connection minimizing the norm of its torsion at each point of the manifold.