Randers metrics with compatible linear connections: a coordinate-free approach

Abstract

A Randers space is a differentiable manifold equipped with a Randers metric. It is the sum of a Riemannian metric and a one-form on the base manifold. The compatibility of a linear connection with the metric means that the parallel transports preserve the Randers norm of tangent vectors. The existence of such a linear connection is not guaranteed in general. If it does exist then we speak about a generalized Berwald Randers metric. In what follows we give a necessary and sufficient condition for a Randers metric to be a generalized Berwald metric and we describe some distinguished compatible linear connections. The method is based on the solution of constrained optimization problems for tensors that are in one-to-one correspondence to the compatible linear connections. The solutions are given in terms of explicit formulas by choosing the free tensor components to be zero. Throughout the paper we use a coordinate-free approach to keep the geometric feature of the argumentation as far as possible.