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

TL;DR

The work investigates when a Randers metric $F=\sqrt{\alpha(v,v)}+\beta(v)$ on a manifold $M$ admits a linear connection compatible with $F$, meaning parallel transport preserves the Randers norm. It recasts compatibility as $\nabla\alpha=0$ and $\nabla\beta=0$, and solves constrained tensor optimization to obtain a explicit difference tensor $A$ yielding a compatible connection; solvability requires the dual $\beta^{\sharp}$ to have constant Riemannian length $K$, with $K=0$ recovering the Riemannian case and $K>0$ giving a concrete generalized Berwald Randers metric. The paper then minimizes torsion by introducing a skew part $B$, derives the extremal connection condition (compatibility with an integrable projected distribution), and provides a torsion formula together with local component expressions. Overall, the approach is coordinate-free and constructive, delivering explicit criteria and formulas for generalized Berwald Randers metrics and clarifying the geometric/topological obstructions to their existence.

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.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Theorem 5
• Remark 2