On compatible linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds

Abstract

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory (Vincze in J AMAPN 21:199–204, 2005) such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann–Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in terms of quantities of the Finsler manifold. In the paper we are going to give explicit formulas for the linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds (Theorem 2). The results are based on averaging of (intrinsic) Finslerian quantities by integration over the indicatrix surfaces. They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric (Theorems 3 and 4). The possible cases are Riemannian spaces of constant zero curvature, constant positive curvature or Riemannian spaces admitting Killing vector fields of constant Riemannian length.