On Randers manifolds with semi-symmetric compatible linear connections

Let MM be a differentiable manifold equipped with a Riemannian metric tensor. If we apply translations in the tangent spaces to the Riemannian unit balls such that the translated bodies are keeping the origin in their interiors then they are working as unit balls for a Randers manifold. The unit balls allow us to measure the length of tangent vectors with the help of the induced Minkowski functionals. Analytically these Minkowski functionals are coming from a Riemannian metric tensor by using one-form perturbation in the tangent spaces. Manifolds equipped with a smoothly varying family of Minkowski functionals are called Finsler manifolds. The one-form perturbation of a Riemannian metric in the tangent spaces results in the class of Randers manifolds introduced by G. Randers in 1941. They occur naturally in physical applications related to electron optics, navigation problems or the Lagrangian of relativistic electrons. In this paper we are interested in Randers manifolds with semi-symmetric compatible linear connections. Compatibility means that the parallel transports preserve the length of tangent vectors with respect to the perturbed metric. If the torsion is decomposable in a special way then we speak about a semi-symmetric linear connection. Up to local isometries we characterize Riemannian manifolds admitting a one-form perturbation such that the resulting Randers manifold has a compatible semi-symmetric linear connection. As a paraphrase of our previous work (Vincze, 2006) communicated by Professor J.J. Duistermaat we present an existence theorem of generalized Berwald manifolds with semi-symmetric compatible linear connections. The terminology goes back to M. Matsumoto, M. Hashiguchi and S. Bácsó’s work (Bácsó et al., 1997).