On the 1953 Barthel connection of a Finsler space and its mathematical and physical interpretation

Abstract

A precise formulation of the Barthel connection in Finsler spaces is given, based on the general theory of connections in fibre bundles. It is shown that, if a nonzero vector field Y is given, the Barthel connection is closely related to the Cartan connection and that then an adjoint Riemannian space can be defined and the Barthel connection is metrical in the Riemannian space. A short discussion of the Finsler spaces with (α,β)-metric is given and the problem of physical interpretation is briefly remarked. It seems, in the light of the result of our paper, that the Barthel connection can be considered as the connection of a Finsler space with a distinguished vector field, similarly as the Levi-Civita connection is the connection of a Riemannian space.