Isometries, submetries and distance coordinates on Finsler manifolds

This paper considers fundamental issues related to Finslerian iso- metries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Us- ing distance coordinates, a simple proof is given for the Finslerian version of the Myers-Steenrod theorem and for the differentiability of Finslerian submetries

Finslerian Lie derivative and Landsberg manifolds

In this paper we take a close look at Lie derivatives on a Finsler bundle and give a geometric meaning to the vanishing of the mixed curvature of certain covariant derivatives on a Finsler bundle. As an application, we obtain some characterizations of Landsberg manifolds

Several ways to a Berwald manifold - and some steps beyond

After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present several equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald’s classical definition to the existence of a torsion-free covariant derivative on the base manifold compatible with the Finsler function, the vanishing of the h-Berwald differential of the Cartan tensor and Aikou’s characterization of Berwald manifolds. Finally, we study some implications of V. Matveev’s observation according to which quadratic convexity may be omitted from the definition of a Berwald manifold. These include, among others, a generalization of Z.I. Szab´o’s well-known metrization theorem, and also lead to a natural generalization of Berwald manifolds, to Berwald { Matveev manifolds.The first two authors were supported by Hungarian Scientific Research Fund OTKA No. NK 81402.peerReviewe