ON LANDSBERG SPACES AND THE LANDSBERG-BERWALD PROBLEM

This paper is concerned with the geometry of a class of Finsler spaces called Landsberg spaces. A Landsberg space may be characterized by the fact that its fundamental tensor is covariant constant along horizontal curves with respect to its Berwald connection. A Finsler space whose Berwald connection is affine is called a Berwald space. Berwald spaces are necessarily Landsbergian, but whether there are y-global Landsberg spaces which are not of Berwald type is not known. Resolving this question is the Landsberg-Berwald problem of the title. The paper deals with several topics in Landsberg geometry which are related mainly by the possibility that the results obtained may throw light on the Landsberg-Berwald problem. It is assumed throughout that the dimension of the base manifold is at least 3. It is shown that a Landsberg space over a compact base, which is R-quadratic, is necessarily Berwaldian. A model for the holonomy algebra of a Landsberg space is proposed. Finally, the technique of averaging the fundamental tensor over the indicatrix is discussed, and it is shown that for a Landsberg space, with the correct interpretations, the averaged Berwald connection is the Levi-Civita connection of the averaged metric.