On a class of Landsberg metrics in Finsler geometry

Abstract

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We show that a regular Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, φ = φ(s), for which there are a Riemannian metric α and a 1-form β on a manifold M such that the scalar function F = αφ(β/α) on TM is an almost regular Landsberg metric, but not a Berwald metric. © Canadian Mathematical Society 2009.