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# On a Class of Landsberg Metrics in Finsler Geometry

Published:2009-12-01
Printed: Dec 2009
• Zhongmin Shen
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## 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 \emph{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, $\phi=\phi(s)$, for which there are a Riemannian metric $\alpha$ and a $1$-form $\beta$ on a manifold $M$ such that the scalar function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.
 MSC Classifications: 53B40 - Finsler spaces and generalizations (areal metrics) 53C60 - Finsler spaces and generalizations (areal metrics) [See also 58B20]