On a Class of Landsberg Metrics in Finsler Geometry

http://dx.doi.org/10.4153/CJM-2009-064-9
Canad. J. Math. 61(2009), 1357-1374

  Published:2009-12-01
 Printed: Dec 2009

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.