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On a Class of Landsberg Metrics in Finsler Geometry

Published online by Cambridge University Press:  20 November 2018

Zhongmin Shen
Zhongmin Shen*
Affiliation:
Department of Mathematical Sciences, Indiana University Purdue University Indianapolis (IUPUI), 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA and Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang Province 310027, P.R. China email: zshen@math.iupui.edu
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Abstract

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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, $\phi \,=\,\phi \left( s \right)$ , for which there are a Riemannian metric $\alpha $ and a 1-form $\beta $ on a manifold $M$ such that the scalar function $F\,=\,\alpha \phi \left( \beta /\alpha \right)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.

Keywords

53B4053C60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 6 , 01 December 2009 , pp. 1357 - 1374
Copyright
Copyright © Canadian Mathematical Society 2009

References

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