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ON THE FLAG CURVATURE OF FINSLER METRICS OF SCALAR CURVATURE

Published online by Cambridge University Press:  17 November 2003

XINYUE CHEN ,
XIAOHUAN MO  and
ZHONGMIN SHEN
XINYUE CHEN
Affiliation:
Department of Mathematics, Chongqing Institute of Technology, Chongqing 400050, Chinachenxy58@163.net
XIAOHUAN MO
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, Chinamoxh@pku.edu.cn
ZHONGMIN SHEN
Affiliation:
Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USAzshen@math.iupui.edu
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Abstract

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.

Keywords

53B4053C60 (primary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 68 , Issue 3 , December 2003 , pp. 762 - 780
Copyright
The London Mathematical Society 2003

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Footnotes

The first author was supported by the National Natural Science Foundation of China (10171117). The second author was supported by the National Natural Science Foundation of China (10171002).