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An Algebraic Approach to Weakly Symmetric Finsler Spaces

Part of: Lie groups Infinite-dimensional manifolds Global differential geometry

Published online by Cambridge University Press:  20 November 2018

Shaoqiang Deng
Shaoqiang Deng*
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China, e-mail: dengsq@nankai.edu.cn
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Abstract

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In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing $\text{S}$ -curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing $\text{S}$ -curvature may exist at large. Hence the generalized volume comparison theorems due to $\text{Z}$ . Shen are valid for a rather large class of Finsler spaces.

Keywords

weakly symmetric Finsler spacesweakly symmetric Lie algebrasBerwald spacesS-curvature

MSC classification

Secondary: 53C60: Finsler spaces and generalizations (areal metrics) 58B20: Riemannian, Finsler and other geometric structures 22E46: Semisimple Lie groups and their representations 22E60: Lie algebras of Lie groups

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 1 , 01 February 2010 , pp. 52 - 73
Copyright
Copyright © Canadian Mathematical Society 2010

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