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# An Algebraic Approach to Weakly Symmetric Finsler Spaces

Published:2009-12-04
Printed: Feb 2010
• Shaoqiang Deng,
Nankai University, China
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## Abstract

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 S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.
 Keywords: weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature
 MSC Classifications: 53C60 - Finsler spaces and generalizations (areal metrics) [See also 58B20] 58B20 - Riemannian, Finsler and other geometric structures [See also 53C20, 53C60] 22E46 - Semisimple Lie groups and their representations 22E60 - Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx}

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