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 weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature

MSC Classifications:

53C60, 58B20, 22E46, 22E60 show english descriptions Finsler spaces and generalizations (areal metrics) [See also 58B20]
Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]
Semisimple Lie groups and their representations
Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 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}

top of page | contact us | social media | privacy | site map