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An algebraic approach to weakly symmetric Finsler spaces. (English) Zbl 1205.53078
A (connected) Finsler space $$(M,F)$$ is called weakly symmetric if, for any $$p, q \in M$$, there is an isometry $$\sigma \in I(M,F)$$ with $$\sigma(p)=q$$ and $$\sigma(q) = p$$. Weakly symmetric Finsler spaces are a generalization of weakly symmetric Riemannian spaces. Aiming at an algebraic description of weak symmetry, the author introduces the notion of a Riemannian weakly symmetric Lie algebra (a pair $$(\mathfrak{g},\mathfrak{h})$$ of Lie algebras satisfying certain conditions) and proves that, for every weakly symmetric Finsler space $$(M,F)$$, the pair $$(\mathfrak{g},\mathfrak{h})$$ where $$\mathfrak{g}$$ and $$\mathfrak{h}$$ are the Lie algebras of the group $$G = I(M,F)$$ and the isotropy group $$H$$ of some $$x \in M$$, respectively, is a Riemannian weakly symmetric Lie algebra. Conversely, any Riemannian weakly symmetric Lie algebra gives rise to a (non-uniquely defined) weakly symmetric Finsler space.
Examples of Riemannian weakly symmetric Lie algebras are given and weakly symmetric Finsler spaces of dimension $$\leq 3$$ are classified. The spaces found are reversible non-Berwaldian with vanishing S-curvature.

##### MSC:
 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 22E46 Semisimple Lie groups and their representations 22E60 Lie algebras of Lie groups
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