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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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