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On generalized symmetric Berwald spaces. (English) Zbl 1197.53096

A connected Riemannian manifold \((M, g)\) is said to be generalized symmetric if, at each point \(x\) of \(M\), there is an isometry \(s_x\) of \((M,g)\) with \(x\) as an isolated fixed point. The family \(\{s_x\mid x\in M\}\) is called an s-structure on \((M,g)\). In this paper, the authors generalize this notion to Berwald spaces and study some fundamental properties of such spaces. They prove that each parallel s-structure on a Berwald space must be regular and that any generalized symmetric Berwald surface must be symmetric.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C35 Differential geometry of symmetric spaces
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