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Locally symmetric Finsler spaces in negative curvature. (English. Abridged French version) Zbl 0882.53051

A Finsler metric \(F:TM \to\mathbb{R}\) is called reversible, if \(F(x,v) =F(x,-v)\) holds, and locally symmetric, if, for any point, the geodesic reflection is a local isometry. A reversible, locally symmetric and \(C^3\) Finsler metric is parallel, that is, the curvature \(R\) satisfies \(D_x R=0\) for a connection \(D_x\). The main purpose of the present note is to announce that a compact parallel Finsler space with negative curvature is isometric to a locally symmetric negatively curved Riemannian space.
Reviewer’s remark: We call attention to Ding-Kia Shing’s paper [Chinese Math. 9(1967), 498-506 (1958; Zbl 0164.52502)]. The main result of this paper is: In a domain \(\Omega\), in order for the symmetric translation to realize precisely the parallel translation along a geodesic passing through a fixed point, it is necessary and sufficient that \(A^i_{jk|h} =0\) and \(R^i_{jkl |h} =0\) in \(\Omega\). It seems to the reviewer that Shing’s paper has some mistakes.

MSC:

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

Citations:

Zbl 0164.52502
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