Foulon, Patrick Locally symmetric Finsler spaces in negative curvature. (English. Abridged French version) Zbl 0882.53051 C. R. Acad. Sci., Paris, Sér. I 324, No. 10, 1127-1132 (1997). 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. Reviewer: M.Matsumoto (Kyoto) Cited in 3 ReviewsCited in 10 Documents MSC: 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) Keywords:Finsler metric; reversible; locally symmetric; parallel Finsler space; negative curvature Citations:Zbl 0164.52502 × Cite Format Result Cite Review PDF Full Text: DOI