Abstract.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.
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supported by the National Natural Science Foundation of China (10371138).
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Shen, Z. Finsler manifolds with nonpositive flag curvature and constant S-curvature. Math. Z. 249, 625–639 (2005). https://doi.org/10.1007/s00209-004-0725-1
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DOI: https://doi.org/10.1007/s00209-004-0725-1