Finsler Manifolds with Nonpositive Flag Curvature and Constant S-curvature

Details Finsler Manifolds with Nonpositive Flag Curvature and Constant S-curvature Finsler Manifolds with Nonpositive Flag Curvature and Constant S-curvature

2003-11-14

arxiv.org/abs/math/0311232v1

Mathematics

Differential Geometry

15 pages

Scientific paper

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 R^n 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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