The flag curvature is a natural extension of the sectional curvature inRiemannian geometry, and the S-curvature is a non-Riemannian quantity whichvanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finslermetrics on an open subset in R^n with negative flag curvature and constantS-curvature. In this paper, we are going to show a global rigidity theorem thatevery Finsler metric with negative flag curvature and constant S-curvature mustbe Riemannian if the manifold is compact. We also study the nonpositive flagcurvature case.
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