  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.
