Finsler manifolds with nonpositive flag curvature and constant S-curvature
Finsler manifolds with nonpositive flag curvature and constant S-curvature
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DOI：
10.1007/s00209-004-0725-1
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发表时间：
2003-11
影响因子：
0.8
通讯作者：
Z. Shen
中科院分区：
文献类型：
--
作者：
Z. Shen
旗曲率是黎曼几何中截面曲率的自然推广，S-曲率是一个非黎曼量，对于黎曼度量为零。Rn中具有负旗曲率和常S曲率的开子集上存在（不完全）非Riemann Finsler度量.本文证明了一个整体刚性定理，即如果流形是紧的，则每个具有负旗曲率和常S曲率的Finsler度量必是黎曼度量。我们还研究了非正旗曲率的情况。
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 Rnwith 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.