An elementary approach to gap theorems
An elementary approach to gap theorems
复制标题
间隙定理的基本方法
DOI：
10.1007/s12044-009-0020-5
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
H. Seshadri
中科院分区：
文献类型：
--
作者：
H. Seshadri
利用初等比较几何的方法证明了：设(M，g)是一维≥3的单连通完备黎曼流形.设截面曲率K满足−1−S(R)≤K≤−1，其中r表示到M中不动点的距离.如果Limr→∞e2Rs(R)=0，则(M，g)必与ℍn等距.同样证明了如果K满足−S(R)≤K≤0其中Limr→∞R2S(R)=0，则(M，g)与ℝn等距.我们的第二个结果是局部的：设(M，g)是任意黎曼流形。对于∈ℝ，如果K≤a在M中的测地球BP(R)上且K=a在∂BP(R)上，则K=a在BP(R)上。
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If limr → ∞ e2rs(r) = 0, then (M, g) has to be isometric to ℍn.The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where limr → ∞r2s(r) = 0, then (M, g) is isometric to ℝn, a result due to Greene and Wu.Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball Bp(R) in M and K = a on ∂Bp(R), then K = a on Bp(R).