An Elementary Approach to Gap Theorems

Proceedings Mathematical Sciences

Harish Seshadri¹ 

[1] Department of Mathematics, Indian Institute of Science, Bangalore 0 0, India$$

关键词: Riemannian manifold;    sectional curvature;    volume comparison;    hyperbolic space.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $lim_{r→∞} e^{2r}s(r)=0$, then (𝑀, 𝑔) has to be isometric to $mathbb{H}^n$.The same proof also yields that if 𝐾 satisfies $-s(r)≤ K≤ 0$ where $lim_{r→∞}r^2 s(r)=0$, then (𝑀, 𝑔)) is isometric to $mathbb{R}^n$, a result due to Greene and $Wu$.Our second result is a local one: Let (𝑀, 𝑔) be any Riemannian manifold. For a $inmathbb{R}$, if $K≤ a$ on a geodesic ball $B_p(R)$ in 𝑀 and $K=a$ on $𝜕 B_p(R)$, then $K=a$ on $B_p(R)$.

【 授权许可】

Unknown