An elementary approach to some rigidity theorems
| dc.creator | Seshadri, Harish | |
| dc.date | 2008-01-01 | |
| dc.date.accessioned | 2026-07-07T08:52:04Z | |
| dc.date.available | 2026-07-07T08:52:04Z | |
| dc.description | Using elementary comparison geometry, we prove: Let $(M,g)$ be a simply-connected complete Riemannian manifold of dimension $\ge 3$. Suppose that the sectional curvature $K$ satisfies $ -1-s(r) \le K \le -1$, where $r$ denotes distance to a fixed point in $M$. If $\lim_{r \rt \infty} e^{2r}s(r) =0$, then $(M,g)$ has to be isometric to ${\mathbb H}^n$. The same proof also yields that if $K$ satisfies $-s(r) \le K \le 0$ where $\lim_{r \rt \infty} r^2s(r)=0$, then $(M,g)$ is isometric to $\R^n$, a result due to Greene and Wu. Our second result is a local one: Let $(M,g)$ be any Riemannian manifold. For $a \in \R$, if $K \le a$ on a geodesic ball $B_p(R)$ in $M$ and $K = a$ on $\partial B_p(R)$, then $K= a $ on $B_p(R)$. | |
| dc.description | 5 Pages | |
| dc.identifier | https://arxiv.org/abs/0801.0285 | |
| dc.identifier | http://arxiv.org/abs/0801.0285 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/145155 | |
| dc.subject | Differential Geometry | |
| dc.title | An elementary approach to some rigidity theorems | |
| dc.type | text |