An elementary approach to some rigidity theorems

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 $B_p(R)$.