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 ${\lim }_{r\text{rt}\mathrm{\infty }}{e}^{2r}s(r)=0$, 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 ${\lim }_{r\text{rt}\mathrm{\infty }}{r}^{2}s(r)=0$, 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 $a\in \mathbb{R}$, if $K\le a$ on a geodesic ball $B_p(R)$ in $M$ and $K=a$ on $\mathrm{\partial }{B}_p(R)$, then $K=a$ on $B_p(R)$.