  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 $B_p(R)$.
