An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simplyconnected 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 limr→∞ e2r s(r) = 0, then (M, g) has to be isometric to H. The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where limr→∞ r2 s(r) = 0, then (M, g) is isometric to R, a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ R, if K ≤ a on a geodesic ball Bp(R) in M and K = a on ∂Bp(R), then K = a on Bp(R).