An elementary approach to some rigidity theorems

TL;DR

This paper employs elementary comparison geometry to establish new rigidity theorems for Riemannian manifolds with curvature bounds, characterizing when such manifolds are isometric to hyperbolic space or Euclidean space based on decay conditions.

Contribution

It provides elementary proofs of rigidity theorems for manifolds with curvature bounds, extending previous results and offering local curvature characterization.

Findings

Manifolds with curvature decay conditions are isometric to hyperbolic space or Euclidean space.

Elementary comparison geometry suffices for proving these rigidity results.

Local curvature conditions imply global curvature constancy under certain boundary conditions.

Abstract

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 $B_p(R)$.