# An elementary approach to gap theorems

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## Abstract

Using elementary comparison geometry, we prove: Let (*M, g*) be a simply-connected 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 lim_{ r → ∞} e^{2r } *s*(*r*) = 0, then (*M, g*) has to be isometric to ℍ^{ n }.

The same proof also yields that if *K* satisfies −*s*(*r*) ≤ *K* ≤ 0 where lim_{ r → ∞} *r* ^{2} *s*(*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* ∈ ℝ, if *K* ≤ *a* on a geodesic ball *B* _{ p }(*R*) in *M* and *K* = *a* on *∂B* _{ p }(*R*), then *K* = *a* on *B* _{ p }(*R*).

## Keywords

Riemannian manifold sectional curvature volume comparison hyperbolic space## Preview

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