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Proceedings - Mathematical Sciences

, Volume 119, Issue 2, pp 197–201 | Cite as

An elementary approach to gap theorems

  • Harish SeshadriEmail author
Article
  • 65 Downloads

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 → ∞ e2r 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 Ka 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 

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Copyright information

© Indian Academy of Sciences 2009

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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