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An elementary approach to gap theorems

Seshadri, Harish (2009) An elementary approach to gap theorems. In: Proceedings Of The Indian Academy Of Sciences-Mathematical Sciences, 119 (2). pp. 197-201.

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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 ->infinity) 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 ->infinity) 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 E R, if K < a on a geodesic ball Bp (R) in M and K = a on partial derivative B-p (R), then K = a on B-p (R).

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Proceedings Of The Indian Academy Of Sciences.
Keywords: Riemannian Manifold;Sectional Curvature;Volume Comparison; Hyperbolic Space.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 10 Aug 2009 11:35
Last Modified: 19 Sep 2010 05:40
URI: http://eprints.iisc.ernet.in/id/eprint/22064

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