An elementary approach to gap theorems |
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Authors: | Harish Seshadri |
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Affiliation: | (1) Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India |
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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 → ∞ 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 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). |
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Keywords: | Riemannian manifold sectional curvature volume comparison hyperbolic space |
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