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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Ricci Curvature ♦ Xue-mei Li ♦ Complete Riemannian Manifold ♦ Finite Fundamental Group ♦ Recent Result ♦ Negative Curvature ♦ Positive Curvature ♦ Myers Theorem ♦ Assume Ric ♦ Similar Result ♦ Recent Work ♦ Small Diameter ♦ Positive Number ♦ Potential Kernel ♦ Quick Proof ♦ Noncompact Manifold ♦ Compact Riemannian Manifold ♦ Early Result ♦ Little Bit ♦ Quick Probabilistic Proof ♦ Smooth Func-tion ♦ Bismut-witten Laplacian ♦ Small Volume ♦ Smooth Real-valued Function
Abstract Let M be a compact Riemannian manifold and h a smooth func-tion on M. Let ρh(x) = inf v =1 (Ricx(v, v) − 2Hess(h)x(v, v)). Here Ricx denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has finite fundamental group if ∆h − ρh < 0. Here ∆h =: ∆ + 2L∇h is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers ’ theorem to mani-folds with mostly positive curvature. There is also a similar result for noncompact manifolds. An early result of Myers says a complete Riemannian manifold with Ricci curvature bounded below by a positive number is compact and has finite fundamental group. See e.g. [9]. Since then efforts have been made to get the same type of result but to allow a little bit of negativity of the curvature (see Bérard and Besson[2]). Wu [12] showed that Myers ’ theorem holds if the manifold is allowed to have negative curvature on a set of small diameter, while Elworthy and Rosenberg [8] considered manifolds with some negative curvature on a set of small volume, followed by recent work of Rosenberg and Yang [10]. We use a method of Bakry [1] to obtain a result given in terms of the potential kernel related to ρ(x) = inf v =1Ricx(v, v), which gives a quick probabilistic proof of recent results on extensions of Myers ’ theorem. Here Ricx denotes the Ricci curvature at x. Let M be a complete Riemannian manifold, and h a smooth real-valued function on it. Assume Ric−2Hess(h) is bounded from below, where Hess(h)
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