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Finiteness of \(\pi _{1}\) and geometric inequalities in almost positive Ricci curvature. (English) Zbl 1141.53034

This paper shows boundedness of the diameter and finiteness of the fundamental group of a complete \(n\)-manifold under a global \(L^p\)-control of the Ricci curvature for \(p > n/2\). It is a generalization of the S. B. Myers theorem [Duke Math. J. 8, 401–404 (1941; Zbl 0025.22704)] without unnatural extra hypotheses. It is also shown that metrics with similar \(L^{n/2}\)-control of their Ricci curvature are dense in the set of complete metrics of any compact differentiable manifold by considering the Gromov-Hausdorff distance.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
58D17 Manifolds of metrics (especially Riemannian)

Citations:

Zbl 0025.22704
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References:

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