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Integral curvature bounds and bounded diameter. (English) Zbl 0984.53018
The paper generalizes the result of Myers that a complete n-dimensional Riemannian manifold (M,g) with Ricci curvature Ric(M)(n-1)k>0 is compact, has diam(M)π/k and fundamental group π 1 (M) finite by using an inequality of Cheeger and Colding. The author proves that if (M,g) is compact and Ric(M)0, given δ>0 and a sufficiently small (depending on δ and n) integral involving the lowest eigenvalue of the Ricci tensor, then diam(M)<π+δ. If M is either not compact or the Ricci curvature is bounded nonpositively below, that is, Ric(M)(n-1)k with k0 but satisfies a similar integral curvature bound, then M is compact with bounded diameter and finite fundamental group. The integral curvature condition replaces bounds on diam(M) from above and vol(M) from below as were assumed in previous extensions of Myers’ result that π 1 (M) is finite.
MSC:
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)