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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 $\left(M,g\right)$ with Ricci curvature $\text{Ric}\left(M\right)\ge \left(n-1\right)k>0$ is compact, has $\text{diam}\left(M\right)\le \pi /\sqrt{k}$ and fundamental group ${\pi }_{1}\left(M\right)$ finite by using an inequality of Cheeger and Colding. The author proves that if $\left(M,g\right)$ is compact and $\text{Ric}\left(M\right)\ge 0,$ given $\delta >0$ and a sufficiently small (depending on $\delta$ and $n\right)$ integral involving the lowest eigenvalue of the Ricci tensor, then $\text{diam}\left(M\right)<\pi +\delta$. If $M$ is either not compact or the Ricci curvature is bounded nonpositively below, that is, $\text{Ric}\left(M\right)\ge \left(n-1\right)k$ with $k\le 0$ 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 $\text{diam}\left(M\right)$ from above and $\text{vol}\left(M\right)$ from below as were assumed in previous extensions of Myers’ result that ${\pi }_{1}\left(M\right)$ is finite.
##### MSC:
 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)