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Volume, Cheeger and Gromov. (English) Zbl 0957.53012

In this short paper the author considers complete Riemannian manifolds with all sectional curvatures bounded from below, and injectivity radius bigger than \(r_0>0\). He proves the following theorem: Given \(h,\delta>0\), assume the Cheeger constant of \(M\) is bigger than \(h\) and the Gromov hyperbolicity constant of \(M\) is smaller than \(\delta\), then either \(M\) has infinite volume or its diameter is bounded by \(f(h,\delta)<\infty\), a function which depends only on \(\delta\) and \(h\) and the bounded geometry parameters.

MSC:

53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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