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Closed geodesics and volume growth of Riemannian manifolds

2011
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Proceedings of the American Mathematical Society
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In this paper, we study the relation between the existence of closed geodesics and the volume growth of open Riemannian manifolds with nonnegative curvature. Proof. If M n contains a closed geodesic, by Lemma 2.1, we have mes(Σ) = 0 (induced measure of the unit sphere). By Fubini's theorem, for any r > 0 we have Since exp is C ∞ , by Sard's theorem [3] , for any r > 0 we have Then by Lemma 2.3, we have α M = 0. 3. An application of Theorem 1.1 Combining with the Cheeger-Gromoll soul theorem

doi:10.1090/s0002-9939-2010-10549-x
fatcat:kmq3ey5zabh3pkq3vxn3r3dzbm