Closed geodesics and volume growth of Riemannian manifolds

Jianming Wan
2011 Proceedings of the American Mathematical Society  
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
more » ... ll soul theorem (see [2]), we get another proof of Marenich and Toponogov's beautiful theorem (see [5] ): Proof. If M n is not diffeomorphic to R n , by the Cheeger-Gromoll soul theorem, the soul (a totally geodesic submanifold) of M n is not a point. Then the soul must contain a closed geodesic (since any compact manifold contains at least one closed geodesic [4] ). It is also the closed geodesic of M n , which is a contradiction to Theorem 1.1. Remark 3.2. By a different method, Theorem 3.1 is also a consequence of Perelman's celebrated flat strip theorem (cf. [7] ).
doi:10.1090/s0002-9939-2010-10549-x fatcat:kmq3ey5zabh3pkq3vxn3r3dzbm