On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

Pan, Jiayin

doi:10.2140/gt.2021.25.1059

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On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

Jiayin Pan

Geometry & Topology 25 (2021) 1059–1085

DOI: 10.2140/gt.2021.25.1059

Abstract

A consequence of the Cheeger–Gromoll splitting theorem states that for any open manifold $(M, x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_{1} (M, x)$ are contained in a bounded ball, then $π_{1} (M, x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_{1} (M, x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_{1} (M, x)$ is virtually abelian.

Keywords

Ricci curvature, fundamental groups

Mathematical Subject Classification 2010

Primary: 53C20, 53C23

Secondary: 53C21, 57S30

References

Publication

Received: 6 March 2020

Revised: 17 May 2020

Accepted: 17 June 2020

Published: 27 April 2021

Proposed: Tobias H Colding

Seconded: András I Stipsicz, David M Fisher

Authors

Jiayin Pan
Department of Mathematics University of California, Santa Barbara Santa Barbara, CA United States

Volume 25, issue 2 (2021)