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

2020/03/31 by Jiayin Pan
Mathematics · #Abelian group #Ball (mathematics) #Bounded function #Combinatorics #Curvature #Geodesic #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Ricci curvature #math.DG

paper · pdf · doi:10.2140/gt.2021.25.1059

published as Geom. Topol. 25 (2021) 1059-1085 · Changed Question 3.13 to Conjecture 3.13 with a modified statement. Fixed some typos. To appear in Geometry & Topology

openalex created_date 2020/03/13 · arxiv created 2020/06/22 · openalex publication_date 2021/04/27 · arxiv updated 2021/05/05 · openalex updated_date 2026/08/05

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 π₁(M,x) are contained in a bounded ball, then π₁(M,x) is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of π₁(M,x) escape from any bounded metric balls at a sublinear rate with respect to their lengths, then π₁(M,x) is virtually abelian.

Citations

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