Vol. 10, No. 2, 2017

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Characterizations of the round two-dimensional sphere in terms of closed geodesics

Lee Kennard and Jordan Rainone

Vol. 10 (2017), No. 2, 243–255

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.

closed geodesics, surface of revolution
Mathematical Subject Classification 2010
Primary: 53C20, 58E10
Received: 30 August 2015
Revised: 7 March 2016
Accepted: 25 March 2016
Published: 10 November 2016

Communicated by Kenneth S. Berenhaut
Lee Kennard
Department of Mathematics
University of Oklahoma
Norman, OK 73019
United States
Jordan Rainone
Department of Mathematics
Stony Brook University
100 Nicolls Road
Stony Brook, NY 11794
United States