Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 2 (2017), 243-255.

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

Lee Kennard and Jordan Rainone

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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.

Article information

Involve, Volume 10, Number 2 (2017), 243-255.

Received: 30 August 2015
Revised: 7 March 2016
Accepted: 25 March 2016
First available in Project Euclid: 13 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 58E10: Applications to the theory of geodesics (problems in one independent variable)

closed geodesics surface of revolution


Kennard, Lee; Rainone, Jordan. Characterizations of the round two-dimensional sphere in terms of closed geodesics. Involve 10 (2017), no. 2, 243--255. doi:10.2140/involve.2017.10.243.

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