## 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

#### Abstract

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

**Source**

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

**Dates**

Received: 30 August 2015

Revised: 7 March 2016

Accepted: 25 March 2016

First available in Project Euclid: 13 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513135629

**Digital Object Identifier**

doi:10.2140/involve.2017.10.243

**Mathematical Reviews number (MathSciNet)**

MR3574299

**Zentralblatt MATH identifier**

1352.53034

**Subjects**

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

**Keywords**

closed geodesics surface of revolution

#### Citation

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. https://projecteuclid.org/euclid.involve/1513135629