---
abstract: 'In this article, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder $M\simeq S^1\times{{\mathbb{R}}}$ or a complete Riemannian plane $M\simeq{{\mathbb{R}}}^2$ leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.'
address:
- 'Simon Allais, École Normale Supérieure de Lyon, UMPA46 allée d’Italie, 69364 Lyon Cedex 07, France'
- 'Tobias Soethe, Ruhr-Universität, Fakultät für Mathematik Universitätsstr. 150, 44780 Bochum, Deutschland'
author:
- Simon Allais
- Tobias Soethe
bibliography:
- '../biblio.bib'
date: 'May 11, 2020'
title: Homologically visible closed geodesics on complete surfaces
---

Introduction
============

The problem of the existence and multiplicity of closed geodesics plays an important role in both Riemannian geometry and dynamics. Going back to Hadamard and Poincaré [@Had98; @Poi05], it is still open for a lot of Riemannian manifolds. Given a complete Riemannian manifold $(M,g)$, a famous question is whether it possesses a closed geodesic for every Riemannian metric $g$. This is always true if $M$ is closed [@Bir66; @LF51; @Fet52]. We can then ask whether the number 