Homologically visible closed geodesics on complete surfaces

TL;DR

This paper investigates conditions under which the existence of one or two closed geodesics on certain complete surfaces implies infinitely many, focusing on cylinders and planes, and addresses a question posed by Abbondandolo.

Contribution

It establishes new results linking the number of closed geodesics to the topology of complete surfaces, especially cylinders and planes, and provides conditions for infinitely many geodesics.

Findings

Complete cylinders with isolated geodesics have 0, 1, or infinitely many homologically visible geodesics.

Presence of one or two geodesics can imply infinitely many under certain conditions.

Answers a question of Alberto Abbondandolo regarding geodesic multiplicity.

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.