Two loops $\alpha,\beta\in\Lambda M$ are said to be geometrically distinct if their images are distinct: $\alpha(S^1)\neq\beta(S^1)$. Throughout the article, by writing that two closed geodesics are distinct we will always mean that they are geometrically distinct. A closed geodesic $\gamma\in\Lambda M$ is said to be homologically visible if the local homology of the critical circle $S^1\cdot\gamma \subset \Lambda M$ of the energy functional is non-zero (see Section \[se:preliminaries\] for precise definitions).

\[thm:main\] Let $M$ be a complete Riemannian cylinder with isolated closed geodesics and assume one of the following hypothesis:

1.  \[it:contractible\] there exists a contractible closed geodesic,

2.  \[it:selfintersecting\] there exists a self-intersecting closed geodesic,

3.  \[it:intersecting\] there exist two distinct closed geodesics that intersect,

4.  \[it:mind\] there exists a closed geodesic of non-zero average index,

5.  \[it:homological\] there exist two homologically visible closed geodesics.

Then $M$ contains infinitely many geometrically distinct homologically visible closed geodesics.

Notice that according to Bott iteration theory, a closed geodesic $c$ has a non-zero average index if and only if some iterate $c^m$ has a non-zero index. The fact that hypothesis \[it:homological\] implies that there exists infinitely many closed geodesics proves a conjecture of Abbondandolo:

Any complete Riemannian cylinder with isolated closed geodesics 