the sake of completeness we give a comprehensive proof in the paper. The proof of Theorem \[thm:plane\] is quite similar and relies extensively on the analogous theorem of Bangert when $M$ is a plane with isolated closed geodesics: if there exists an open neighborhood of infinity $C\neq M$ with a boundary $\partial C$ which is not totally geodesic, $M$ contains infinitely many homologically visible closed geodesics [@Ban80 Theorem 3]. These two theorems were originally used by Bangert to prove that any complete Riemannian plane of finite area has infinitely many closed geodesics.

In fact, Theorem \[thm:main\] extends *verbatim* to the case where $M$ is a complete reversible Finsler manifold as we will essentially use variational properties of geodesics in our proof with no concern for geometric notion specific to Riemannian manifold. However, nothing can be said concerning the more general case of a complete (asymmetrical) Finsler manifold. The major issue is that, in the asymmetrical case, a closed subset of $M$ which is bounded by a geodesic is not locally convex. In this direction, we point out that the related question of whether or not infinitely many closed geodesics exist on every irreversible Finsler cylinder of finite area is still 