of closed geodesics is infinite or not. It is known that every closed surface has infinitely many geometrically distinct closed geodesics [@Fra92; @Ban93; @Hin93]. However, this question is still open for spheres of higher dimension. In this article, we are interested in non-compact complete Riemannian surfaces for which even the existence of one closed geodesic fails in general: planes and cylinders. For instance, the Euclidean plane does not possess any closed geodesic. Nevertheless, under specific geometric conditions, interesting results can be stated. In 1980, Bangert proved that any complete Riemannian cylinder, plane or Möbius band of finite area has infinitely many closed geodesics [@Ban80]. For the plane and the cylinder he proved the same result even under the weaker assumption of just the existence of a convex neighborhood of infinity. We will discuss this result in greater depth as it is used extensively in our proofs. The purpose of this article is to give simple conditions under which the existence of one or two distinct closed geodesics implies that a complete Riemannian cylinder or plane contains infinitely many geometrically distinct closed geodesics.

Let $S^1:={{\mathbb{R}}}/{{\mathbb{Z}}}$ and let $M\simeq S^1\times{{\mathbb{R}}}$ be a complete Riemannian cylinder. Let $\Lambda M$ be its loop space. 