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This paper is concerned with the existence of closed geodesics on a non–compact manifold M . There are very few papers on such a problem, see [3, 13, 14]. In particular, Tanaka deals with the manifod M = R×S , endowed with a metric g(s, ξ) = g0(ξ) + h(s, ξ), where g0 is the standard product metric on R × S N . Under the assumption that h(s, ξ) → 0 as |s| → ∞, he proves the existence of a closed geodesic, found as a critical point of the energy functional

A Note on Closed Geodesics for a Class of Non-Compact Riemannian Manifolds