A note on closed geodesics for a class of non-compact Riemannian manifolds
The author studies the existence of closed geodesics on \({\mathbb R}\times S^N\) with respect to a perturbation of the standard product metric. More generally he considers the same problem for perturbations of the product metric on \({\mathbb R}\times M_0\) where \(M_0\) is a compact Riemannian manifold either having a nondegenerate closed geodesic or under the assumption that \(M_0\) is not simply connected and has only isolated closed geodesics. The proofs rely on a perturbation result of \textit{A. Ambrosetti} and \textit{M. Badiale} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, No. 2, 233-252 (1998; Zbl 1004.37043)]. The results of the paper generalize some of the results of \textit{K. Tanaka} [Ann. Inst. Henri Poincaré Anal. Non Linéaire 17, No. 1, 1-33 (2000; Zbl 0955.37040)]. Previously closed geodesics on noncompact manifolds were studied by \textit{G. Thorbergsson} in [Math. Z. 159, 249-258 (1978; Zbl 0369.53045)] and \textit{V. Benci} and \textit{F. Giannoni} in [Duke Math. J. 68, 195-215 (1992; Zbl 0789.53028)].
scientific article; zbMATH DE number 1675807