On the existence of infinitely many closed geodesics on non-compact manifolds

Luca Asselle (1) , Marco Mazzucchelli (2)

1 Ruhr-Universitat Bochum, Fakultat fur Mathematik
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We prove that any complete (and possibly non-compact) Rie-mannian manifold M possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than dim(M), and there are no close conjugate points at infinity. Our argument builds on an existence result due to Benci and Giannoni, and generalizes the celebrated theorem of Gromoll and Meyer for closed manifolds.

Domaines: Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]

Luca Asselle, Marco Mazzucchelli. On the existence of infinitely many closed geodesics on non-compact manifolds. Proceedings of the American Mathematical Society, 2016, 145, pp.2689 - 2697. ⟨10.1090/proc/13398⟩. ⟨ensl-01520339⟩