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

Asselle, Luca; Mazzucchelli, Marco

doi:10.1090/proc/13398

On the existence of infinitely many closed geodesics on non-compact manifolds
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by Luca Asselle and Marco Mazzucchelli PDF

Proc. Amer. Math. Soc. 145 (2017), 2689-2697 Request permission

Abstract:

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than $\mathrm {dim}(M)$ 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.

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