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

Luca Asselle; Marco Mazzucchelli

doi:10.1090/proc/13398

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

Differential Geometry 2017-03-21 v2 Symplectic Geometry

Authors: Luca Asselle , Marco Mazzucchelli

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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 the dimension of $M$ , 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.

Keywords

closed geodesics non-compact manifolds Riemannian manifold free loop space Betti numbers conjugate points Gromoll-Meyer theorem

Cite

@article{arxiv.1602.03679,
  title  = {On the existence of infinitely many closed geodesics on non-compact manifolds},
  author = {Luca Asselle and Marco Mazzucchelli},
  journal= {arXiv preprint arXiv:1602.03679},
  year   = {2017}
}

Comments

9 pages; Version 2: minor corrections

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

Abstract

Keywords

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