Morse theory, closed geodesics, and the homology of free loop spaces

Abstract

This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations using Morse theory, $SO(2)$- vs. $O(2)$-symmetries, Katok's examples and Finsler metrics, relations to symplectic geometry, and a guide to the literature. The Appendix written by Umberto Hryniewicz gives an account of the problem of the existence of infinitely many closed geodesics on the $2$-sphere.

arXiv comment45 pages, 5 figures. Appendix by Umberto Hryniewicz

References in corpus (2)

• Local Floer homology and infinitely many simple Reeb orbits
• The space of paths in complex projective space with real boundary conditions

Cited by in corpus (5)

• Homotopy properties of horizontal loop spaces and applications to closed sub-riemannian geodesics
• The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
• Global surfaces of section for Reeb flows in dimension three and beyond
• Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}{P}^n$
• The cohomology of free loop spaces of homogeneous spaces