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

2014/06/12 by Alexandru Oancea, Oancea, Alexandru
Mathematics · #37J05 #55P35 #58E05 #58E10 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:37J05 #msc:55P35 #msc:58E05 #msc:58E10

paper · pdf · doi:10.48550/arxiv.1406.3107

45 pages, 5 figures. Appendix by Umberto Hryniewicz

arxiv created 2014/06/12 · arxiv updated 2014/06/13

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.

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