Closed geodesics and the first Betti number

2024/07/03 by Contreras, Gonzalo, Mazzucchelli, Marco · 1 citation
#53C22 #58E10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2407.02995

Abstract

We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a C^∞ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable C^∞-close Riemannian metric.

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