Closed geodesics and the first Betti number

Differential Geometry Seminar

Organizers：

Jialong Deng, Akito Futaki

Speaker：

Marco Mazzucchelli (ENS Lyon)

Time：

Tues., 21:00-22:00

Feb. 25, 2025

Title:

Closed geodesics and the first Betti number

Abstract：

In this talk, based on a joint work with Gonzalo Contreras, I will present the following result: on a closed Riemannian manifold of dimension at least two with non-trivial first Betti number, the existence of a minimal closed geodesic, 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 -close Riemannian metric. As a corollary of this result and of existing literature, we infer that on any closed manifold of dimension at least two with non-trivial first Betti number, a generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length.