On closed geodesics in Lorentz manifolds

Author:

Souheib Allout, Abderrahmane Belkacem, Abdelghani Zeghib

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG)

journal:

date:

2023-07-30 16:00:00

Abstract

We construct a compact semi-Riemannian manifold without periodic geodesics. We also provide examples of three-dimensional Lorentz manifolds without closed geodesics. All these examples are of the form $\Gamma\backslash G$ where $G$ is a Lie group endowed with a left invariant metric and $\Gamma\subset G$ is a cocompact lattice. Unlike the locally homogeneous case, we show that a homogeneous semi-Riemannian manifold admits closed geodesics.

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