On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime

TL;DR

This paper studies the topological structure of the set of pairs in a globally hyperbolic spacetime connected by multiple maximizing geodesics, proving local contractibility and homotopy equivalences involving Lorentzian Aubry sets.

Contribution

It extends previous work by establishing the local contractibility of the locus of multiple maximizing geodesics and relating it to Lorentzian Aubry sets through homotopy equivalences.

Findings

The locus of multiple maximizing geodesics is locally contractible.

Homotopy equivalences are established between the locus, cut locus, and Lorentzian Aubry set.

The set of causally related pairs with multiple geodesics has specific topological properties.

Abstract

Extending the recent work of Cannarsa, Cheng and Fathi, we investigate topological properties of the locus ${\cal NU}(M,g)$ of multiple maximizing geodesics on a globally hyperbolic spacetime $(M,g)$, i.e.\ the set of causally related pairs $(x,y)$ for which there exists more than one maximizing geodesic (up to reparametrization) from $x$ to $y$. We will prove that this set is locally contractible. We will also define the notion of a Lorentzian Aubry set ${\cal A}$ and prove that the inclusions ${\cal NU}(M,g)\hookrightarrow \operatorname{Cut}_M\hookrightarrow J^+\backslash {\cal A}$ are homotopy equivalences.