On the Alexandrov Topology of sub-Lorentzian Manifolds

TL;DR

This paper explores how the Alexandrov topology can be defined on sub-Lorentzian manifolds, analyzing its properties and relationships to causality and the manifold's original topology, extending concepts from Lorentzian geometry.

Contribution

It introduces three methods to define the Alexandrov topology on sub-Lorentzian manifolds and studies their interrelations and connection to causality.

Findings

The three definitions of Alexandrov topology usually differ but coincide in Lorentzian case.

The topologies are related to the manifold's original topology and causality properties.

The study clarifies the topological structure of sub-Lorentzian manifolds in relation to causality.

Abstract

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.