Sub-Lorentzian Geometry on Anti-De Sitter Space

TL;DR

This paper extends sub-Riemannian geometry to the pseudo-Riemannian setting of anti-de Sitter space, developing sub-Lorentzian structures, analyzing horizontal curves, and deriving geodesics using Hamiltonian and Lagrangian formalisms.

Contribution

It introduces the first study of sub-Lorentzian geometry on anti-de Sitter space, exploring two distributions and their geodesics, bridging sub-Riemannian and pseudo-Riemannian geometries.

Findings

Existence of non-empty sets of timelike and spacelike horizontal curves

Development of Hamiltonian and Lagrangian methods for geodesics

Analysis of horizontal connectivity in anti-de Sitter space

Abstract

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a large amount of literature developing sub-Riemannian Geometry. However, very few is known about its natural extension to pseudo-Riemannian analogues. It is natural to begin such a study with some low-dimensional manifolds. Based on ideas from sub-Riemannian geometry we develop sub-Lorentzian geometry over the classical 3-D anti-de Sitter space. Two different distributions of the tangent bundle of anti-de Sitter space yield two different geometries: sub-Lorentzian and sub-Riemannian. It is shown that the set of timelike and spacelike `horizontal' curves is non-empty and we study the problem of horizontal connectivity in anti-de Sitter space. We also use Lagrangian and Hamiltonian formalisms for both sub-Lorentzian and sub-Riemannian geometries to find geodesics.