Lorentzian distance on the Lobachevsky plane
type: Item
instance of: scholarly article
title: Lorentzian distance on the Lobachevsky plane
author: Yu L. Sachkov
published in: Nonlinearity
publication date: 2024-08-13T00:00:00Z
review text: The problem considered is that of admissible geodesics of a Lorentzian structure; here admissible means time-like, in relativistic terms. The problem is formulated as an optimal control problem, with the time-like constraint then formulated as a constraint on the controls. Many ideas are initially presented in a general manner (with references to their more complete presentation), but the new results in the paper are those pertaining to Lorentzian structures on the space of orientation-preserving affine mappings of \(\mathbb{R}\), this space being a Lie group diffeomorphic to the upper half-plane. The Lorentzian structures considered are left-invariant.\N\NThe author uses the Pontryagin maximum principle to study the problem of the existence and character of length minimizing time-like curves. The emphasis is on explicit characterization of geometric properties like curvature, geodesic connectedness, completeness, distance, spheres, isometries, etc. The paper closes with a particularly detailed analysis of a few particular left-invariant Lorentzian structures on the upper half-plane.\N\NFor a reader who is familiar with the Pontryagin maximum principle, the paper is quite easy to follow. The techniques are all ``elementary'' (not intended in a pejorative sense), and the paper is very well written.
reviewed by: Andrew D. Lewis
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 49K15
zbMATH DE Number: 7896366
zbMATH Keywords: Lorentzian manifolds
zbMATH Keywords: Lobachevski plane
zbMATH Keywords: time-like geodesics
zbMATH Keywords: Pontryagin maximum principle
zbMATH Keywords: length minimizers
MaRDI profile type: Publication
cites work: Q4871706
cites work: An invitation to Lorentzian geometry
cites work: General Relativity
cites work: Control theory from the geometric viewpoint.
cites work: Introduction to geometric control. Translated from the Russian
cites work: Lorentzian geometry on the Lobachevsky plane
cites work: Q3062972
cites work: Q3147594
cites work: Reachable sets for the Heisenberg sub-Lorentzian structure on \(\mathbb R^3\). An estimate for the distance function
cites work: Sub-Lorentzian distance and spheres on the Heisenberg group
cites work: Q4210022
cites work: Q4003523
cites work: Q3312869
cites work: Q3285814
cites work: Geodesic connectedness of affine manifolds
cites work: Geodesic connectedness of semi-Riemannian manifolds.
DOI: 10.1088/1361-6544/AD67A0
description: scientific article; zbMATH DE number 7896366