An introduction to Lorentz surfaces
type: Item
instance of: scholarly article
title: An introduction to Lorentz surfaces
zbMATH Open document ID: 0881.53001
published in: De Gruyter Expositions in Mathematics
publication date: 1996-08-18T00:00:00Z
review text: ``This book describes an emerging field of research, and includes detailed proofs of numerous results, many of which have not yet appeared elsewhere.'' Nevertheless, the book is well suited as a text for a special topics course or a seminar for students, too: besides some elementary knowledge in differential geometry, topology, and complex analysis, no qualifications are required to understand the text. The author emphasizes the presentation of the methods and gives elementary proofs where possible; and not only for the few results that are stated without proofs, a comprehensive list of references is given. The book is enriched with a large number of exercises and examples. Lorentz surfaces are the signature analogs of Riemann surfaces: defining two Lorentzian metrics \(h\), \(\widetilde h\), i.e., metrics of signature (1,1), on a (2-dimensional) surface \(S\) to be conformally equivalent if there is a positive function \(\lambda\) on \(S\) with \(\widetilde h= \lambda h\), a Lorentz surface \({\mathcal L}\) is a surface \(S\) equipped with an equivalence class of conformally equivalent metrics \({\mathcal L}= (S,[h])\). Obviously, the causal character \(\text{sign} h(v,v)\) of a tangent vector \(v\) is an invariant of a conformal class \([h]\) of Lorentzian metrics on a surface -- thus, in the theory of Lorentz surfaces the notion of causal character replaces the notion of angles in Riemann surface theory. In fact, the system of null lines on a surface \(S\) determines the corresponding conformal class of Lorentzian metrics as soon as the two systems of null lines are ordered by means of an orientation of \(S\). This is the reason for the prominent position of the study of null line systems in the book: the basic facts are discussed in the first chapter ``Null lines on Lorentz surfaces''. In the second chapter ``Box surfaes, yardsticks and global properties of Lorentzian metrics'', the analog for the familiar interpretation of a Riemann surface as a complex curve i…
reviewed by: Udo Hertrich-Jeromin
Mathematics Subject Classification ID: 53-01
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 917504
zbMATH Keywords: Minkowski space
zbMATH Keywords: associated minimal surfaces
zbMATH Keywords: Lorentz surfaces
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Poincaré index
zbMATH Keywords: conformal boundary
zbMATH Keywords: timelike minimal surface
zbMATH Keywords: Weierstrass representation
author: Tilla Weinstein
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 917504