An Introduction to Lorentz Surfaces

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Walter de Gruyter, Jun 24, 2011 - Mathematics - 226 pages

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

 

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Contents

Introduction
1
Chapter 1 Null lines on Lorentz surfaces
4
12 Metrics and null direction fields
8
13 Lorentz surfaces and proper null coordinates
12
14 A first look at null lines
15
15 The Euclidean plane E2 and the Minkowski plane E21
17
Chapter 2 Box surfaces yardsticks and global properties of Lorentzian metrics
19
22 Yardsticks and timeorientability
22
52 Spans on ℒ
87
53 A special ℋ+ chart on the span of a null curve
95
54 Characterization of C0 smoothability of the conformal boundary
103
55 Kulkarnis use of the conformal boundary
110
Chapter 6 Conformal invariants on Lorentz surfaces
119
62 Conformal indices associated with ℒ and more properties of ℒ
124
63 Some notions of symmetry
131
64 Smyths digraph determining sets and some other conformal invariants
134

23 Intrinsic curvature and a first look at the example in our logo
25
24 Geodesics and pregeodesics
27
25 Completeness inextendibility and causality conditions
30
Chapter 3 Conformal equivalence and the Poincaré index
36
32 Cj conformally equivalent Lorentz surfaces need not be Cj+1 conformally equivalent
39
33 The Poincaré index
47
34 The Poincaré Index Theorem
50
Chapter 4 Kulkarnis conformal boundary
54
42 The points on the conformal boundary
61
43 The topology on the conformal boundary
68
44 Some properties of the conformal boundary
73
Chapter 5 Using the conformal boundary
84
Chapter 7 Classical surface theory and harmonically immersed surfaces
139
72 A quick review of local surface theory in Minkowski 3space
149
73 Contrasting the behavior of surfaces in E3 and E31
162
74 The HilbertHolmgren theorem for harmonically immersed surfaces
166
Chapter 8 Conformal realization of Lorentz surfaces in Minkowski 3space
175
82 Associate families of minimal surfaces
182
83 Some conformal realizations of Lorentz surfaces in E31
188
84 Some last remarks on conformal imbeddings and immersions
196
Bibliography
201
Index
205
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