Null Curves And Hypersurfaces Of Semi-riemannian Manifolds

by Krishan L Duggal, Ho Jin Dae

Overview

This is a first textbook that is entirely focused on the up-to-date developments of null curves with their applications to science and engineering. It fills an important gap in a second-level course in differential geometry, as well as being essential for a core undergraduate course on Riemannian curves and surfaces. The sequence of chapters is arranged to provide in-depth understanding of a chapter and stimulate further interest in the next. The book comprises a large variety of solved examples and rigorous exercises that range from elementary to higher levels. This unique volume is self-contained and unified in presenting:

Product Details

ISBN-13:	9789812706478
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/04/2007
Pages:	304
Product dimensions:	6.60(w) x 9.80(h) x 0.90(d)

Table of Contents

Preface     V
The concept of null curves     1
Smooth manifolds     1
Semi-Riemannian manifolds     5
Introduction to null curves     7
Screen and null transversal bundles     9
Quasi-orthonormal basis along a null curve     10
Brief notes and exercises     14
Null curves in Lorentzian manifolds [Characters not reproducible]     19
Frenet frames along null curves     19
Invariance of Frenet frames     25
Null Cartan curves in [Characters not reproducible], [Characters not reproducible] and [Characters not reproducible]     29
Null Cartan curves in [Characters not reproducible]     35
Fundamental theorem of null curves in [Characters not reproducible]     42
Brief notes and exercises     45
Null curves in [Characters not reproducible]     51
Type 1 general Frenet frames     52
Type 2 general Frenet frames     55
Natural Frenet equations     63
Invariance of Frenet equations     74
Geometry of null curves in [Characters not reproducible]     81
Geometry of null curves in [Characters not reproducible]     86
Fundamental theorem of null curves in [Characters not reproducible]     91
Brief notes and exercises     95
Null curves in [Characters not reproducible]     97
Frenet equations of Type 1 and Type 2     97
Frenet equations of Type q ([greater than or equal] 3)     100
Natural Frenet equations     108
Invariance of Frenet frames     119
Geometry of null curves in [Characters not reproducible]     129
Fundamental theorem of null curves in [Characters not reproducible]     135
Geometry of null Cartan curves     141
Null Cartan curves in Lorentzian manifolds     141
Null Cartan helices     146
Brief notes and research problems     157
Applications of null curves     165
Soliton solutions in 3D Lorentzian space forms     165
Soliton solutions in Minkowski spaces     169
Mechanical systems and 3D null curves     173
Lightlike hypersurfaces     185
Introduction     185
Screen and transversal bundles     190
Lightlike second fundamental forms     198
Integrable screen distribution     210
Induced curvature and Ricci tensors     220
Scalar curvature of genus zero     232
Brief notes and exercises      235
Geometry and physics of null geodesics     241
Introduction     241
Globally null manifolds     246
Frenet frames along null geodesics     252
Scalar curvature and null warped products     263
Null geodesics and photon surfaces     268
Brief notes on research papers     275
Bibliography     281
Index     289