Synopses & Reviews
This book has been written with a two-fold approach in mind: firstly, it adds to the theory of submanifolds the missing part of lightlike (degenerate) submanifolds of semi-Riemannian manifolds, and, secondly, it applies relevant mathematical results to branches of physics. It is the first-ever attempt in mathematical literature to present the most important results on null curves, lightlike hypersurfaces and their applications to relativistic electromagnetism, radiation fields, Killing horizons and asymptotically flat spacetimes in a consistent way. Many striking differences between non-degenerate and degenerate geometry are highlighted, and open problems for both mathematicians and physicists are given. Audience: This book will be of interest to graduate students, research assistants and faculty working in differential geometry and mathematical physics.
Synopsis
This book is about the light like (degenerate) geometry of submanifolds needed to fill a gap in the general theory of submanifolds. The growing importance of light like hypersurfaces in mathematical physics, in particular their extensive use in relativity, and very limited information available on the general theory of lightlike submanifolds, motivated the present authors, in 1990, to do collaborative research on the subject matter of this book. Based on a series of author's papers (Bejancu 3], Bejancu-Duggal 1,3], Dug- gal 13], Duggal-Bejancu 1,2,3]) and several other researchers, this volume was conceived and developed during the Fall '91 and Fall '94 visits of Bejancu to the University of Windsor, Canada. The primary difference between the lightlike submanifold and that of its non- degenerate counterpart arises due to the fact that in the first case, the normal vector bundle intersects with the tangent bundle of the submanifold. Thus, one fails to use, in the usual way, the theory of non-degenerate submanifolds (cf. Chen 1]) to define the induced geometric objects (such as linear connection, second fundamental form, Gauss and Weingarten equations) on the light like submanifold. Some work is known on null hypersurfaces and degenerate submanifolds (see an up-to-date list of references on pages 138 and 140 respectively). Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of an up-to-date information on null curves, lightlike hypersur- faces and submanifolds, consistent with the theory of non-degenerate submanifolds.
Table of Contents
Preface.
1. Algebraic Preliminaries.
2. Differential-Geometric Structures on Manifolds.
3. Geometry of Null Curves in Lorentz Manifolds.
4. Lightlike Hypersurfaces of Semi-Riemannian Manifolds.
5. Lightlike Submanifolds of Semi-Riemannian Manifolds.
6. CR-Lightlike Submanifolds of Indefinite Kaehler Manifolds.
7. Lightlike Hypersurfaces of Lorentz Framed Manifolds.
8. Lightlike Hypersurfaces and Electromagnetism.
9. Lightlike Hypersurfaces and General Relativity. References. Author Index. Subject Index.