Differential Geometry of Lightlike Submanifolds

ISBN-13: 9783034602501 / Angielski / Miękka / 2010 / 488 str.

Krishan L. Duggal;Bayram Sahin

Opis

Since the second half of the 20th century, the Riemannian and semi-Riemannian geometries have been active areas of research in di?erential geometry and its - plications to a variety of subjects in mathematics and physics. A recent survey in Marcel Berger s book 60] includes the major developments of Riemannian ge- etry since 1950, citing the works of di?erential geometers of that time. During the mid 1970s, the interest shifted towards Lorentzian geometry, the mathematical theory used in general relativity. Since then there has been an amazing leap in the depth of the connection between modern di?erential geometry and mathematical relativity, both from the local and the global point of view. Most of the work on global Lorentzian geometry has been described in a standard book by Beem and Ehrlich 34] and in their second edition in 1996, with Easley. As for any semi-Riemannian manifold there is a natural existence of null (lightlike)subspaces, in 1996, Duggal-Bejancupublished a book 149] on the lig- like (degenerate) geometry of submanifolds needed to ?ll an important missing part in the general theory of submanifolds. Since then the large number of papers published on lightlike hypersurfaces and general theory of submanifolds of semi- Riemannian manifolds has created a demand for publication of this volume as an update on the study of lightlike geometry. The objective is to focus on all new geometric results (in particular, those availableonlyafterpublicationoftheDuggal Bejancubook)onlightlikegeometry with proofs and their physical applications in mathematical physics."

Recenzje

From the reviews:

"It provides a reasonably self-contained and very comprehensive account of all aspects of the subject. ... The book contains more than 400 references to the literature, as well as a wealth of applications to physics (general relativity). Written by two of the main contributors to the field this comprehensive presentation is certain to be the standard work for the foreseeable future." (M. Kunzinger, Monatshefte für Mathematik, Vol. 167 (1), July, 2012)

Spis treści

Preface.- Notations.- 1 Preliminaries.- 2 Lightlike hypersurfaces.- 3 Applications of lightlike hypersurfaces.- 4 Half-lightlike submanifolds.- 5 Lightlike submanifolds.- 6 Submanifolds of indefinite Kähler manifolds.- 7 Submanifolds of indefinite Sasakian manifolds.- 8 Submanifolds of Indefinite quaternion Kähler manifolds.- 9 Applications of lightlike geometry.- Bibliography.- Index.

Blurb

This is the first systematic account of the main results in the theory of lightlike submanifolds of semi-Riemannian manifolds which have a geometric structure, such as almost Hermitian, almost contact metric or quaternion Kähler. Using these structures, the book presents interesting classes of submanifolds whose geometry is very rich.

The book also includes hypersurfaces of semi-Riemannian manifolds, their use in general relativity and Osserman geometry, half-lightlike submanifolds of semi-Riemannian manifolds, lightlike submersions, screen conformal submersions, and their applications in harmonic maps.

Basic constructions and definitions are presented as preliminary background in every chapter. The presentation explores applications and suggests several open questions.

This self-contained monograph provides up-to-date research in lightlike geometry and is intended for graduate students and researchers just entering this field.