Differential Geometry of Lightlike Submanifolds

This book presents research on the latest developments in differential geometry of lightlike (degenerate) subspaces. The main focus is on hypersurfaces and a variety of submanifolds of indefinite Kählerian, Sasakian and quaternion Kähler manifolds.

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.

There does not exist any other book covering the material presented in this volume. This makes the book a uniquely comprehensive work in the field This is the first book which contains unique existence theorems (with proofs) for the screen distributions of lightlike submanifolds Applications are focused on time-dependent realistic models of black hole horizons, lightlike versions of Osserman geometry, harmonic maps and morphisms, CR and contact structures in physics For readers who wish to do further research, there is an extensive bibliography including many papers and books on the Riemannian geometry of submanifolds Includes supplementary material: sn.pub/extras

Zusammenfassung

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)

Inhalt
Preliminaries.- Lightlike hypersurfaces.- Applications of lightlike hypersurfaces.- Half-lightlike submanifolds.- Lightlike submanifolds.- Submanifolds of indefinite Kähler manifolds.- Submanifolds of indefinite Sasakian manifolds.- Submanifolds of indefinite quaternion Kähler manifolds.- Applications of lightlike geometry.