Overview

With special emphasis on new techniques based on the holonomy of the normal connection, this book provides a modern, self-contained introduction to submanifold geometry. It offers a thorough survey of these techniques and their applications and presents a framework for various recent results to date found only in scattered research papers. The treatment introduces all the basics of the subject, and along with some classical results and hard-to-find proofs, presents new proofs of several recent results. Appendices furnish the necessary background material, exercises give readers practice in using the techniques, and discussion of open problems stimulates readers' interest in the field.

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9781584883715

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The book provides a very comprehensive monograph on the modern geometry of submanifolds emphasizing the normal holonomy as a powerful tool in this theory. - Zentralblatt MATH, 1043 This book is a valuable addition to the literature on the geometry of submanifolds. It gives a comprehensive presentation of several recent developments in the theory, including submanifolds with parallel second fundamental form, isoparametric submanifolds and their Coxeter groups, and the Normal Holonomy Theorem. Of particular importance are the isotropy representations of semisimple symmetric spaces, which play a unifying role in the text and have several notable characterizations. The book is well organized and carefully written, and it provides an excellent treatment of an important part of modern submanifold theory. -Thomas E. Cecil, Professor of Mathematics, College of the Holy Cross, Worcester, Massachusetts, USA The study of submanifolds of Euclidean space and more generally of spaces of constant curvature has a long history. While usually only surfaces or hypersurfaces are considered the emphasis of this monograph is on higher codimension. Exciting beautiful results have emerged in recent years in this area and are all presented in this volume, many of them for the first time in book form. One of the principal tools of the authors is the holonomy group of the normal bundle of the submanifold and the surprising result of C. Olmos, which parallels Marcel Berger's classification in the Riemannian case. Great efforts have been made to develop the whole theory from scratch and simplify existing proofs. The book will surely become an indispensable tool for anyone seriously interested in submanifoldgeometry. -Professor Ernst Heintze, Institut fuer Mathematik, Universitaet Augsburg, Germany This book is carefully written, it contains some new proofs and open problems, many exercises and references, and an appendix for basic materials, and so it would be very useful for not only researchers but also graduate students in geometry. -Mathematical Reviews, Issue 2004e

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