Overview

Submanifolds and Holonomy, Second Edition explores recent progress in the submanifold geometry of space forms, including new methods based on the holonomy of the normal connection. This second edition reflects many developments that have occurred since the publication of its popular predecessor. New to the Second Edition New chapter on normal holonomy of complex submanifolds New chapter on the Berger–Simons holonomy theorem New chapter on the skew-torsion holonomy system New chapter on polar actions on symmetric spaces of compact type New chapter on polar actions on symmetric spaces of noncompact type New section on the existence of slices and principal orbits for isometric actions New subsection on maximal totally geodesic submanifolds New subsection on the index of symmetric spaces The book uses the reduction of codimension, Moore’s lemma for local splitting, and the normal holonomy theorem to address the geometry of submanifolds. It presents a unified treatment of new proofs and main results of homogeneous submanifolds, isoparametric submanifolds, and their generalizations to Riemannian manifolds, particularly Riemannian symmetric spaces.

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9781482245158

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Praise for the First Edition: 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 not only for researchers but also graduate students in geometry. -Mathematical Reviews, Issue 2004e 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 co-dimension. 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 submanifold geometry. -Professor Ernst Heintze, Institut fur Mathematik, Universitaet Augsburg, Germany

Praise for the First Edition: 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 not only for researchers but also graduate students in geometry. -Mathematical Reviews, Issue 2004e 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 co-dimension. 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 submanifold geometry. -Professor Ernst Heintze, Institut fur Mathematik, Universitaet Augsburg, Germany

Praise for the First Edition: 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 not only for researchers but also graduate students in geometry. -Mathematical Reviews, Issue 2004e 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 co-dimension. 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 submanifold geometry. -Professor Ernst Heintze, Institut fur Mathematik, Universitaet Augsburg, Germany

Author Information

Jürgen Berndt is a professor of mathematics at King’s College London. He is the author of two research monographs and more than 50 research articles. His research interests encompass geometrical problems with algebraic, analytic, or topological aspects, particularly the geometry of submanifolds, curvature of Riemannian manifolds, geometry of homogeneous manifolds, and Lie group actions on manifolds. He earned a PhD from the University of Cologne. Sergio Console (1965–2013) was a researcher in the Department of Mathematics at the University of Turin. He was the author or coauthor of more than 30 publications. His research focused on differential geometry and algebraic topology. Carlos Enrique Olmos is a professor of mathematics at the National University of Cordoba and principal researcher at the Argentine Research Council (CONICET). He is the author of more than 35 research articles. His research interests include Riemannian geometry, geometry of submanifolds, submanifolds, and holonomy. He earned a PhD from the National University of Cordoba.

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