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2nd Edition

Submanifolds and Holonomy




ISBN 9781482245158
Published February 8, 2016 by Chapman and Hall/CRC
494 Pages 8 B/W Illustrations

 
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Book Description

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.

Table of Contents

Basics of Submanifold Theory in Space Forms
The fundamental equations for submanifolds of space forms
Models of space forms
Principal curvatures
Totally geodesic submanifolds of space forms
Reduction of the codimension
Totally umbilical submanifolds of space forms
Reducibility of submanifolds

Submanifold Geometry of Orbits
Isometric actions of Lie groups
Existence of slices and principal orbits for isometric actions
Polar actions and s-representations
Equivariant maps
Homogeneous submanifolds of Euclidean spaces
Homogeneous submanifolds of hyperbolic spaces
Second fundamental form of orbits
Symmetric submanifolds
Isoparametric hypersurfaces in space forms
Algebraically constant second fundamental form

The Normal Holonomy Theorem
Normal holonomy
The normal holonomy theorem
Proof of the normal holonomy theorem
Some geometric applications of the normal holonomy theorem
Further remarks

Isoparametric Submanifolds and Their Focal Manifolds
Submersions and isoparametric maps
Isoparametric submanifolds and Coxeter groups
Geometric properties of submanifolds with constant principal curvatures
Homogeneous isoparametric submanifolds
Isoparametric rank

Rank Rigidity of Submanifolds and Normal Holonomy of Orbits
Submanifolds with curvature normals of constant length and rank of homogeneous submanifolds
Normal holonomy of orbits

Homogeneous Structures on Submanifolds
Homogeneous structures and homogeneity
Examples of homogeneous structures
Isoparametric submanifolds of higher rank

Normal Holonomy of Complex Submanifolds
Polar-like properties of the foliation by holonomy tubes
Shape operators with some constant eigenvalues in parallel manifolds
The canonical foliation of a full holonomy tube
Applications to complex submanifolds of Cn with nontransitive normal holonomy
Applications to complex submanifolds of CPn with nontransitive normal holonomy

The Berger–Simons Holonomy Theorem
Holonomy systems
The Simons holonomy theorem
The Berger holonomy theorem

The Skew-Torsion Holonomy Theorem
Fixed point sets of isometries and homogeneous submanifolds
Naturally reductive spaces
Totally skew one-forms with values in a Lie algebra
The derived 2-form with values in a Lie algebra
The skew-torsion holonomy theorem
Applications to naturally reductive spaces

Submanifolds of Riemannian Manifolds
Submanifolds and the fundamental equations
Focal points and Jacobi fields
Totally geodesic submanifolds
Totally umbilical submanifolds and extrinsic spheres
Symmetric submanifolds

Submanifolds of Symmetric Spaces
Totally geodesic submanifolds
Totally umbilical submanifolds and extrinsic spheres
Symmetric submanifolds
Submanifolds with parallel second fundamental form

Polar Actions on Symmetric Spaces of Compact Type
Polar actions — rank one
Polar actions — higher rank
Hyperpolar actions — higher rank
Cohomogeneity one actions — higher rank
Hypersurfaces with constant principal curvatures

Polar Actions on Symmetric Spaces of Noncompact Type
Dynkin diagrams of symmetric spaces of noncompact type
Parabolic subalgebras
Polar actions without singular orbits
Hyperpolar actions without singular orbits
Polar actions on hyperbolic spaces
Cohomogeneity one actions — higher rank
Hypersurfaces with constant principal curvatures

Appendix: Basic Material

Exercises appear at the end of each chapter.

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Author(s)

Biography

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.

Reviews

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 für Mathematik, Universitaet Augsburg, Germany