# Submanifolds and Holonomy

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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 C* ^{n}* with nontransitive normal holonomy

Applications to complex submanifolds of C

*P*with nontransitive normal holonomy

^{n}

**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.

## 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