494 Pages 8 B/W Illustrations
    by Chapman & Hall

    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.

    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.

    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.

    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