Submanifolds and Holonomy (2^nd Ed.) Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series

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. Submanifold Geometry of Orbits. The Normal Holonomy Theorem. Isoparametric Submanifolds and Their Focal Manifolds. Rank Rigidity of Submanifolds and Normal Holonomy of Orbits. Homogeneous Structures on Submanifolds. Normal Holonomy of Complex Submanifolds. The Berger–Simons Holonomy Theorem. The Skew-Torsion Holonomy Theorem. Submanifolds of Riemannian Manifolds. Submanifolds of Symmetric Spaces. Polar Actions on Symmetric Spaces of Compact Type. Polar Actions on Symmetric Spaces of Noncompact Type. Appendix.

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.