Submanifolds and Holonomy, Second Edition [Hardcover]

Submanifolds and Holonomy, Second Editionexplores 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 BergerSimons 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, Moores 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-represenl“g