Submanifolds and Holonomy

; Sergio Console ; Carlos Enrique Olmos

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. Les mer
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Om boka

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.

Fakta

Innholdsfortegnelse

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.

Om forfatteren

Jurgen 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.