Submanifolds and Holomomy

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