The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.

Real Hypersurfaces in Complex Space Forms

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric
 
$_g=cI+D$
 for some 
$c\in{mathbb R}$
 and some derivationD of the Lie algebra 
${\mathfrak n}$
 ofN, where Ric
 
$_g$
 denotes the Ricci operator of 
$(N,g)$
 . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold 
$(S,\tilde{g})$
 is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set.

Ricci soliton homogeneous nilmanifolds

General theory of topological groups: Basic topological group theory Some examples Integration and convolution Representation theory and compact groups: Basic representation theory Representations of compact groups Compact Lie groups and homogeneous spaces Discrete co-compact subgroups Introduction to commutative spaces: Basic theory of commutative spaces Spherical transforms and Plancherel formulae Special case: Commutative groups Structure and analysis for commutative spaces: Riemannian symmetric spaces Weakly symmetric and reductive commutative spaces Structure of commutative nilmanifolds Analysis on commutative nilmanifolds Classification of commutative spaces Bibliography Subject index Symbol index Table index.

Harmonic analysis on commutative spaces

Manifolds all of whose geodesics are closed

and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

A short survey on biharmonic maps between Riemannian manifolds

Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

Since E. Cartan's work in the late 30's the classification problem of hypersurfaces with constant principal curvatures is known to be far from trivial. In real space forms it leads to the well-known classification problem of isoparametric hypersurfaces, which has been solved in euclidean space by T. Levi-Civita [6] and B. Segre [11] and in hyperbolic space by E. Cartan [1]; in the sphere, however, a complete classification has not been obtained until now (for essential results see [1], [2], [3], [8], [9], [10], and the literature cited there).

Real hypersurfaces with constant principal curvatures in complex hyperbolic space.

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.

Submanifolds and Holonomy

The complex two-plane Grassmannian G
2(C

 m+2
 in equipped with both a Kahler and a quaternionic Kahler structure. By applying these two structures to the normal bundle of a real hypersurface M in G
2(C

 m+2
 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G
2
C

 m+2
 , m≥3, for which these two distributions are invariant under the shape operator of M.

Real Hypersurfaces in Complex Two-Plane Grassmannians

Obviously, every totally umbilical hypersurface of a Riemannian manifold is curvature-adapted. In spaces of constant sectional curvature every hypersurface is curvature-adapted. But in other ambient spaces our definition is restrictive. For example, in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurfaces (see [3] for the notion of Hopf hypersurfaces). In locally Symmetrie spaces it turns out that for the investigation of the geometry of curvature-adapted hypersurfaces Jacobi field theory may be very useful ( s can be seen in section 5).

Jurgen Berndt

Papers

Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

Real hypersurfaces with constant principal curvatures in complex hyperbolic space.

Submanifolds and Holonomy

Real Hypersurfaces in Complex Two-Plane Grassmannians

Real hypersurfaces in quaternionic space forms.