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

One derives a local classification of all three-dimensional Riemannian manifolds whose Ricci tensor satisfies the equation ▿(ric– 1 4 sg) = 1 20 ds ⊙ g.

Three-dimensional Einstein-like manifolds

The k -number of a complex flag manifold and the index number of a real flag manifold is known to be equal to the sum of the Z 2 -Betti numbers of the manifold. We give an interpretation and alternative proofs of these results in the framework of symplectic topology and Morse theory

On index number and topology of flag manifolds

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct many examples which arise from the theory of polars and centrioles in Riemannian symmetric spaces of compact type.

Compact homogeneous Riemannian manifolds with low co-index of symmetry

We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.

Jurgen Berndt

Papers

Three-dimensional Einstein-like manifolds

On index number and topology of flag manifolds

Compact homogeneous Riemannian manifolds with low co-index of symmetry

Homogeneous polar foliations of complex hyperbolic spaces

Surveys in Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck