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

We present without proofs some basic theory about contact hypersurfaces in Kahler manifolds. We then discuss the classification problem for some Hermitian symmetric spaces.

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Contact hypersurfaces in Kähler manifolds

We classify all totally geodesic submanifolds of connected irreducible Riemannian symmetric spaces of noncompact type which arise as a singular orbit of a cohomogeneity one action on the symmetric space.

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Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit

Weakly symmetric spaces are particular Riemannian homogeneous spaces which have been introduced by Selberg [21] in 1956 in the framework of his trace formula. They attracted only little interest until the author and Vanhecke [7] found a simple geometric characterization of weakly symmetric spaces which lead to a large number of new examples. The purpose of this note is to present a survey about this topic.

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Geometry of weakly symmetric spaces

One studies two classes of Riemannian manifolds which extend the class of locally symmetric spaces: manifolds all of whose Jacobi operators Rγ have constant eigenvalues ( C -spaces) or parallel eigenspaces ( B -spaces) along geodesics γ. One gives several examples, derives equivalent characterizations and treats classifications for the two- and the three-dimensional case.

Two natural generalizations of locally symmetric spaces

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kahler structure or a quaternionic Kahler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU2, m/S(U2Um) of noncompact type.

Jurgen Berndt

Papers

Contact hypersurfaces in Kähler manifolds

Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit

Geometry of weakly symmetric spaces

Two natural generalizations of locally symmetric spaces

Hypersurfaces in noncompact complex grassmannians of rank two