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 classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G
2 (ℂ
 m+2
 ) of all 2-dimensional linear subspaces in ℂ
 m+2
 , m ≥ 3.

Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH", n ≥ 3. For the quaternionic hyperbolic spaces MH n , n > 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.

/pdf/cohomogeneity-one-actions-on-noncompact-symmetric-spaces-of-8ul64vv6u9.pdf

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.

We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures

/pdf/real-hypersurfaces-with-constant-principal-curvatures-in-3ekywnsg2u.pdf

Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces

We determine the isometric congruence classes of homogeneous Riemannian foliations of codimension one on connected irreducible Riemannian symmetric spaces of noncompact type. As an application we show that on each connected irreducible Riemannian symmetric space of noncompact type and rank greater than two there exist noncongruent homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.

/pdf/homogeneous-codimension-one-foliations-on-noncompact-1cqoznheqn.pdf

Jurgen Berndt

Papers

Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces

Homogeneous Codimension One Foliations on Noncompact Symmetric Spaces