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 introduce the pseudo-Einstein structure on real hypersurfaces in a K\"ahlerian manifold, namely, the Ricci curvature tensor for the generalized Tanaka-Webster connection (restricted) on the Levi subbundle $D$ is proportional to the Levi form. In particular, we give a classification of pseudo-Einstein Hopf-hypersurfaces in a non-flat complex space form.

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Pseudo-Einstein CR-structures on real hypersurfaces in a complex space form

In this paper, we classify the real hypersurfaces in a non-flat complex space form with η-parallel shape operator.

Real hypersurfaces in a complex space form with η-parallel shape operator

This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector 4 is principal, the focal map has constant rank, and VyS = 0, where S is the Ricci tensor of the real hypersurface.

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Ricci tensors of real hypersurfaces in a complex projective space

Abstract In this paper we study real hypersurfaces in a non-flat complex space form with $\eta $ -parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

Real hypersurfaces in a complex space form with eta-parallel shape operator

Let Mn(c) denote an n-dimensional complex space form with constant holomorphic sectional curvature c. It is well known that a complete and simplyconnected complex space form consists of a complex projective space CPn, a complex Euclidean space Cn or a complex hyperbolic space CHn, according as c>0, c=0 or c<0. In this paper we consider a real hypersurface M of CPn or CHn. The study of real hypersurfaces of CPn was initiated by Takagi [10], who proved that all homogeneous hypersurfaces of CPn could be divided into six types which are said to be of type Au A2, B, C, D and E. Moreover, he showed that if a real hypersurface M of CPn has two or three distinctconstant principal curvatures, then M is locally congruent to one of the homogeneous ones of type Au A2 and B ([11]). Recently, to give another charac terization of homogeneous hypersurfaces of type Alt A2 and B in CPn Kimura and Maeda [6] introduced the notion of an ^-parallel second fundamental form, which was defined by g((FxA)Y, Z)=0 for any vector fields X, Y and Z orthogonal to the structure vector field£,where A means the second fundamental form of M in CPn, and g and V denote the induced Riemannian metric and the induced Riemannian connection, respectively. On the other hand, real hypersurfaces of CHn have also been investigated by many authors (Berndt [1], Montiel [8], Montiel and Romero [9]). Using some results about focal sets, Berndt [1] proved the following.

Young JinSuh

Papers

On real hypersurfaces of a complex space form with ^-parallel ricci tensor

ON TYPE NUMBER OF REAL HYPERSURFACES IN Pn(C)