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

. In this paper, ﬂrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C

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The ricci tensor of real hypersurfaces in complex two-plane grassmannians

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

If a compact real hypersurface of contact-type in a complex number space admits a Ricci soliton, then it is a sphere. A compact Hopf hypersurface in a non-flat complex space form does not admit a Ricci soliton. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Ricci solitons of compact real hypersurfaces in Kähler manifolds

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type (A) in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator

We study real hypersurfaces of a complex space form such that the Jacobi operator with respect to the structure vector field and the structure tensor $\phi$ on the real hypersurface commute.

On real hypersurfaces of a complex space form in terms of jacobi operators

We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽S = 0 and Sξ = ξ for a smooth function , then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.

REAL HYPERSURFACES WITH ξ-PARALLEL RICCI TENSOR IN A COMPLEX SPACE FORM

Let M be a real hypersurface with almost contact metric structure in a nonflat complex space form . In this paper, we prove that if the structure Jacobi operator commutes with both the structure tensor and the Ricc tensor S, then M is a Hopf hypersurface in provided that the mean curvature of M is constant or is constant.

The structure jacobi operator on real hypersurfaces in a nonflat complex space form

The purposer of this paper is to prove a rigidity theorem for real hypersurfaces in a complex projective spece.

A rigidity theorem for real hypersurfaces in a complex projective space

In this paper, We characterize a semi-invariant sub- manifold of codimension 3 satisfying r»A = 0 in a complex pro- jective space CP n+1 , where r»A is the covariant derivative of the shape operator A in the direction of the distinguished normal with respect to the structure vector field ».

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Seong-Baek Lee

Papers

On real hypersurfaces of a complex space form in terms of jacobi operators

REAL HYPERSURFACES WITH ξ-PARALLEL RICCI TENSOR IN A COMPLEX SPACE FORM

The structure jacobi operator on real hypersurfaces in a nonflat complex space form

A rigidity theorem for real hypersurfaces in a complex projective space

On some semi-invariant submanifolds of codimension 3 in a complex projective space