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

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given

On Ruled Real Hypersurfaces in Complex Space Forms

The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemannian manifolds. After some remarks on the general case, timelike and lightlike totally umbilic submanifolds of Lorentzian manifolds are discussed, along with their physical interpretation in view of general relativity. In particular, the mathematical notion of totally umbilic submanifolds is linked to the notions of photon surfaces and of null strings which have been used in the physics literature.

On totally umbilic submanifolds of semi-Riemannian manifolds

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

A complex n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). A complete and simply connected complex space form consists of a complex projective space PnC, a complex Euclidean space Cn or a complex hyperbolic space HnC, according as c>0, c=0 or c<0. The induced almost contact metric structure of a real hypersurface M of Mn{c) is denoted by (<ft, t-,rj,g). In his study of real hypersurfaces of a complex projective space PnC, Takagi [10] classifiedall homogeneous real hypersurfaces and Cecil-Ryan [2] showed also that they are realized as the tubes of constant radius over Kaehlerian submanifolds if the structure vector field£is principal. On the other hand, real hypersurfaces of a complex hyperbolic space HnC also investigated by Berndt [1], Montiel [7], Montiel and Romero [8] and so on. Berndt [1] classifiedall homogeneous real hypersurfaces of HnC and showed that they are realized as the tubes of constant radius over certain submanifolds. According to Takagi's classificationtheorem and Berndt's one the principal curvatures and their multiplicitiesof homogeneous real hypersurfaces of Mn{c) are all determined. In particular,Maeda [6] and Okumura [9] (resp. Montiel [7] and MontielRomero [8]) considered real hypersurfaces of Mn{c), c>0 (resp. c<0) whose second fundamental tensor A of M in MJc), c^O, satisfies

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On ruled real hypersurfaces in a complex space form

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

A totally umbilic submanifold of a pseudo-Riemanian manifold is a submanifold whose first fundamental form and second fundamental form are proportiona. An ordinary hypersphere $S^n(r)$ of an affine (n + 1)-space of the Euclidean space $E^m$ is the best known example of totally umbilic submanifolds of $E^m$.

Totally umbilic lorentzian submanifolds

The purpose of this paper is to give another new characterization of ruled real hypersurfaces in a complex space form (c), c0 in terms of the covariant derivative of its Weingarten map in the direction of the structure vector .

A new characterization of ruled real hypersurfaces in complex space forms

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

Seong-Soo Ahn

Papers

On ruled real hypersurfaces in a complex space form

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

Totally umbilic lorentzian submanifolds

A new characterization of ruled real hypersurfaces in complex space forms

A rigidity theorem for real hypersurfaces in a complex projective space