U-Hang Ki

Totally real submanifolds of a complex space form

Totally real submanifolds of a complex space form are studied. In particular, totally real submanifolds of a complex number space with parallel mean curvature vector are classified.

Compact minimal generic submanifolds with parallel normal section in a complex projective space

Generic submanifold have been investigated by many authors (e.g. [5], [7], [8], [9], [21]). Here a submanifold M in a Kaehlerian manifold is called generic if each normal space of M is mapped into the tangent space of M by the complex structure of the ambient space (cf. [2], [4], [22]). Any real hypersurface in a Kaehlerian manifold is a typical example of the generic submanifold. In particular, the model space of the so called AI , A^ B, C, D and E-type are typical examples of a real hypersurface in a complex projective space P(C). Recently, the third named author, B. H. Kim and I.-B. Kim [19] proved that those model spaces exhaust all intrinsic homogeneous real hypersurfaces in P(C). On the other hand, the model spaces of the type AI and A 2 was frist introduced by Law son [13], and he gave a characterization of them. Moreover, Choe and Okumura [5] gave a generalization of Lawsons theorem in [13] from a viewpoint of the CR-submanifold (see §1 for the definiton). The purpose of the present paper is to give another generalization (Theorem A) of Lawsons theorem, from a viewpoint of the generic submanifold, and to give new examples of a generic submanifold in P(C). The authors would like to thank the refree for his suggestions, which resulted in many improvments of the present paper.

Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonflat Complex Space Form

Let M be a real hypersurface of a complex space form with almost contact metric structure (φ, ξ, η, g). In this paper, we prove that if the structure Jacobi operator Rξ = R(·, ξ)ξ is φ∇ξξ-parallel and Rξ commute with the structure tensor φ, then M is a homogeneous real hypersurface of Type A provided that TrRξ is constant.

JACOBI OPERATORS ALONG THE STRUCTURE FLOW ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM II

Let M be a real hypersurface of a complex space form with almost contact metric structure (ϕ; �; �; g ). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator R = R (� ; � )� is � -parallel. In particular, we prove that the condition ∇ R = 0 characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when R ϕS = R Sϕ holds on M , where S denotes the Ricci tensor of type (1,1) on M.

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM USED BY THE ζ-PARALLEL STRUCTURE JACOBI OPERATOR

Let M be a real hypersurface of a complex space form with almost contact metric structure . In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator is -parallel. In particular, we prove that the condition characterize the homogeneous real hypersurfaces of type A in a complex: projective space or a complex hyperbolic space when is constant and not equal to -c/24 on M, where c is a constant holomorphic sectional curvature of a complex space form.

Complete and non-compact conformally flat manifolds with constant scalar curvature

Non-compact conformally flat manifolds with constant scalar curvature and non-compact Kaehler manifolds with vanishing Bochner curvature are studied and classified.

Structure Jacobi Operators of Real Hypersurfaces with Constant Mean Curvature in a Complex Space Form

Let M be a real hypersurface with constant mean curvature in a complex space form Mn(c), c ̸= 0. In this paper, we prove that if the structure Jacobi operator Rξ = R(·, ξ)ξ with respect to the structure vector field ξ is φ∇ξξ-parallel and Rξ commute with the structure tensor field φ, then M is a homogeneous real hypersurface of Type A.