In-Bae Kim

Characterizations of Real Hypersurfaces in a Complex Space Form

We study a real hypersurface M in a complex space form Mn, c \neq 0, whose shape operator and structure tensor commute each other on the holomorphic distribution of M.

ON CHARACTERIZATIONS OF REAL HYPERSURFACES WITH

We shall give a characterization of a real hypersurface M in a complex space form Mn(c), , whose Ricci operator and structure tensor commute each other on the holomorphic distribution of M, and the Ricci operator is .

{\eta}-PARALLEL

Let M be a real hypersurface in a complex space form Mn(c), c ̸= 0. In this paper we prove that if RξLξ = LξRξ holds on M , then M is a Hopf hypersurface, where Rξ and Lξ denote the structure Jacobi operator and the induced operator from the Lie derivative with respect to the structure vector field ξ, respectively. We characterize such Hopf hypersurfaces of Mn(c).

RICCI OPERATORS IN A COMPLEX SPACE FORM

Let (C) be a complex hyperbolic space of complex dimension ( ≧ 2) endowed with the metric of constant holomorphic sectional curvatur e 4 , and be the identity component of the group of all isometries of ( C). A submanifold in ( C) is said to be extrinsically homogeneous if is an orbit under a closed subgroup of . As proposed also in R. Niebergall and P.J. Ryan ([7]), the fol lowing is an open problem:Classify all extrinsically homogeneous real hypersurface s in (C). As a partial answer of this problem, J. Berndt ([1]) classified all ex trinsically homogeneous real hypersurfaces in ( C) whose structure vector fields are principal, where an eigen vector of the shape operator is called principal. Recently he constructed in [2] a subgroup of for each ( ≧ 2) such that a certain orbit under in ( C) has three distinct principal curvatures 1, −1 and 0 with multiplicities 1, 1 and 2− 3 respectively and the structure vector field on is not principal. We shall call this group the Berndt subgroupof . The following is due to J. Berndt and H. Tamaru.

On characterizations of Hopf hypersurfaces in a nonflat complex space form with commuting operators

AbstractIn the class of real hypersurfaces M2n−1 isometrically immersed into a nonflat complex space form

A characterization of a certain real hypersurface of type (A 2 ) in a complex projective space

\widetilde {{M_n}}\left( c \right)

Ricci-Pseudo-Symmetric Real Hypersurfaces in Complex Space Forms

Mn˜(c) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ℂPn(c) or a complex hyperbolic space ℂHn(c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in