• Kyungpook Mathematical Journal
• /
• v.46 no.4
• /
• pp.463-476
• /
• 2006

It is known that there are no real hypersurfaces with parallel Ricci tensor in a nonflat complex space form ([6], [9]). In this paper we investigate real hypersurfaces in a complex projective space $P_n\mathbb{C}$ using some conditions of the Ricci tensor S which are weaker than ${\nabla}S=0$. We characterize Hopf hypersurfaces of $P_n\mathbb{C}$.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.2
• /
• pp.315-336
• /
• 1999

In the paper [12] we have introduced the new kind of pseudo-einstein ruled real hypersurfaces in complex space forms $M_n(c), c\neq0$, which are foliated by pseudo-Einstein leaves. The purpose of this paper is to give a geometric condition for non-proper pseudo-Einstein ruled real hypersurfaces to be totally geodesic in the sense of Kimura [8] for c> and Ahn, Lee and the present author [1] for c<0.

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.307-326
• /
• 2007

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]). In this paper we investigate real hypersurfaces M in a nonflat complex space form $M_n(c)$ under the condition that ${\nabla}_{\varepsilon}S=0\;and\;{\nabla}_{\varepsilon}R_{\varepsilon}=0,\;where\;S\;and\;R_{\varepsilon}$ respectively denote the Ricci tensor and the structure Jacobi operator of M in $M_n(c)$.

• The Pure and Applied Mathematics
• /
• v.17 no.1
• /
• pp.51-63
• /
• 2010

In this paper, we prove two characterization theorems for real half lightlike submanifold (M,g,S(TM)) of an indefinite Kaehler manifold $\bar{M}$ or an indefinite complex space form $\bar{M}$(c) subject to the conditions : (a) M is totally umbilical in $\bar{M}$, or (b) its screen distribution S(TM) is totally umbilical in M.

• Honam Mathematical Journal
• /
• v.44 no.3
• /
• pp.339-359
• /
• 2022

In this paper, we investigate k-Ricci curvature and k-scalar curvature on lightlike hypersurfaces of a real space form ${\tilde{M}}$(c) of constant sectional curvature c, endowed with semi-symmetric non-metric connection. Using this curvatures, we establish some inequalities for screen homothetic lightlike hypersurface of a real space form ${\tilde{M}}$(c) of constant sectional curvature c, endowed with semi-symmetric non-metric connection. Using these inequalities, we obtain some characterizations for such hypersurfaces. Considering the equality case, we obtain some results.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.173-184
• /
• 1997

An n-dimensional complex space form $M_n(c)$ is a Kaehlerian manifold of constant holomorphic sectional curvature c. As is well known, complete and simply connected complex space forms are a complex projective space $P_n C$, a complex Euclidean space $C_n$ or a complex hyperbolic space $H_n C$ according as c > 0, c = 0 or c < 0.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.195-202
• /
• 2007

We prove that a real hypersurface M in a complex space form Mn(c), $c{\neq}0$, whose Ricci operator and structure tensor commute each other on the holomorphic distribution and the Ricci operator is ${\eta}-parallel$, is a Hopf hypersurface. We also give a characterization of this hypersurface.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.327-335
• /
• 2005

For a compact and orientable minimal real hypersurface $M\;in\;QP^n$, we prove that if the minimum of the sectional curvatures of Mis 3/(4n - 1), then M is isometric to the geodesic minimal hypersphere $M_{0,n-1}^Q$.

• East Asian mathematical journal
• /
• v.36 no.3
• /
• pp.389-415
• /
• 2020

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, ξ, η, g) in a complex space form M_n+1(c), c ≠ 0. We denote by R_ξ and R'_X be the structure Jacobi operator with respect to the structure vector ξ and be R'_X = (∇_XR)(·, X)X for any unit vector field X on M, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R_ξ𝜙 = 𝜙R_ξ and at the same time R'_ξ = 0, then M is a Hopf real hypersurfaces of type (A), provided that the scalar curvature ${\bar{r}}$ of M holds ${\bar{r}}-2(n-1)c{\leq}0$.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.337-358
• /
• 2005

Let M be a real hypersurface with almost contact metric structure $(\phi,\;\xi,\;\eta,\;g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor S, then M is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of M is constant or $g(S\xi,\;\xi)$ is constant.