• 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.50 no.4
• /
• pp.1087-1098
• /
• 2013

We find a $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on $\mathbb{R}^k$, $k{\geq}3$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^k$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.581-588
• /
• 2012

We find a $C^{\infty}$ one-parameter family of Riemannian metrics $g_t$ on $\mathbb{R}^3$ for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^3$, the scalar curvatures of $g_t$ are strictly decreasing in t in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball.

• East Asian mathematical journal
• /
• v.38 no.1
• /
• pp.43-63
• /
• 2022

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). We denote by R_ξ the structure Jacobi operator with respect to the structure vector field ξ and by ${\bar{r}}$ the scalar curvature of M. Suppose that R_ξ is φ∇_ξξ-parallel and at the same time 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, then M is a Hopf hypersurface of type (A) in M_n+1(c) provided that ${\bar{r}-2(n-1)c}$ ≤ 0.

• East Asian mathematical journal
• /
• v.38 no.5
• /
• pp.549-581
• /
• 2022

Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form M_n+1(c), c≠ 0. We denote by S and R_𝜉 be the Ricci tensor of M and the structure Jacobi operator in the direction of the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a certain scalar 𝜃(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that M satisfies R_𝜉S = SR_𝜉 and at the same time R_𝜉𝜙 = 𝜙R_𝜉, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.3
• /
• pp.763-774
• /
• 2023

If a paraSasakian manifold of dimension (2n + 1) represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric g has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if r = -6; shrinking if r > -6; expanding if r < -6.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.189-201
• /
• 2005

In this article, we establish relations between the sectional curvature function K and the shape operator, and also relationship between the k-Ricci curvature and the shape operator for slant submanifolds in generalized complex space forms with arbitrary codimension.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.243-251
• /
• 1996

In a Sasakian manifold, a C-Bochner curvature tensor is constructed from the Bochner curvature tensor in a Kaehlefian manifold by the fibering of Boothby-wang[2]. Many subjects for vanishing C-Bocher curvature tensor with constant scalar curvature were studied in [3], [6], [7], [9], [10], [11] and so on.

• Journal of the Korean Mathematical Society
• /
• v.31 no.4
• /
• pp.629-632
• /
• 1994

Let (M, g) be a Riemannian manifold. The relation between a (pointwise) conformal metric of the metric g and the scalar curvature of this new metrics is investigated by Kazdan, Warner and Schoen (cf. [1, 4]).

• Honam Mathematical Journal
• /
• v.30 no.4
• /
• pp.693-701
• /
• 2008

In this paper, when N is a compact Riemannian manifold of class (A), we consider the nonexistence of some warping functions on space-times M = $[a,{\infty}){\times}_fN$ with prescribed scalar curvatures.