• 대한수학회논문집
• /
• 제25권2호
• /
• pp.245-249
• /
• 2010

We show that on a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ (dim $M^{2n}\;=\;2n\;{\geq}\;4$), $M^{2n}$ is K$\ddot{a}$hler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of $M^{2n}$ everywhere. As a consequence, if a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ is both conformally flat and scalar flat, then $M^{2n}$ is K$\ddot{a}$hler. In contrast with the compact case, we show that there exists a locally conformal K$\ddot{a}$hler manifold with k equal to s, which is not K$\ddot{a}$hler.

• 대한수학회보
• /
• 제56권1호
• /
• pp.219-228
• /
• 2019

In this paper it is proved that on an almost $coK{\ddot{a}}hler$ 3-manifold M, (i) M is h-semisymmetric, (ii) the curvature condition $Q{\cdot}R=0$ and (iii) M is $coK{\ddot{a}}hler$ are equivalent.

• 대한수학회지
• /
• 제44권4호
• /
• pp.757-766
• /
• 2007

In this note we construct an anti-symplectic involution on the non-$K\ddot{a}hler$, symplectic 4-manifold which is constructed by Thurston and show that the quotient of the Thurston's 4-manifold is not symplectic. Also we construct a non-$K\ddot{a}hler$, symplectic 4-manifold using the Gomph's symplectic sum method and an anti-symplectic involution on the non-$K\ddot{a}hler$, symplectic 4-manifold.

• 대한수학회보
• /
• 제55권3호
• /
• pp.789-798
• /
• 2018

Let M be a compact almost $coK{\ddot{a}}hler$ 5-manifold with $K{\ddot{a}}hlerian$ leaves. In this paper, we prove that M is locally symmetric if and only if it is locally isometric to a Riemannian product of a unit circle $S^1$ and a locally symmetric compact $K{\ddot{a}}hler$ 4-manifold.

• Korean Journal of Mathematics
• /
• 제5권1호
• /
• pp.75-82
• /
• 1997

We show that the normal bundle of a Lagrangian submanifold in a K$\ddot{a}$hler manifold has a symplectic structure and provide the equivalent conditions for the normal bundle of such to be K$\ddot{a}$hler.

• 대한수학회보
• /
• 제56권1호
• /
• pp.253-263
• /
• 2019

Let M be a curvature homogeneous or ball-homogeneous non-$coK{\ddot{a}}hler$ almost $coK{\ddot{a}}hler$ 3-manifold. In this paper, we prove that M is locally isometric to a unimodular Lie group if and only if the Reeb vector field ${\xi}$ is an eigenvector field of the Ricci operator. To extend this result, we prove that M is homogeneous if and only if it satisfies ${\nabla}_{\xi}h=2f{\phi}h$, $f{\in}{\mathbb{R}}$.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제20권3호
• /
• pp.199-206
• /
• 2013

We present a 4-dimensional nil-manifold as the first example of a closed non-K$\ddot{a}$hlerian symplectic manifold with the following property: a function is the scalar curvature of some almost K$\ddot{a}$hler metric iff it is negative somewhere. This is motivated by the Kazdan-Warner's work on classifying smooth closed manifolds according to the possible scalar curvature functions.

• 대한수학회보
• /
• 제55권3호
• /
• pp.871-879
• /
• 2018

The purpose of the present paper is to study on nearly $paraK{\ddot{a}}hler$ manifolds. Firstly, to investigate some properties of the Ricci tensor and the $Ricci^*$ tensor of nearly $paraK{\ddot{a}}hler$ manifolds. Secondly, to define a special metric connection with torsion on nearly $paraK{\ddot{a}}hler$ manifolds and present its some properties.

• Korean Journal of Mathematics
• /
• 제21권4호
• /
• pp.473-481
• /
• 2013

We discuss on the classification problem of symplectic manifolds into three families according to the scalar curvature functions of almost K$\ddot{a}$hler metrics they admit. We also present a 4-dimensional solv-manifold as an example which belongs to one of the three families.

• 대한수학회논문집
• /
• 제29권4호
• /
• pp.549-554
• /
• 2014

We present a smooth simply connected closed eight dimensional manifold with distinct symplectic deformation equivalence classes [[${\omega}_i$]], i = 1, 2 such that the symplectic Z invariant, which is defined in terms of the scalar curvatures of almost K$\ddot{a}$hler metrics in [5], satisfies $Z(M,[[{\omega}_1]])={\infty}$ and $Z(M,[[{\omega}_2]])$ < 0.