• 대한수학회지
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• 제44권4호
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• pp.757-766
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• 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.

• Korean Journal of Mathematics
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• 제32권1호
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• pp.83-95
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• 2024

The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin ${\frac{1}{2}}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.

• 충청수학회지
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• 제20권2호
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• pp.157-161
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• 2007

We show that there is an equivariant symplectic embedding of a two-torus with a nontrivial action into a symplectic manifold with a symplectic circle action if and only if the circle action on the manifold is non-Hamiltonian. This is a new equivalent condition for non-Hamiltonian action and gives us a new insight to solve the famous conjecture by Frankel and McDuff.

• 대한수학회보
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• 제36권1호
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• pp.209-216
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• 1999

Let X be a minimal symplectic four-manifold with ${b_2}^{+}$=1 and $c_1(K)^2\;\geq\;0$. Then we show that there are no symple tic structures $\omega$ such that $$c_1(K)$\cdot\omega$ > 0, if X contains an embedded symplectic submanifold $\Sigma$ satisfying $\int_\Sigmac_1$(K)<0.

• 대한수학회논문집
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• 제29권4호
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• pp.549-554
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• 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.

• 대한수학회보
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• 제38권4호
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• pp.741-751
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• 2001

In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1｜{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

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

• 대한수학회지
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• 제47권4호
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• pp.675-689
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• 2010

For a closed symplectic 4-manifold X, let $Diff_0$(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of $Diff_0$(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {$n_1$, $n_2$, $\ldots$, $n_k$} and any non-negative integer m, there exists a closed symplectic (or K$\ddot{a}$hler) 4-manifold X with $b_2^+$ (X) > m such that the homologies $H_i$ of the quotient space $Diff_0$(X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = $2n_1$ - 1, $\ldots$, $2n_k$ - 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati$\acute{c}$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제3권1호
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• pp.19-32
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• 1996

A symplectic manifold is a pair (M, $\omega$) consisting of a smooth manifold M and a non-degenerate closed 2-form $\omega$ on M. Locally, $\omega$ = (equation omitted) and d$\omega$ = 0, when n = dimM. The condition d$\omega$ = 0 implies that locally $\omega$ = d${\alpha}$ with ${\alpha}$ = (equation omitted). There are three main sources of symplectic manifolds.(omitted)

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권4호
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• pp.359-364
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• 2015

We present smooth simply connected closed 4k-dimensional manifolds N := N_k, for each k ∈ {2, 3, ⋯}, with distinct symplectic deformation equivalence classes [[ω_i]], i = 1, 2. To distinguish [[ω_i]]’s, we used the symplectic Z invariant in [4] which depends only on the symplectic deformation equivalence class. We have computed that Z(N, [[ω₁]]) = ∞ and Z(N, [[ω₂]]) < 0.