• Honam Mathematical Journal
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• 제43권4호
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• pp.701-715
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• 2021

In this paper, we study almost complex Norden Silver manifolds and Kaehler-Norden Silver manifolds. We define adapted connections of first, second and third type to an almost complex Norden Silver manifold and establish the necessary and sufficient conditions for the integrability of almost complex Norden Silver structure. Moreover, we investigate that a complex Norden Silver map is a harmonic map between Kaehler-Norden Silver manifolds.

• Bulletin of the Korean Mathematical Society
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• 제41권1호
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• pp.137-145
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• 2004

In [13], we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the Kahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of M is ${\geq}\;c^2\;>\;0$, then M is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for Kahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

• Journal of the Korean Mathematical Society
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• 제40권3호
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• pp.391-408
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• 2003

In this paper we discuss complex manifolds of dimension $n{\ge}2$ that admit effective actions of either $U_n$\;or\;$SU_n$ by biholomorphic transformations.

• Bulletin of the Korean Mathematical Society
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• 제21권2호
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• pp.61-66
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• 1984

It is well-known that the most interesting non-integrable almost Hermitian manifold are the nearly Kaehlerian manifolds ([2] and [3]), and that there exists a complex but not a Kaehlerian structure on Riemannian product manifolds of two normal contact manifolds [4]. The purpose of the present paper is to study nearly Kaehlerian product manifolds of two almost contact metric manifolds and investigate the geometrical structures of these manifolds. Unless otherwise stated, we shall always assume that manifolds and quantities are differentiable of class $C^{\infty}$. In Paragraph 1, we give brief discussions of almost contact metric manifolds and their Riemannian product manifolds. In paragraph 2, we investigate the perfect conditions for Riemannian product manifolds of two almost contact metric manifolds to be nearly Kaehlerian and the non-existence of a nearly Kaehlerian product manifold of contact metric manifolds. Paragraph 3 will be devoted to a proof of the following; A conformally flat compact nearly Kaehlerian product manifold of two almost contact metric manifolds is isomatric to a Riemannian product manifold of a complex projective space and a flat Kaehlerian manifold..

• Journal of the Korean Mathematical Society
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• 제46권3호
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• pp.513-521
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• 2009

We first introduce a complex hyperbolic space and a complex hyperbolic manifold. After defining the canonical horoball and the canonical cusp on the complex hyperbolic manifold, we estimate the volumes of canonical cusps of complex hyperbolic manifolds. Finally, we deal with cusped, complex hyperbolic 2-manifolds, and in particular, the ones with only one cusp.

• Bulletin of the Korean Mathematical Society
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• 제59권5호
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• pp.1069-1091
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• 2022

We study almost complex metallic Norden manifolds and their adapted connections with respect to an almost complex metallic Norden structure. We study various connections like special connection of the first type, special connection of the second type, Kobayashi-Nomizu metallic Norden type connection, Yano metallic Norden type connection etc., on almost complex metallic Norden manifolds. We establish classifications of almost complex metallic Norden manifolds by using covariant derivative of the almost complex metallic Norden structure and also by using torsion tensor on the canonical connections.

• Bulletin of the Korean Mathematical Society
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• 제39권1호
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• pp.133-140
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• 2002

On compact complex hyperbolic manifolds of complex dimension two, we show that the dimension of the space of infinitesimal deformations of self-dual conformal structures is smaller than that of the deformation obstruction space and that every self-dual metric with covariantly constant Ricci tensor must be a standard one upto rescalings and diffeomorphisms.

• East Asian mathematical journal
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• 제35권5호
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• pp.529-534
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• 2019

The purpose of this article is to present The $Carath{\acute{e}}odory$-Cartan-Kaup-Wu theorem for connected Kobayashi hyperbolic almost complex manifolds.

• Honam Mathematical Journal
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• 제5권1호
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• pp.45-69
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• 1983

The hermitian structures on complex manifolds have been studied by several mathematicians ([1], [2], and [3]), and the Kähler structure on hermitian manifolds have been so much too ([6], [12], and [15]). There has been some gradual progress in studying the invariant forms on Grassmann manifolds ([17]). The purpose of this dissertation is to prove the Theorem 3.4 and the Theorem 4.7, with relation to the nature of complex Grassmann manifolds. In $\S$ 2. in order to prove the Theorem 4.7, which will be explicated further in $\S$ 4, the concepts of the hermitian structure, connection and curvature have been defined. and the characteristic nature about these were proved. (Proposition 2.3, 2.4, 2.9, 2.11, and 2.12) Two characteristics were proved in $\S$ 3. They are almost not proved before: particularly. we proved the Theorem 3.3 : $G_{k}(C^{n+k})=\frac{GL(n+k,C)}{GL(k,n,C)}=\frac{U(n+k)}{U(k){\times}U(n)}$ In $\S$ 4. we explained and proved the Theorem 4. 7 : i) Complex Grassmann manifolds are Kahlerian. ii) This Kähler form is $\pi$-fold of curvature form in hyperplane section bundle. Prior to this proof. some propositions and lemmas were proved at the same time. (Proposition 4.2, Lemma 4.3, Corollary 4.4 and Lemma 4.5).

• Bulletin of the Korean Mathematical Society
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• 제42권2호
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• pp.405-413
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• 2005

We consider the connections $\nabla$ on the Rizza manifold (M, J, L) satisfying ${\nabla}G=0\;and\;{\nabla}J=0$. Among them, we derive a Lichnerowicz connection from the Cart an connection and characterize it in terms of torsion. Generalizing Kahler condition in Hermitian geometry, we define a Kahler condition for Rizza manifolds. For such manifolds, we show that the Cartan connection and the Lichnerowicz connection coincide and that the almost complex structure J is integrable.