• Journal for History of Mathematics
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• v.14 no.1
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• pp.109-114
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• 2001

Using Fermi coordinate and Jacobi operator on Kahler manifold, we characterize partially conformal geodesic transformation in Kahler geometry.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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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.

• Journal for History of Mathematics
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• v.16 no.3
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• pp.109-114
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• 2003

In this paper, we deal with partially conformal geodesic transformations in Kahler geometry by using Fermi coordinates when tile submanifold is a geodesic sphere. We derive the necessary and sufficient condition for tile existence of such transformation in terms of the Jacobi operator and its derivative.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.303-315
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• 2001

In this paper, we investigate the topology of complex Finsler manifolds. For a complex Finsler manifold (M, F), we introduce a certain condition on the Finsler metric F on M. This is a generalization of Kahler condition for the Hermitian metric. Under this condition, we can produce a Kahler metric on M. This enables us to use the usual techniques in the Kahler and Riemannian geometry. We show that if the holomorphic sectional curvature of $ M is\geqC^2>0\; for\; some\; c>o,\; then\; diam(M)\leq\frac{\pi}{c}$ and hence M is compact. This is a generalization of the Bonnet\`s theorem in the Riemannian geometry.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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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.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.47-59
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• 2024

The purpose of the Nash embedding theorem was to take extrinsic help for studying the intrinsic Riemannian geometry. To realize this aim in actual practice there is a need for optimal relationships between the known intrinsic invariants and the main extrinsic invariants for Riemannian submanifolds. This paper aims to provide an optimal relationship for CR-warped product submanifolds of locally conformal Kähler manifolds.

• Honam Mathematical Journal
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• v.44 no.1
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• pp.62-77
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• 2022

This study presents an 𝛼-Sasakian structure on the product manifold M₁ × 𝛽(I), where M₁ is a Kähler manifold with an exact 1-form, and 𝛽(I) is an open curve. It then defines a new type warped product metric to study the warped product of almost Hermitian manifolds with almost contact metric manifolds, contact metric manifolds, and K-contact manifolds.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.384-390
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• 2022

In this paper, we study the obstruction on the jets of an almost complex structure J to the existence of a symplectic form ω such that J is compatible with ω. We describe some almost complex structures on ℝ⁴ and on ℝ⁶, respectively, that cannot be calibrated by any symplectic forms. In particular, these examples pertain to the model almost complex structure on ℝ⁴ in [3], and the simple model structure on ℝ⁶ in [7].