• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.205-214
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• 2016

Any open Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. This result along with [5] now shows that an open Riemann surface admits conformal models in any Riemannian manifold of dimension ${\geq}3$.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.35-41
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• 2005

Let (M, $g$) be a compact oriented 4-dimensional Riemannian manifold whose sectional curvature $k$ satisfies $1{\geq}k{\geq}0.1714$. We show that M is topologically $S^4$ or ${\pm}\mathbb{C}\mathbb{P}^2$.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1261-1269
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• 2023

For a fixed parametrization of a curve in an orientable two-dimensional Riemannian manifold, we introduce and investigate a new frame and curvature function. Due to the way of defining this new frame as being the time-dependent rotation in the tangent plane of the standard Frenet frame, both these new tools are called flow.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.605-639
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• 2006

Lower dimensional cases of Einstein's connection were already investigated by many authors for n = 2,3,4,5,6,7. This paper is the first part of the following series of two papers, in which we obtain a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor, with main emphasis on the derivation of powerful and useful recurrence relations which hold in 8-dimensional Einstein's unified field theory(i.e., 8-g-UFT): I. The recurrence relations in 8-g-UFT II. The Einstein's connection in 8-g-UFT All considerations in these papers are restricted to the first class only of the generalized 8-dimensional Riemannian manifold $X_8$.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.561-577
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• 1998

We study (n＋3)-dimensional contact three CR submanifolds of a Riemannian manifold with Sasakian three structure and investigate some characterizations of $S^{4r＋3}$(a) $\times$ $S^{4s＋3}$(b) ($a^2$＋ $b^2$=1, 4(r ＋ s) = n - 3) as a contact three CR sub manifold of a (4m＋3)-dimensional unit sphere.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.793-805
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• 2017

In this paper, we prove that the Ricci tensor of a three-dimensional almost Kenmotsu manifold satisfying ${\nabla}_{\xi}h=0$, $h{\neq}0$, is ${\eta}$-parallel if and only if the manifold is locally isometric to either the Riemannian product $\mathbb{H}^2(-4){\times}\mathbb{R}$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1081-1088
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• 2012

We shall prove the universality of the curvature identity for the 4-dimensional Riemannian manifold using a different method than that used by Gilkey, Park, and Sekigawa [5].

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.231-240
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• 2013

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and of a round 4-sphere.

• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.55-63
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• 1998

A Gauss map of m-dimensional distribution on a Riemannian manifold M is called a harmonic Gauss map if it is a harmonic map from the manifold into its Grassmann bundle $G_m$(TM) of m-dimensional tangent subspace. We calculate the tension field of the Gauss map of m-dimensional distribution and especially show that the Hopf fibrations on $S^{4n＋3}$ are the harmonic Gauss map of 3-dimensional distribution.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.613-618
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• 1996

In this paper, M will always denote an n- dimensional complete Riemannian manifold and $\omega_n$ is the volume of $S^n$.