• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.109-124
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• 2011

The main objective of the paper is to provide a full classification of quasi-conformally recurrent Riemannian manifolds with harmonic quasi-conformal curvature tensor. Among others it is shown that a quasi-conformally recurrent manifold with harmonic quasi-conformal curvature tensor is any one of the following: (i) quasi-conformally symmetric, (ii) conformally flat, (iii) manifold of constant curvature, (iv) vanishing scalar curvature, (v) Ricci recurrent.

• Honam Mathematical Journal
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• v.41 no.2
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• pp.301-319
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• 2019

The aim of the paper is to characterize some curves with the help of their harmonic curvature functions. First of all, we have defined harmonic curvature function of an arbitrary curve and have re-determined the position vectors of helices in terms of their harmonic curvature functions in Galilean 3-space. Then, we have investigated the relation between rectifying curves and Salkowski (or anti-Salkowski) curves in Galilean 3-space. Furthermore, the position vectors of them are obtained via the serial approach of the curves. Finally, we have given some illustrated examples of helices and rectifying curves with some assumptions.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.285-289
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• 1999

In this paper, we estimate the total curvature of non-parametric minimal surfaces by using the properties of univalent harmonic mappings defined on ${\Delta}=\{z:{\mid}z:{\mid}>1\}$.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.803-814
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• 1995

Let $\Sigma$ be the class of all complex-valued, harmonic, orientation-preserving, univalent mappings defined on $\Delta = {z : $\mid$z$\mid$ > 1}$ that map $\infty$ to $\infty$.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.205-212
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• 1998

We investigate the behavior of $L^2$ harmonic one forms on complete manifolds and as an application, we show the space of $L^2$harmonic one forms on a complete Riemannian manifold of nonnegative Ricci curvature outside a compact set with bounded $n/2$-norm of Ricci curvature satisfying the Sobolev inequality is finite dimensional.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.623-636
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• 2023

In this paper, we prove some vanishing theorems under the assumptions of weighted BiRic curvature or m-Bakry-Émery-Ricci curvature bounded from below.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.789-796
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• 1996

A complete open manifold with nonnegative curvature is diffeomorphic to the normal bundle of the soul, and the projection map is a Riemannian submersion. Under certain circumstances, we prove that a harmonic map from a compact manifold followed by the projection is again harmonic. Therefore we obtain a harmonic map onto the soul when there is a harmonic map into an open manifold.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.911-920
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• 2019

In this paper we characterize semiconformally flat spacetimes and a spacetime with harmonic semiconformal curvature tensor. At first in a semiconformally flat perfect fluid spacetime we obtain a state equation and prove that in particular for dimension n = 4, the spacetime represents a model for incoherent radiation. Next we prove that perfect fluid spacetime with harmonic semiconformal curvature tensor is of Petrov type I, D or O and the spacetime is a GRW spacetime. As a consequence we obtain several corollaries.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.977-991
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• 2012

Let (M,F) and (M',F') be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F' is nonpositive, then any transversally harmonic map ${\phi}:(M,F){\rightarrow}(M^{\prime},F^{\prime})$ is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ${\phi}$ is transversally constant.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1461-1466
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• 2010

aSuppose that (M, g) is a complete Riemannian manifold with Ricci curvature bounded below by -K < 0 and (N, $\bar{b}$) is a complete Riemannian manifold with sectional curvature bounded above by a constant $\mu$ > 0. Let u : $M{\times}[0,\;{\infty}]{\rightarrow}B_{\tau}(p)$ is a heat equation for harmonic map. We estimate the energy density of u.