• Honam Mathematical Journal
• /
• v.32 no.4
• /
• pp.651-661
• /
• 2010

In this paper, we obtain a necessary and sufficient condition for a left invariant connection in the tangent bundle over a closed Lie group with a left invariant metric to be a Yang-Mills connection. Moreover, we have a necessary and sufficient condition for a left invariant connection with a torsion-free Weyl structure in the tangent bundle over SU(2) with a left invariant Riemannian metric g to be a Yang-Mills connection.

• Honam Mathematical Journal
• /
• v.38 no.1
• /
• pp.1-8
• /
• 2016

In this paper, we get a complete condition for a group homomorphism of a compact Lie group with an arbitrarily given left invariant Riemannian metric into another Lie group with a left invariant metric to be a harmonic map, and then obtain a necessary and sufficient condition for a group homomorphism of (SU(2), g) with a left invariant metric g into the Heisenberg group (H, $h_0$) to be a harmonic map.

• Honam Mathematical Journal
• /
• v.36 no.1
• /
• pp.141-146
• /
• 2014

In this paper, we obtain a necessary and sufficient condition for the second variation of an arbitrarily given smooth variation of a geodesic on a Riemannian manifold to be 0.

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

• Honam Mathematical Journal
• /
• v.39 no.4
• /
• pp.465-473
• /
• 2017

In this paper, we define a totally umbilic hypersurface of first order and show that a totally umbilic hypersurface of first order in an Einstein manifold has a parallel second fundamental form. Furthermore we prove that a complete, simply connected and totally umbilic hypersurface of first order in a space of constant curvature is a Riemannian product of Einstein manifolds. Finally we show a proper example which is a totally umbilic hypersurface of first order but not a totally umbilic hypersurface.

• Journal of the Korean Mathematical Society
• /
• v.55 no.5
• /
• pp.1075-1089
• /
• 2018

We define a new connection on semi-Riemannian manifolds, which is a non-symmetric and non-metric connection. We say that this connection is an (${\ell}$, m)-type connection. Semi-symmetric non-metric connection and non-metric ${\phi}$-symmetric connection are two important examples of this connection such that (${\ell}$, m) = (1, 0) and (${\ell}$, m) = (0, 1), respectively. In this paper, we study lightlike hypersurfaces of an indefinite trans-Sasakian manifold with an (${\ell}$, m)-type connection.

• Journal of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.449-463
• /
• 1998

In this paper we introduce the notions of orbital Lipschitz stability (in variation) and orbital exponential asymptotic stability (in variation) of $C^{r}$ dynamical systems (or $C^{r}$ diffeomor-phisms) on Riemannian manifolds, and study the embedding problem of those concepts in $C^{r}$ dynamical systems.stems.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.3
• /
• pp.475-487
• /
• 2003

A generalized even-dimensional Riemannian manifold defined by the ME-connection which is both Einstein and of the form (3.3) is called an even-dimensional ME-manifold and we denote it by $MEX_{2n}$. The purpose of this paper is to study a necessary and sufficient condition that there is an ME-connection, to derive the useful properties of some tensors, and to investigate a representation of the ME-vector in $MEX_{2n}$.

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

• Communications of the Korean Mathematical Society
• /
• v.13 no.3
• /
• pp.539-544
• /
• 1998

Let (M, g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvature S, which is close to O. With conditions on a conformal invariant and scalar curvature of (M, g), we show that there exists a conformal metric (equation omitted), near g, whose scalar curvature (equation omitted) = 0 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_{i}$ with ∪$K_{i}$ = M.