• Bulletin of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.819-835
• /
• 1998

The main purpose of this paper is to prove that if a real hypersurfaces M of a complex space form satisfies ${\nabla}_{\xi}S$=0 and $S{\xi}=\sigma\xi$ for some constant on $\sigma$ on M, then the structure vector field $\xi$ is principal, where S denotes the Ricci tensors of M.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.647-653
• /
• 2012

In this paper, we obtain a generalized Lorentzian splitting theorem by weakening the assumption of nonnegative Ricci curvature.

• Journal of the Korean Mathematical Society
• /
• v.46 no.3
• /
• pp.449-461
• /
• 2009

The object of the present paper is to study $(LCS)_n$-manifolds. Several interesting results on a $(LCS)_n$-manifold are obtained. Also the generalized Ricci recurrent $(LCS)_n$-manifolds are studied. The existence of such a manifold is ensured by several non-trivial new examples.

• Journal of the Korean Mathematical Society
• /
• v.40 no.1
• /
• pp.129-167
• /
• 2003

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concircular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in [18] and [19].

• Honam Mathematical Journal
• /
• v.33 no.4
• /
• pp.547-556
• /
• 2011

In this paper, we estimated the Ricci curvature and the scalar curvature on SU(3)/T (k, l) under the condition (k, l) ${\in}\mathbb{R}^2$ (${\mid}k{\mid}+{\mid}l{\mid}{\neq}0$), where the four isotropy irreducible representations in SU(3)/T (k, l) are, not necessarily, mutually equivalent or inequivalent.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.1087-1094
• /
• 2016

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

• Proceedings of the Korean Statistical Society Conference
• /
• 2002.05a
• /
• pp.45-48
• /
• 2002

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of Stochastic Differential Equation(SDE) for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng(1995). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.

• Journal of the Korean Mathematical Society
• /
• v.42 no.4
• /
• pp.655-672
• /
• 2005

The purpose of this paper is to study n-dimensional QR-submanifolds of maximal QR-dimension isometrically immersed in a quaternionic projective space and to give sufficient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.

• The Pure and Applied Mathematics
• /
• v.18 no.1
• /
• pp.79-86
• /
• 2011

Suppose that (M,g) is a complete and noncompact Riemannian mani-fold with Ricci curvature bounded below by $-K{\leq}0$ and (N, $\bar{g}$) is a complete Riemannian manifold with nonpositive sectional curvature. Let u : $M{\times}[0,{\infty}){\rightarrow}N$ be the solution of a heat equation for harmonic map with a bounded image. We estimate the energy density of u.

• Journal of the Korean Mathematical Society
• /
• v.61 no.1
• /
• pp.183-205
• /
• 2024

In this paper, for an m-dimensional (m ≥ 5) complete non-compact submanifold M immersed in an n-dimensional (n ≥ 6) simply connected Riemannian manifold N with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of M, the norm of its weighted mean curvature vector |H_f| and the weighted real-valued function f, we can obtain: • several one-end theorems for M; • two Liouville theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.