• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1607-1620
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• 2023

In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of ℓ^p-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.737-767
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• 2016

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

• Proceedings of the Korean Statistical Society Conference
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• 2002.05a
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• pp.45-48
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• 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
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• v.44 no.6
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• pp.1363-1372
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• 2007

We investigate the containment measure of one domain to contain in another domain in a plane $X^{\kappa}$ of constant curvature. We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in $X^{\kappa}$.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.587-599
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• 2000

In this paper we study some nonpositively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. Among other results it is proved that if the universal covering manifold satisfies some conditions then every nonexceptional singular orbit is a totally geodesic submanifold. When M is flat and is not toruslike, it is proved that either each orbit is isometric to $R^k\timesT^m$or there is a singular orbit. If the singular orbit is unique and nonexceptional, then it is isometric to $R^k\timesT^m$.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.835-848
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• 1995

Let $M_n$(c) be an n-dimensional complete and simply connected Kahlerian manifold of constant holomorphic sectional curvature c, which is called a complex space form. Then according to c > 0, c = 0 or c < 0 it is a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.285-292
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• 2001

The authors compute the Ricci curvature of the GRW space-time to obtain two conditions for the conjugate points which appear as the Timelike Convergence Condition(TCG) and the Jacobi inequality. Moreover, under such two conditions, we obtain a lower bound of the length of a unit timelike geodesic for focal points emanating form the immersed spacelike hypersurface, the graph over the fiber in the GRW space-time.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.173-187
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• 2021

In this paper, we deal with the notion of a v-semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. We investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. Given such a map with totally umbilical fibers, we have a condition for the fibers of the map to be minimal. We also obtain an inequality of a proper v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and a v-semi-slant angle. Moreover, we give some examples of such maps and some open problems.

• East Asian mathematical journal
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• v.16 no.1
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• pp.135-145
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• 2000

The purpose of this paper is to study the Chern-type problem of the complete space-like submanifolds of an indefinite complex space form.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.665-678
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• 1997

In this paper we prove that if the minimum of the sectional curvatures of a compact n-dimensional minimal generic submanifold M of a complex projective space is 1/n, then M is the geodesic minimal hypersphere.