• 한국수학사학회지
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• 제24권3호
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• pp.23-35
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• 2011

본 논문에서는 어떤 기하학적 양이 핀치되어 있으면 위상적 또는 미분위상적인 구면이 된다는 구면정리의 발전과 역사를 다루었다. 단면곡률의 핀칭과 관련하여, 고전적 핀칭 구면 정리에서 최근에 증명된 기념비적인 미분 핀칭 구면정리로 발전하는 과정의 역사를 기술하였다. 또 직경, 반경, 부피 등과 관련하여 계량불변량 구면정리와 미분 계량불변량 구면정리의 발전의 과정을 소개하였고, 구면정리와 관련된 미해결문제에 대한 역사를 기술하였다.

• 대한수학회지
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• 제33권4호
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• pp.1101-1114
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• 1996

In Riemannian geometry, it is one of the important things to study the topological or geometric invariants of manifolds since they characterize the topology of manifolds.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제8권1호
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• pp.33-46
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• 2001

The purpose of this paper is to study totally umbilical coisotropic sub-manifold(M. g, SM) of a semi-Riemannian manifold(M,g)

• 대한수학회보
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• 제50권3호
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• pp.705-717
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• 2013

In this paper, we study the geometry of half lightlike submanifolds M of a semi-Riemannian manifold $\tilde{M}$ with a semi-symmetric non-metric connection subject to the conditions; (1) the characteristic vector field of $\tilde{M}$ is tangent to M, the screen distribution on M is totally umbilical in M and the co-screen distribution on M is conformal Killing, or (2) the screen distribution is integrable and the local lightlike second fundamental form of M is parallel.

• Transactions on Control, Automation and Systems Engineering
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• 제4권4호
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• pp.330-340
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• 2002

In this Paper. a non-linear approach to a design of model reference adaptive control is presented. The approach is demonstrated by a case study of a simple single-pole and no zero, linear, discrete-time plant. The essence of the idea is to generate a full non-linear model of the plant dynamics and the parameter adaptation dynamics as a gradient descent algorithm with respect to a Riemannian metric. It is shown how a Riemannian metric can be chosen so that the modelled plant dynamics do in fact match the true plant dynamics. The performance of the proposed scheme is compared to a traditional model reference adaptive control scheme using the classical sensitivity derivatives (Euclidean gradients) for the descent algorithm.

• 대한수학회지
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• 제36권5호
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• pp.959-1008
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• 1999

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

• 대한수학회지
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• 제60권1호
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• pp.115-141
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• 2023

We study a class of left-invariant pseudo-Riemannian Sasaki metrics on solvable Lie groups, which can be characterized by the property that the zero level set of the moment map relative to the action of some one-parameter subgroup {exp tX} is a normal nilpotent subgroup commuting with {exp tX}, and X is not lightlike. We characterize this geometry in terms of the Sasaki reduction and its pseudo-Kähler quotient under the action generated by the Reeb vector field. We classify pseudo-Riemannian Sasaki solvmanifolds of this type in dimension 5 and those of dimension 7 whose Kähler reduction in the above sense is abelian.

• 대한수학회보
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• 제26권1호
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• pp.47-52
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• 1989

Methods of Riemannian geometry has played an important role in the study of compact transformation groups. Every effective action of a compact Lie group on a differential manifold leaves a Riemannian metric invariant and the study of such actions reduces to the one involving the group of isometries of a Riemannian metric on the manifold which is, a priori, a Lie group under the compact open topology. Once an action of a compact Lie group is given an invariant metric is easily constructed by the averaging method and the Lie group is naturally imbedded in the group of isometries as a Lie subgroup. But usually this invariant metric has more symmetries than those given by the original action. Therefore the first question one may ask is when one can find a Riemannian metric so that the given action coincides with the action of the full group of isometries. This seems to be a difficult question to answer which depends very much on the orbit structure and the group itself. In this paper we give a sufficient condition that a subgroup action of a compact Lie group has an invariant metric which is not invariant under the full action of the group and figure out some aspects of the action and the orbit structure regarding the invariant Riemannian metric. In fact, according to our results, this is possible if there is a larger transformation group, containing the oringnal action and either having larger orbit somewhere or having exactly the same orbit structure but with an orbit on which a Riemannian metric is ivariant under the orginal action of the group and not under that of the larger one. Recently R. Saerens and W. Zame showed that every compact Lie group can be realized as the full group of isometries of Riemannian metric. [SZ] This answers a question closely related to ours but the situation turns out to be quite different in the two problems.

• 대한수학회논문집
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• 제21권2호
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• pp.321-335
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• 2006

Generic submanifolds of a Riemannian manifold endowed with a hypercosymplectic 3-structure are studied. Integrability conditions for certain distributions on the generic submanifold are discussed. Geometry of leaves of certain distributions are also studied.

• 호남수학학술지
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• 제34권2호
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• pp.273-278
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• 2012

Using a comparison of the Jacobi differential equations instead of volume comparison by Itokawa in [3], the focal radius is estimated under the suitable Ricci and mean curvature conditions.