• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1823-1834
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• 2018

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component $x^T$ of the position vector field is the most natural vector field tangent to the Euclidean submanifold M. We simply call the vector field $x^T$ the canonical vector field of the Euclidean submanifold M. In earlier articles [4,5,9,11,12], we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1525-1537
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• 2019

Let ${\tilde{M}}$ be a Riemannian manifold equipped with a concircular vector field ${\tilde{X}}$ and M a submanifold (with its induced metric) of ${\tilde{M}}$. Denote by X the restriction of ${\tilde{X}}$ on M and by $X^T$ the tangential component of X, called the canonical field of M. In this article we study submanifolds of ${\tilde{M}}$ whose canonical field $X^T$ is also concircular. Several characterizations and classification results in this respect are obtained.

• Journal of the Korean Mathematical Society
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• v.36 no.5
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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.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.393-400
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• 1994

Let M(f,η,ξ,g) be a (2m ＋ 1)-dimensional Sasakian manifold with soldering form dp ∈ ΓHom(Λ/sup q/TM, TM) (dp: canonical vector-valued 1-form) where f,η,ξ and g are the (1,1)-tensor field, the structure 1-form, the structure vector field and the metric tensor of M, respectively.(omitted)

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1077-1100
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• 2016

We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.

• 제어로봇시스템학회:학술대회논문집
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• 1988.10b
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• pp.939-942
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• 1988

In order to construct a nonlinear observer, change of coordinate system is necessary. However, as in the case of feedback linearizable system it is not easy to obtain a coordinate transformation map. ln this paper, a canonical structure is proposed for observable systems with an objective of finding a vector field which is necessary for the generation of a new coordinate system.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.7
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• pp.653-658
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• 2001

In this paper, we find the necessary and sufficient conditions for the discrete time nonlinear system to be transformed into observable canonical form by state coordinates change. Unlike the continuous time case, our theorems give the desired state coordinates change without solving partial differential equations. Also, our approach is applicable to both autonomous systems and control systems by slight change of the definition of the vector field.