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

• Journal of applied mathematics & informatics
• /
• 제20권1_2호
• /
• pp.575-584
• /
• 2006

In this paper we introduce a discrete tangent vector of a polygon defined on each vertex by a linear combination of forward difference and backward difference, and show that if the polygon is originated from a smooth curve then direction of the discrete tangent vector is a second order approximation of the direction of the tangent vector of the original curve. Using this discrete tangent vector, we also introduced the geometric cubic Hermite interpolation of a polygon with controlled initial and terminal speed of the curve segments proportional to the edge length. In this case the whole interpolation is $C^1$. Experiments suggest that about $90\%$ of the edge length is the best fit for the initial and terminal speeds.

• 대한수학회보
• /
• 제31권2호
• /
• pp.215-236
• /
• 1994

The purpose of the present paper is to study generic submanifolds of a Sasakian space form with nonvanishing parallel mean curvature vector field such that the shape operator in the direction of the mean curvature vector field commutes with the structure tensor field induced on the submanifold. In .cint. 1 we state general formulas on generic submanifolds of a Sasakian manifold, especially those of a Sasakian space form. .cint.2 is devoted to the study a generic submanifold of a Sasakian manifold, which is not tangent to the structure vector. In .cint.3 we investigate generic submanifolds, not tangent to the structure vector, of a Sasakian space form with nonvanishing parallel mean curvature vactor field. In .cint.4 we discuss generic submanifolds tangent to the structure vector of a Sasakian space form and compute the restricted Laplacian for the shape operator in the direction of the mean curvature vector field. As a applications of these, in the last .cint.5 we prove our main results.

• 한국수학교육학회지시리즈A:수학교육
• /
• 제53권3호
• /
• pp.313-327
• /
• 2014

'Tangent' is one of the most important concepts in the middle and high school mathematics, especially in dealing with calculus. The concept of tangent in the current textbook consists of the ways which make use of discriminant or differentiation. These ways, however, do not present dynamic view points, that is, the concept of variation. In this paper, after applying 'Roberval's way of finding tangent using vectors in terms of kinematics to parabola, ellipse, circle, hyperbola, cycloid, hypocycloid and epicycloid, we will identify that this is the tangent of those curves. This trial is the educational link of mathematics and physics, and it will also suggest the appropriate example of applying vector. We will also help students to understand the tangent by connecting this method to the existing ones.

• 디지털콘텐츠학회 논문지
• /
• 제16권4호
• /
• pp.575-581
• /
• 2015

본 논문에서는 3차원 벡터필드의 탄젠트 곡선을 계산하는 효율적이고 정확한 방법을 제안한다. 탄젠트 곡선 상의 정확한 값을 구하지 못하고 단지 탄젠트 곡선의 근사치를 구하는 Runge-Kutta 같은 기존의 방법과는 달리 여기서 제안한 방법은 3D 사면체 도메인에서 벡터필드가 선형적으로 변한다는 가정하에 탄젠트 곡선 상의 정확한 값을 계산한다. 새로 제안한 방법은 벡터필드가 3D 사면체 도메인에서 선형적으로 변한다고 가정한다. 우선 이 방법은 3차원 벡터필드에서 육면체 셀을 5 또는 6개의 사면체 셀로 분해하는 것을 요구한다. 임계점은 각 사면체의 간단한 선형 시스템을 풀어서 간단하게 구할 수 있다. 이 방법은 이전 사면체에서 계산된 탄젠트 곡선상의 점들을 기초로 다음 사면체에서 탄젠트 곡선상의 계속적인 점들을 생성함으로써 출구 점을 구한다. 탄젠트 곡선상의 점들은 각 사면체의 명시해에 의해서 계산되었기 때문에 새로운 방법은 3D 벡터필드를 가시화하는데 정확한 위상을 마련한다.

• 충청수학회지
• /
• 제34권1호
• /
• pp.37-53
• /
• 2021

In this paper we define scaled visual curvature and visual Frenet frame that can be visually accepted for discrete space curves. Scaled visual curvature is relatively simple compared to multi-scale visual curvature and easy to control the influence of noise. We adopt scaled minimizing directions of height functions on each neighborhood. Minimizing direction at a point of a curve is a direction that makes the point a local minimum. Minimizing direction can be given by a small noise around the point. To reduce this kind of influence of noise we exmine the direction whether it makes the point minimum in a neighborhood of some size. If this happens we call the direction scaled minimizing direction of C at p ∈ C in a neighborhood Br(p). Normal vector of a space curve is a second derivative of the curve but we characterize the normal vector of a curve by an integration of minimizing directions. Since integration is more robust to noise, we can find more robust definition of discrete normal vector, visual normal vector. On the other hand, the set of minimizing directions span the normal plane in the case of smooth curve. So we can find the tangent vector from minimizing directions. This lead to the definition of visual tangent vector which is orthogonal to the visual normal vector. By the cross product of visual tangent vector and visual normal vector, we can define visual binormal vector and form a Frenet frame. We examine these concepts to some discrete curve with noise and can see that the scaled visual curvature and visual Frenet frame approximate the original geometric invariants.

• 대한수학회논문집
• /
• 제35권3호
• /
• pp.1019-1035
• /
• 2020

In this paper, we introduce the deformed-Sasaki metric on the tangent bundle TM over an m-dimensional Riemannian manifold (M, g), as a new natural metric on TM. We establish a necessary and sufficient conditions under which a vector field is harmonic with respect to the deformed-Sasaki Metric. We also construct some examples of harmonic vector fields.

• 한국멀티미디어학회논문지
• /
• 제12권11호
• /
• pp.1623-1628
• /
• 2009

본 논문에서는 2차원 벡터 필드의 탄젠트 곡선을 계산하는 효율적이고 정확한 방법을 제안한다. 탄젠트 곡선 상의 정확한 값을 구하지 못하고 단지 탄젠트 곡선의 근사치를 구하는 Runge-Kutta 같은 종래의 방법과는 달리 여기서 제안한 방법은 2D 삼각형에서 벡터 필드가 선형적으로 변한다는 가정 하에 탄젠트 곡선상의 정확한 값을 계산한다. 새로 제안한 방법은 벡터 필드가 2D 삼각형에서 선형적으로 변한다고 가정한다. 우선 이 방법은 2D에서 사각형 셀을 2개의 삼각형 셀로 분해하는 것을 요구한다. 임계점은 각 삼각형의 간단한 선형 시스템을 풀어서 간단하게 구할 수 있다. 이 방법은 이전 삼각형에서 계산된 탄젠트 곡선상의 점들을 기초로 다음 삼각형에서 탄젠트 곡선상의 계속적인 점들을 생성함으로써 출구 점을 구한다. 탄젠트 곡선상의 점들은 각 삼각형의 명시해에 의해서 계산되었기 때문에 새로운 방법은 2D 벡터 필드를 가시화하는데 정확한 위상을 마련한다.

• 호남수학학술지
• /
• 제45권2호
• /
• pp.300-315
• /
• 2023

In this article, we deal with the biharmonicity of a vector field X viewed as a map from a pseudo-Riemannian manifold (M, g) into its tangent bundle TM endowed with the Sasaki metric g_S. Precisely, we characterize those vector fields which are biharmonic maps, and find the relationship between them and biharmonic vector fields. Afterwards, we study the biharmonicity of left-invariant vector fields on the three dimensional Heisenberg group endowed with a left-invariant Lorentzian metric. Finally, we give examples of vector fields which are proper biharmonic maps on the Gödel universe.

• 호남수학학술지
• /
• 제43권3호
• /
• pp.483-501
• /
• 2021

We study the complete lifts of projectable linear connection for semi-tangent bundle. The aim of this study is to establish relations between these and complete lift already known. In addition, the relations between infinitesimal linear transformations and projectable linear connections are studied. We also have a new example for good square in this work.