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Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen's work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point χ0 in R3. Secondly, some formulas of curvature and torsion for sphere curves are identified. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratio of the torsion to the curvature is a linear function when the curvature is constant. [ABSTRACT FROM AUTHOR]

Following the work of Dr. B.Y. Chen, I summarize some of the recent results on the study of Euclidean submanifolds via the tangent component of their position vector field. I also examine the related generalizations to Riemannian submanifolds. The main objects of study include torqued (and concircular) vector fields, rectifying submanifolds, and Ricci solutions. Chen determined Riemannian manifolds equipped with concircular vector fields have rectifying submanifolds precisely when the induced tangent component of the ambient concircular vector field is also a concircular vector field on the submanifold. Chen also classified Ricci solitons with concircular potential field. I will discuss the recent attempts to extend this work. [ABSTRACT FROM AUTHOR]

The involute of a space curve is a curve whose tangent vector is orthogonal to the tangent vector of the original curve. Though originally studied in Euclidean space, recently the properties of involutes of space curves in Minkowski three-space have been investigated. We study the properties of the involute of a rectifying curve. We also define the secondary involute of a space curve and determine some of its properties in Minkowski three-space. [ABSTRACT FROM AUTHOR]