WITHDRAWN: Geometric invariance for the conformal image of rectifying curve in Euclidean space R3

Abstract

A rectifying curve is a space curve that plays an important role in the field of differential geometry. This article is a follow-up of the work done in two recent papers [4] and [5], where along with various results on normal, rectifying, and osculating curves, several other properties of these curves are investigated. In this paper, we investigate some geometric invariance for the conformal image of a rectifying curves on regular surfaces under conformal transformation in the Euclidean space R3. The main objective of this paper is to discuss the invariant sufficient condition for the conformal image of a rectifying curve under the conformal, homothetic, and isometric transformations. The normal components of the rectifying curves are also computed, and it is demonstrated that these components remained invariant under the isometry of the surfaces in R3. We also investigated that, for a rectifying curve, the Christoffel symbols are invariant under isometry of surfaces. Mathematics Subject Classification (2010). Secondary 53A15, 53A05, 53A04, 51M05.