Conformal image of an osculating curve on a smooth immersed surface

Abstract

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.2010 Mathematics Subject Classification. 53C22, 53A04, 53A05.

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

“…The normal, rectifying, and osculating curves are the most often covered topics in differential geometry, therefore they are typically covered in every basic book on differential geometry of curves and surfaces. For more information on these topics, we can refer the reader to see [1,3,6]. In the Euclidean 3-Dimensional space R 3 , Chen et al [4,10] studied the motion of rectifying curves and investigated some of the basic properties of such curves.…”

“…In the Euclidean 3-Dimensional space R 3 , Chen et al [4,10] studied the motion of rectifying curves and investigated some of the basic properties of such curves. Shaikh et al [1,2,6,13] investigated the sufficient conditions for the invariance of the conformal image of osculating and normal curves on smooth immersed surfaces and found that there are various other properties of such curves that remain invariant under the isometry of surfaces. In the year 2003, Chen [4] came across the following query regarding rectifying curves: What occurs when a space curve's position vector is always within the range of its rectifying plane.…”

“…Further in [1], they investigated the invariant properties of osculating curves under the isometry of surfaces. But a natural question arises: What happens with the geometric properties of rectifying curves with respect to conformal transformation in Euclidean space R 3 .…”

On the position vector of surface curves in the Euclidean space

“…In 2018, Deshmukh et al [4] also characterized rectifying curves by centrodes of an unit speed curve in Euclidean space. The authors of [7,9,10,[12][13][14][15] have studied curves by restricting their position vectors to the rectifying, osculating and normal plane on a surface and obtained their characterization under isometry and conformal maps of surfaces.…”