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.

An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

Warped products play important roles in differential geometry, general relativity, and symmetry science. In this paper, we study the warped product pointwise semi-slant submanifolds that are isometrically immersed into conformal Sasakian space form. We show that there does not exist any proper warped product pointwise semi-slant submanifolds in conformal Sasakian manifolds. We derived some geometric inequalities for squared norm of second fundamental form from a warped product pointwise semi-slant submanifold into a conformal Sasakian manifolds.