# Some characterizations of Rectifying and osculating curves on a smooth immersed surface

## TL;DR

This paper characterizes rectifying and osculating curves on smooth surfaces, examining their properties relative to a specific frame and their invariance under surface isometries.

## Contribution

It provides new characterizations of these curves based on their position vectors and invariance conditions under surface isometries.

## Key findings

- Position vectors' components are computed along the frame vectors.
- Invariance under isometry occurs iff the normal curvature is invariant or the position vector aligns with the tangent.
- The study links curve properties to surface invariance and curvature conditions.

## Abstract

The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface with respect to the reference frame $\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\}$. We have computed the components of position vectors of rectifying and osculating curves along $\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}$ and then investigated their invariancy under isometry of surfaces, and it is shown that they are invariant iff either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.