# Conformal image of an osculating curve on a smooth immersed surface

## TL;DR

This paper studies how osculating curves on smooth surfaces behave under conformal maps, providing conditions for their invariance and relating them to geodesic curves under conformal transformations.

## Contribution

It establishes a sufficient condition for conformal invariance of osculating curves and links them to geodesic curves under conformal motions.

## Key findings

- Conformal invariance of osculating curves is characterized by a specific condition.
- An equivalent system for geodesic curves under conformal transformations is derived.
- Invariance under isometry and homothetic motion is demonstrated.

## 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.