# Curves on a smooth surface with position vectors lie in the tangent plane

## TL;DR

This paper investigates special curves on smooth surfaces where the position vector lies in the tangent plane, demonstrating their invariance under surface isometries and analyzing related geometric properties.

## Contribution

It establishes the invariance of such curves and their geometric quantities under isometries, providing new insights into their properties on smooth surfaces.

## Key findings

- Curves with position vectors in the tangent plane are invariant under surface isometries.
- Length of the position vector remains unchanged under isometry.
- Geodesic curvature of these curves is invariant under isometry.

## Abstract

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length of the position vector, tangential component of the position vector and geodesic curvature of a curve on a surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces.