Invariant Nature of Darboux normal Curves on a Smooth Surface Under Isometry

Abstract

In this article, we explore the relationships between the Darboux frame and the Frenet frame by investigating the invariant sufficient conditions for the isometric image of a Darboux normal curve when using the Darboux frame instead of the Frenet frame. Additionally, we analyze the deviation in the
component of the position vector of a Darboux normal curve on a smooth surface in relation to the provided isometry. The results of this study contribute to the improvement and generalization of certain earlier findings in the literature.