A Study on the Behavior of Osculating and Rectifying Curves on Smooth Immersed Surfaces in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">E</mi><mn>3</mn></msup></semantics></math></inline-formula>

Abstract

This paper presents a detailed investigation into the isometric properties of osculating and rectifying curves on smooth immersed surfaces in E3. We examine the geometric interactions between these curves, specifically when the osculating curve is associated with one surface and the rectifying curve with another. The main objective of this study is to identify the conditions under which these curves exhibit isometric behavior, preserving their intrinsic geometric properties along their respective Frenet frames. Our findings demonstrate that these curves retain isometric characteristics along the tangent, normal, and binormal directions, offering new insights into their structural invariance. This research makes a significant contribution to the broader field of differential geometry, with potential applications in surface theory.

Keywords

• osculating curves
• rectifying curves
• smooth immersed surfaces
• differential geometry
• curves and surfaces