Conformal Interactions of Osculating Curves on Regular Surfaces in Euclidean 3-Space

Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and surfaces. In this paper, we study osculating curves on regular surfaces under conformal transformations. We obtained the conditions required for osculating curves on regular surfaces R and R˜ to remain invariant when subjected to a conformal transformation ψ:R→R˜. The results presented in this paper reveal the specific conditions under which the transformed curve σ˜=ψ∘σ preserves its osculating properties, depending on whether σ˜ is a geodesic, asymptotic, or neither. Furthermore, we analyze these conditions separately for cases with zero and non-zero normal curvatures. We also explore the behavior of these curves along the tangent vector Tσ and the unit normal vector Pσ.