On Anholonomic Coordinates and Their Associated Congruence Surfaces

In this paper, we first establish an anholonomic coordinate system in the Euclidean 3-space by means of the q-frame associated with a regular space curve. The resulting derivative formulas and compatibility equations provide the fundamental framework for constructing three natural congruence surfaces associated with the anholonomic q-frame, namely the tangent, normal and binormal congruence surfaces. We then derive their existence conditions by means of the Frobenius Integrability Theorem. Furthermore, we obtain the corresponding Gauss–Mainardi–Codazzi equations and characterize both the intrinsic and extrinsic geometry of these congruence surfaces through their shape operators, principal curvatures, Gaussian curvatures, and mean curvatures.