On Anholonomic Coordinates and Their Associated Congruence Surfaces

Abstract

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.

1. Introduction

Anholonomic coordinate systems constitute a natural framework for describing the differential geometry of moving frames whose associated vector fields are generally non-integrable. Unlike holonomic coordinates, the corresponding directional derivatives do not commute, and the resulting non-vanishing Lie brackets lead to a rich geometric structure characterized by anholonomy coefficients. Such coordinate systems arise naturally in differential geometry, continuum mechanics, nonlinear physics, and gravitational theories.

The mathematical foundations of anholonomic coordinates were systematically developed by Aris [1], who established the fundamental differential operators, compatibility conditions, and vector identities associated with orthonormal moving frames. In continuum mechanics, Clayton et al. [2] demonstrated that anholonomic configuration spaces provide a natural geometric description of finite elastoplasticity through non-coordinate bases and metric tensors. More recently, anholonomic frame techniques have also become fundamental tools in mathematical physics, particularly in generalized Einstein gravity, Finsler geometry, and nonholonomic geometric flows. Significant contributions in this direction were made by Vacaru [3,4,5], while very recent developments concerning anholonomic frame deformations and generalized geometric structures have been presented by Bubuianu et al. [6].

From the viewpoint of differential geometry of curves, anholonomic coordinate formulations have become an effective tool for investigating the intrinsic properties of orthonormal moving frames. In particular, Gürbüz [7] reformulated the Bishop frame within the framework of anholonomic coordinates and introduced the concept of three anholonomy densities in Euclidean three-space. This formulation was subsequently generalized to hybrid moving frames in Lorentzian space [8], and further extended to geometric phase theory and Hasimoto-type transformations in Lorentzian 3-Lie groups [9]. These developments demonstrate that anholonomic coordinates provide a unified framework for deriving differential invariants and investigating the geometric properties of various moving frames.

Another active direction concerns the interaction between anholonomic coordinates and surface theory. Erdoğdu and Yavuz [10] established an anholonomic formulation for spacelike curves by introducing normal congruence surfaces and deriving their corresponding Gauss–Mainardi–Codazzi equations. Their subsequent works further developed this framework for spacelike curve flows and pseudo-null geometries using anholonomic coordinates [11,12]. Directional energy functionals defined through anholonomic coordinates were recently investigated by Demirkol [13], providing another application of anholonomic formulations in differential geometry. These contributions demonstrate that anholonomic coordinates naturally connect moving frame theory with the local geometry of congruence surfaces.

Besides the recent developments on anholonomic coordinates, various surface constructions generated by moving frames have also attracted considerable attention. Harmonic evolute surfaces associated with quasi-binormal vector fields were investigated in [14], whereas Hasimoto surfaces generated by moving frame transformations were studied in [15]. In this context, Hamouda et al. [16] employed the quasi-frame to study quasi-Hasimoto surfaces associated with solutions of the vortex filament equation in Euclidean 3-space, deriving their fundamental geometric properties and characterizing their parameter curves. More recently, Solouma et al. [17] used the rotation-minimizing Darboux frame to construct RM Hasimoto surfaces in Minkowski 3-space and investigated their curvature properties and evolution equations. These studies highlight the significance of adapted moving frames in the construction and geometric characterization of associated surfaces. More recently, the geometry of non-degenerate Hasimoto-like surfaces associated with pseudo-null curve evolutions and the Darboux triad has also been investigated, providing further applications of moving-frame techniques in surface theory [18]. These studies indicate that moving-frame-based surface constructions provide an effective setting for investigating both intrinsic and extrinsic geometric properties of surfaces, thereby motivating the present anholonomic formulation of congruence surfaces associated with the q-frame.

Motivated by these developments, the present paper investigates anholonomic coordinates associated with the orthonormal q-frame in Euclidean three-space introduced by Dede et al. [19]. Unlike the classical Frenet frame, whose differential geometry is described in terms of curvature and torsion, the q-frame is characterized by three q-curvature functions. Its construction through a prescribed projection vector preserves the tangent direction of the curve while providing additional flexibility in the choice of the normal-plane directions. Moreover, since the q-frame is related to the Frenet frame by a rotation in the normal plane, it retains the underlying Frenet geometry while incorporating the effect of the prescribed projection direction. This additional directional freedom makes the q-frame particularly suitable for the present anholonomic setting, in which the variations of the frame along three distinct directions play a fundamental role.

Within this framework, we first derive the complete directional derivative formulas of the q-frame along the three anholonomic directions by introducing the normal deformation of the tangent vector field and expressing the remaining coefficients in terms of divergence and curl invariants. These formulas yield the corresponding anholonomic mixed-derivative relations together with the compatibility equations governing the q-frame.

2. Construction of the q-Frame in Anholonomic Coordinates

Let $\alpha =\alpha \left(s\right)$ be a unit-speed regular curve in the Euclidean three-space ${\mathbb{E}}^3$, and let $\{\mathbf{T},\mathbf{N},\mathbf{B}\}$ denote its Frenet frame. Thus,

$$\mathbf{T}={\alpha }^{\prime }\left(s\right),\mathbf{N}={\displaystyle \frac{{\mathbf{T}}^{\prime }}{\parallel {\mathbf{T}}^{\prime }\parallel }},\mathbf{B}=\mathbf{T}\times \mathbf{N}.$$

The corresponding Frenet equations are

$$\left(\begin{array}{c}{\mathbf{T}}^{\prime }\\ {\mathbf{N}}^{\prime }\\ {\mathbf{B}}^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& \kappa & 0\\ -\kappa & 0& \tau \\ 0& -\tau & 0\end{array}\right)\left(\begin{array}{c}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{array}\right),$$

where $\kappa$ and $\tau$ denote the curvature and torsion of $\alpha$, respectively.

Let $\mathbf{a}$ be a fixed projection vector satisfying $\mathbf{T}\times \mathbf{a}\ne \mathbf{0}$. The q-frame $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ along $\alpha$ is defined by

$${\mathbf{t}}_q=\mathbf{T},{\mathbf{n}}_q={\displaystyle \frac{\mathbf{T}\times \mathbf{a}}{\parallel \mathbf{T}\times \mathbf{a}\parallel }},{\mathbf{b}}_q={\mathbf{t}}_q\times {\mathbf{n}}_q.$$

Let $\varphi =\varphi \left(s\right)$ denote the oriented angle between the principal normal $\mathbf{N}$ and the quasi-normal vector ${\mathbf{n}}_q$. Then the q-frame and the Frenet frame are related by

$$\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& cos\varphi & sin\varphi \\ 0& -sin\varphi & cos\varphi \end{array}\right)\left(\begin{array}{c}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{array}\right).$$

Differentiating the above relation with respect to s and using the Frenet equations yields

$${\displaystyle \frac{\partial }{\partial s}}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& k_1& k_2\\ -k_1& 0& k_3\\ -k_2& -k_3& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right),$$

(1)

where the q-curvature functions are given by [19]

$$k_1=\kappa cos\varphi ,k_2=-\kappa sin\varphi ,k_3=\tau +{\displaystyle \frac{\partial \varphi }{\partial s}}.$$

We now extend the q-frame from the curve to a local anholonomic orthonormal frame field. Let $U\subset {\mathbb{E}}^3$ be an open neighborhood of $\alpha \left(I\right)$, and assume that the q-frame $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ along $\alpha$ is smoothly extended to an orthonormal frame field on U, denoted by the same symbols. For any smooth scalar or vector field F on U, we define the directional derivatives

$$F_s:=D_{{\mathbf{t}}_q}F,F_n:=D_{{\mathbf{n}}_q}F,F_b:=D_{{\mathbf{b}}_q}F,$$

where D denotes the Euclidean connection. Thus, the subscripts $s,n,b$ refer to differentiation along the frame directions ${\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q$, respectively, and do not in general represent commuting coordinate derivatives.

Since the extended frame is orthonormal,

$$⟨{\mathbf{t}}_q,{\mathbf{t}}_q⟩=⟨{\mathbf{n}}_q,{\mathbf{n}}_q⟩=⟨{\mathbf{b}}_q,{\mathbf{b}}_q⟩=1,$$

and

$$⟨{\mathbf{t}}_q,{\mathbf{n}}_q⟩=⟨{\mathbf{t}}_q,{\mathbf{b}}_q⟩=⟨{\mathbf{n}}_q,{\mathbf{b}}_q⟩=0.$$

We adopt the positive orientation

$${\mathbf{t}}_q\times {\mathbf{n}}_q={\mathbf{b}}_q,{\mathbf{n}}_q\times {\mathbf{b}}_q={\mathbf{t}}_q,{\mathbf{b}}_q\times {\mathbf{t}}_q={\mathbf{n}}_q.$$

On U, we write the derivatives of the extended frame along the ${\mathbf{t}}_q$-direction as

$$D_{{\mathbf{t}}_q}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& k_1& k_2\\ -k_1& 0& k_3\\ -k_2& -k_3& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right),$$

where $k_1,k_2,k_3$ are smooth functions on U whose restrictions to $\alpha$ coincide with the q-curvature functions in (1). The derivatives of the q-frame along the remaining frame directions ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$ are determined by the anholonomic structure and are given in the following theorem.

Theorem 1. 

The directional derivatives of the q-frame along the ${\mathbf{n}}_q$- and ${\mathbf{b}}_q$-directions are given by

$$D_{{\mathbf{n}}_q}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& {\epsilon }_n& ⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\\ -{\epsilon }_n& 0& -div{\mathbf{b}}_q-k_2\\ -\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)& div{\mathbf{b}}_q+k_2& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right),$$

(2)

and

$$D_{{\mathbf{b}}_q}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& -\left(⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3\right)& div{\mathbf{t}}_q-{\epsilon }_n\\ ⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3& 0& div{\mathbf{n}}_q+k_1\\ {\epsilon }_n-div{\mathbf{t}}_q& -\left(div{\mathbf{n}}_q+k_1\right)& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right).$$

(3)

Here,

$${\epsilon }_n=⟨D_{{\mathbf{n}}_q}{\mathbf{t}}_q,{\mathbf{n}}_q⟩$$

denotes the normal deformation of the tangent vector field in the ${\mathbf{n}}_q$-direction.

Proof. 

Since the q-frame is orthonormal, the matrices representing its directional derivatives are skew-symmetric. Hence, there exist smooth functions ${\alpha }_i$ and ${\beta }_i$, $i=1,2,3$, such that

$$D_{{\mathbf{n}}_q}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& {\alpha }_1& {\alpha }_2\\ -{\alpha }_1& 0& {\alpha }_3\\ -{\alpha }_2& -{\alpha }_3& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right),$$

(4)

and

$$D_{{\mathbf{b}}_q}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& {\beta }_1& {\beta }_2\\ -{\beta }_1& 0& {\beta }_3\\ -{\beta }_2& -{\beta }_3& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right).$$

(5)

By definition,

$${\alpha }_1=⟨D_{{\mathbf{n}}_q}{\mathbf{t}}_q,{\mathbf{n}}_q⟩={\epsilon }_n.$$

We first compute the divergences of the frame vector fields. Since $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ is an orthonormal frame on U, we have

$$div{\mathbf{t}}_q=⟨{\mathbf{t}}_q,D_{{\mathbf{t}}_q}{\mathbf{t}}_q⟩+⟨{\mathbf{n}}_q,D_{{\mathbf{n}}_q}{\mathbf{t}}_q⟩+⟨{\mathbf{b}}_q,D_{{\mathbf{b}}_q}{\mathbf{t}}_q⟩.$$

Using the directional derivative formulas, we obtain

$$\begin{array}{cc}\hfill div{\mathbf{t}}_q& =⟨{\mathbf{t}}_q,k_1{\mathbf{n}}_q+k_2{\mathbf{b}}_q⟩+⟨{\mathbf{n}}_q,{\alpha }_1{\mathbf{n}}_q+{\alpha }_2{\mathbf{b}}_q⟩+⟨{\mathbf{b}}_q,{\beta }_1{\mathbf{n}}_q+{\beta }_2{\mathbf{b}}_q⟩\hfill \\ \hfill & ={\alpha }_1+{\beta }_2.\hfill \end{array}$$

Therefore,

$${\beta }_2=div{\mathbf{t}}_q-{\epsilon }_n.$$

(6)

Similarly, applying the same argument to ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$ gives

$${\beta }_3=div{\mathbf{n}}_q+k_1,$$

(7)

and

$${\alpha }_3=-div{\mathbf{b}}_q-k_2.$$

(8)

We next calculate the curls of the frame vector fields. With respect to the orthonormal frame $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$, the curl of a vector field X can be written as

$$curlX={\mathbf{t}}_q\times D_{{\mathbf{t}}_q}X+{\mathbf{n}}_q\times D_{{\mathbf{n}}_q}X+{\mathbf{b}}_q\times D_{{\mathbf{b}}_q}X.$$

Hence,

$$\begin{array}{cc}\hfill curl{\mathbf{t}}_q& ={\mathbf{t}}_q\times (k_1{\mathbf{n}}_q+k_2{\mathbf{b}}_q)+{\mathbf{n}}_q\times ({\alpha }_1{\mathbf{n}}_q+{\alpha }_2{\mathbf{b}}_q)+{\mathbf{b}}_q\times ({\beta }_1{\mathbf{n}}_q+{\beta }_2{\mathbf{b}}_q)\hfill \\ \hfill & =({\alpha }_2-{\beta }_1){\mathbf{t}}_q-k_2{\mathbf{n}}_q+k_1{\mathbf{b}}_q.\hfill \end{array}$$

Therefore,

$$⟨curl{\mathbf{t}}_q,{\mathbf{t}}_q⟩={\alpha }_2-{\beta }_1.$$

(9)

Similarly, applying the same argument to ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$ gives

$${\beta }_1=-\left(⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3\right),$$

(10)

and

$${\alpha }_2=⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3.$$

(11)

Substituting (11) and (8) into (4) yields (2). Similarly, substituting (10), (6), and (7) into (5) yields (3). □

Corollary 1. 

The third q-curvature can be expressed in terms of the abnormalities of the three q-frame vector fields.

Proof. 

Combining (9), (10), and (11), we obtain

$$⟨curl{\mathbf{t}}_q,{\mathbf{t}}_q⟩-⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩-⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩=2k_3.$$

□

Corollary 2. 

Suppose that the orthogonal distributions

$$span\{{\mathbf{n}}_q,{\mathbf{b}}_q\},span\{{\mathbf{t}}_q,{\mathbf{b}}_q\},span\{{\mathbf{t}}_q,{\mathbf{n}}_q\}$$

associated with the orthonormal q-frame $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ are all integrable. Then, by the Frobenius integrability condition,

$$⟨curl{\mathbf{t}}_q,{\mathbf{t}}_q⟩=⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩=⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩=0.$$

Consequently, Corollary 1 yields

$$k_3=0.$$

Moreover, the corresponding surface families, denoted by $X_T$, $X_N$, and $X_B$, with unit normal vector fields ${\mathbf{t}}_q$, ${\mathbf{n}}_q$, and ${\mathbf{b}}_q$, respectively, form a triply orthogonal system. By Dupin’s theorem, the intersection curve $X_N\cap X_B$, whose tangent direction is ${\mathbf{t}}_q$, is a line of curvature of both $X_N$ and $X_B$. Similarly, the intersection curves $X_T\cap X_B$ and $X_T\cap X_N$, whose tangent directions are ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$, respectively, are lines of curvature of the corresponding surfaces. Hence, the pairwise intersection curves are common lines of curvature of the associated surface families.

2.1. Anholonomic Mixed-Derivative Relations

Let $f\in C^{\infty }\left(U\right)$ be a smooth scalar function. With respect to the orthonormal q-frame $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$, its gradient is written as

$$\nabla f={\mathbf{t}}_q{f}_s+{\mathbf{n}}_q{f}_n+{\mathbf{b}}_q{f}_b,$$

(12)

where

$$f_s:=D_{{\mathbf{t}}_q}f,f_n:=D_{{\mathbf{n}}_q}f,f_b:=D_{{\mathbf{b}}_q}f.$$

Here, the subscripts $s,n,b$ denote directional differentiation along ${\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q$, respectively. Using the vector identity

$$curl\left(\phi X\right)=\nabla \phi \times X+\phi curlX,$$

we obtain

$$\begin{array}{cc}\hfill curl(\nabla f)=& \nabla \left(f_s\right)\times {\mathbf{t}}_q+f_{s}curl{\mathbf{t}}_q+\nabla \left(f_n\right)\times {\mathbf{n}}_q+f_{n}curl{\mathbf{n}}_q\hfill \\ \hfill & +\hspace{0.17em}\nabla \left(f_b\right)\times {\mathbf{b}}_q+f_{b}curl{\mathbf{b}}_q.\hfill \end{array}$$

(13)

For the mixed directional derivatives, we adopt the convention

$$f_{ij}:=D_{{\mathbf{e}}_j}\left(D_{{\mathbf{e}}_i}f\right),{\mathbf{e}}_s={\mathbf{t}}_q,{\mathbf{e}}_n={\mathbf{n}}_q,{\mathbf{e}}_b={\mathbf{b}}_q.$$

Thus, for example,

$$f_{sn}=D_{{\mathbf{n}}_q}\left(D_{{\mathbf{t}}_q}f\right),f_{ns}=D_{{\mathbf{t}}_q}\left(D_{{\mathbf{n}}_q}f\right),$$

which need not coincide since the frame is generally anholonomic. The gradients of the directional derivatives of f are

$$\begin{array}{cc}\hfill \nabla \left(f_s\right)& ={\mathbf{t}}_q{f}_{ss}+{\mathbf{n}}_q{f}_{sn}+{\mathbf{b}}_q{f}_{sb},\hfill \\ \hfill \nabla \left(f_n\right)& ={\mathbf{t}}_q{f}_{ns}+{\mathbf{n}}_q{f}_{nn}+{\mathbf{b}}_q{f}_{nb},\hfill \\ \hfill \nabla \left(f_b\right)& ={\mathbf{t}}_q{f}_{bs}+{\mathbf{n}}_q{f}_{bn}+{\mathbf{b}}_q{f}_{bb}.\hfill \end{array}$$

Using

$${\mathbf{t}}_q\times {\mathbf{n}}_q={\mathbf{b}}_q,{\mathbf{n}}_q\times {\mathbf{b}}_q={\mathbf{t}}_q,{\mathbf{b}}_q\times {\mathbf{t}}_q={\mathbf{n}}_q,$$

we obtain

$$\begin{array}{cc}\hfill \nabla \left(f_s\right)\times {\mathbf{t}}_q& =-f_{sn}{\mathbf{b}}_q+f_{sb}{\mathbf{n}}_q,\hfill \\ \hfill \nabla \left(f_n\right)\times {\mathbf{n}}_q& =f_{ns}{\mathbf{b}}_q-f_{nb}{\mathbf{t}}_q,\hfill \\ \hfill \nabla \left(f_b\right)\times {\mathbf{b}}_q& =-f_{bs}{\mathbf{n}}_q+f_{bn}{\mathbf{t}}_q.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill curl(\nabla f)=& (f_{bn}-f_{nb}){\mathbf{t}}_q+(f_{sb}-f_{bs}){\mathbf{n}}_q+(f_{ns}-f_{sn}){\mathbf{b}}_q\hfill \\ \hfill & +\hspace{0.17em}{f}_{s}curl{\mathbf{t}}_q+f_{n}curl{\mathbf{n}}_q+f_{b}curl{\mathbf{b}}_q.\hfill \end{array}$$

(14)

From the proof of Theorem 1, the curls of the q-frame vector fields are

$$\begin{array}{cc}\hfill curl{\mathbf{t}}_q& =({\alpha }_2-{\beta }_1){\mathbf{t}}_q-k_2{\mathbf{n}}_q+k_1{\mathbf{b}}_q,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill curl{\mathbf{n}}_q& ={\alpha }_3{\mathbf{t}}_q-(k_3+{\beta }_1){\mathbf{n}}_q+{\alpha }_1{\mathbf{b}}_q,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill curl{\mathbf{b}}_q& ={\beta }_3{\mathbf{t}}_q-{\beta }_2{\mathbf{n}}_q+({\alpha }_2-k_3){\mathbf{b}}_q.\hfill \end{array}$$

(17)

Substitution of (15)–(17) into (14) yields

$$\begin{array}{cc}\hfill curl(\nabla f)=& \left[(f_{bn}-f_{nb})+f_s({\alpha }_2-{\beta }_1)+f_n{\alpha }_3+f_b{\beta }_3\right]{\mathbf{t}}_q\hfill \\ \hfill & +\left[(f_{sb}-f_{bs})-f_s{k}_2-f_n(k_3+{\beta }_1)-f_b{\beta }_2\right]{\mathbf{n}}_q\hfill \\ \hfill & +\left[(f_{ns}-f_{sn})+f_s{k}_1+f_n{\alpha }_1+f_b({\alpha }_2-k_3)\right]{\mathbf{b}}_q.\hfill \end{array}$$

Since $curl(\nabla f)=0$ and $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ is an orthonormal frame, each of the above coefficients must vanish. Therefore, the mixed directional derivatives satisfy the following relations.

Theorem 2. 

For every smooth scalar function $f\in C^{\infty }\left(U\right)$, one has

$$\begin{array}{cc}\hfill f_{bn}-f_{nb}& =-f_s({\alpha }_2-{\beta }_1)-f_n{\alpha }_3-f_b{\beta }_3,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill f_{bs}-f_{sb}& =-f_s{k}_2+f_n(k_3+{\beta }_1)+f_b{\beta }_2,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill f_{ns}-f_{sn}& =-f_s{k}_1-f_n{\alpha }_1-f_b({\alpha }_2-k_3).\hfill \end{array}$$

(20)

Using

$${\alpha }_1={\epsilon }_n,{\alpha }_2=⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3,{\alpha }_3=-div{\mathbf{b}}_q-k_2,$$

and

$${\beta }_1=-\left(⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3\right),{\beta }_2=div{\mathbf{t}}_q-{\epsilon }_n,{\beta }_3=div{\mathbf{n}}_q+k_1,$$

relations (18)–(20) can equivalently be written as

$$\begin{array}{cc}\hfill f_{bn}-f_{nb}=& -f_s\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+2k_3\right)+f_n(div{\mathbf{b}}_q+k_2)-f_b(div{\mathbf{n}}_q+k_1),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill f_{bs}-f_{sb}=& -f_s{k}_2-f_n⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+f_b(div{\mathbf{t}}_q-{\epsilon }_n),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill f_{ns}-f_{sn}=& -f_s{k}_1-f_n{\epsilon }_n-f_b⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩.\hfill \end{array}$$

(23)

Remark 1. 

Equations (18)–(20) show that, in general, the second-order mixed directional derivatives do not commute. Equivalently, the frame vector fields ${\mathbf{t}}_q$, ${\mathbf{n}}_q$, and ${\mathbf{b}}_q$ need not commute, and hence the extended q-frame is generally anholonomic.

2.2. Compatibility Equations of the Anholonomic q-Frame

For convenience, introduce the abbreviations

$$\begin{array}{cccc}\hfill A& =⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3,\hfill & \hfill C& =⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3,\hfill \\ \hfill D& =div{\mathbf{t}}_q-{\epsilon }_n,\hfill & \hfill P& =div{\mathbf{b}}_q+k_2,\hfill & \hfill Q& =div{\mathbf{n}}_q+k_1.\hfill \end{array}$$

(24)

Then the directional derivative systems obtained previously take the form

$$\begin{array}{cccccc}\hfill D_{{\mathbf{t}}_q}{\mathbf{t}}_q& =k_1{\mathbf{n}}_q+k_2{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{t}}_q}{\mathbf{n}}_q& =-k_1{\mathbf{t}}_q+k_3{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{t}}_q}{\mathbf{b}}_q& =-k_2{\mathbf{t}}_q-k_3{\mathbf{n}}_q,\hfill \end{array}$$

(25)

$$\begin{array}{cccccc}\hfill D_{{\mathbf{n}}_q}{\mathbf{t}}_q& ={\epsilon }_n{\mathbf{n}}_q+A{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{n}}_q}{\mathbf{n}}_q& =-{\epsilon }_n{\mathbf{t}}_q-P{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{n}}_q}{\mathbf{b}}_q& =-A{\mathbf{t}}_q+P{\mathbf{n}}_q,\hfill \end{array}$$

(26)

$$\begin{array}{cccccc}\hfill D_{{\mathbf{b}}_q}{\mathbf{t}}_q& =-C{\mathbf{n}}_q+D{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{b}}_q}{\mathbf{n}}_q& =C{\mathbf{t}}_q+Q{\mathbf{b}}_q,\hfill & \hfill D_{{\mathbf{b}}_q}{\mathbf{b}}_q& =-D{\mathbf{t}}_q-Q{\mathbf{n}}_q.\hfill \end{array}$$

(27)

For a smooth scalar function $f\in C^{\infty }\left(U\right)$, the mixed directional derivatives satisfy

$$\begin{array}{cc}\hfill f_{bn}-f_{nb}& =-(A+C)f_s+P{f}_n-Q{f}_b,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill f_{bs}-f_{sb}& =-k_2{f}_s-C{f}_n+D{f}_b,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill f_{ns}-f_{sn}& =-k_1{f}_s-{\epsilon }_n{f}_n-(A-k_3)f_b.\hfill \end{array}$$

(30)

The same identities are applied componentwise to the vector fields ${\mathbf{t}}_q$, ${\mathbf{n}}_q$, and ${\mathbf{b}}_q$.

Theorem 3. 

The compatibility of the three directional derivative systems (25)–(27) is equivalent to

$$\begin{array}{cc}\hfill {\left({\epsilon }_n\right)}_b+C_n& =(A-C)Q+(D-{\epsilon }_n)P+k_1(A+C),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill A_b-D_n& =(C-A)P+(D-{\epsilon }_n)Q+k_2(A+C),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill P_b+Q_n& =-AC-D{\epsilon }_n-P^2-Q^2-k_3(A+C),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\left(k_1\right)}_b+C_s& =-CD-D{k}_3+Q{k}_2-{\epsilon }_n(C-k_3)+k_1{k}_2,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\left(k_2\right)}_b-D_s& =-A(C-k_3)-C{k}_3+D^2-Q{k}_1+k_2^2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\left(k_3\right)}_b-Q_s& =C{k}_2+DQ+D{k}_1+P(C-k_3)+k_2{k}_3,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\left(k_1\right)}_n-{\left({\epsilon }_n\right)}_s& =-A{k}_3-C(A-k_3)-P{k}_2+{\epsilon }_n^2+k_1^2,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\left(k_2\right)}_n-A_s& =A{\epsilon }_n+D(A-k_3)+P{k}_1+{\epsilon }_n{k}_3+k_1{k}_2,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill P_s+{\left(k_3\right)}_n& =A{k}_1-P{\epsilon }_n+Q(A-k_3)-{\epsilon }_n{k}_2+k_1{k}_3.\hfill \end{array}$$

(39)

Proof. 

For each pair of directions, the compatibility condition decomposes into three scalar equations. Two are obtained from the mixed directional derivatives of ${\mathbf{t}}_q$, whereas the third may be derived from either ${\mathbf{n}}_q$ or ${\mathbf{b}}_q$, since the resulting relations are equivalent. Accordingly, it is sufficient to use ${\mathbf{n}}_q$ for the remaining equation. We consider the three pairs of anholonomic directions separately.

For the $(b,n)$ pair, applying (28) componentwise to $f={\mathbf{t}}_q$ gives

$$D_{{\mathbf{n}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q=-(A+C)D_{{\mathbf{t}}_q}{\mathbf{t}}_q+P{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q-Q{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q.$$

(40)

Using (25)–(27), the right-hand side becomes

$$\begin{array}{cc}\hfill & -(A+C)D_{{\mathbf{t}}_q}{\mathbf{t}}_q+P{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q-Q{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q\hfill \\ \hfill & =\left[-k_1(A+C)+P{\epsilon }_n+CQ\right]{\mathbf{n}}_q+\left[-k_2(A+C)+PA-DQ\right]{\mathbf{b}}_q.\hfill \end{array}$$

(41)

On the other hand,

$$\begin{array}{cc}\hfill D_{{\mathbf{n}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q& =D_{{\mathbf{n}}_q}(-C{\mathbf{n}}_q+D{\mathbf{b}}_q)\hfill \\ \hfill & =-C_n{\mathbf{n}}_q-C{D}_{{\mathbf{n}}_q}{\mathbf{n}}_q+D_n{\mathbf{b}}_q+D{D}_{{\mathbf{n}}_q}{\mathbf{b}}_q\hfill \\ \hfill & =(C{\epsilon }_n-AD){\mathbf{t}}_q+\left[-C_n+DP\right]{\mathbf{n}}_q+\left[CP+D_n\right]{\mathbf{b}}_q,\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill D_{{\mathbf{b}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q& =D_{{\mathbf{b}}_q}({\epsilon }_n{\mathbf{n}}_q+A{\mathbf{b}}_q)\hfill \\ \hfill & ={\left({\epsilon }_n\right)}_b{\mathbf{n}}_q+{\epsilon }_n{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q+A_b{\mathbf{b}}_q+A{D}_{{\mathbf{b}}_q}{\mathbf{b}}_q\hfill \\ \hfill & =(C{\epsilon }_n-AD){\mathbf{t}}_q+\left[{\left({\epsilon }_n\right)}_b-AQ\right]{\mathbf{n}}_q+\left[A_b+{\epsilon }_{n}Q\right]{\mathbf{b}}_q.\hfill \end{array}$$

(43)

Therefore,

$$\begin{array}{cc}\hfill D_{{\mathbf{n}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q=& \left[-C_n+DP-{\left({\epsilon }_n\right)}_b+AQ\right]{\mathbf{n}}_q\hfill \\ \hfill & +\left[CP+D_n-A_b-{\epsilon }_{n}Q\right]{\mathbf{b}}_q.\hfill \end{array}$$

(44)

Comparing the ${\mathbf{n}}_q$- and ${\mathbf{b}}_q$-components in (41) and (44) yields (31) and (32), respectively. The ${\mathbf{t}}_q$-component vanishes identically. To obtain the third relation, we apply (28) componentwise to $f={\mathbf{n}}_q$:

$$D_{{\mathbf{n}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{n}}_q=-(A+C)D_{{\mathbf{t}}_q}{\mathbf{n}}_q+P{D}_{{\mathbf{n}}_q}{\mathbf{n}}_q-Q{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q.$$

(45)

Direct differentiation gives

$$\begin{array}{cc}\hfill D_{{\mathbf{n}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q& =D_{{\mathbf{n}}_q}(C{\mathbf{t}}_q+Q{\mathbf{b}}_q)\hfill \\ \hfill & =\left[C_n-AQ\right]{\mathbf{t}}_q+\left[C{\epsilon }_n+PQ\right]{\mathbf{n}}_q+\left[AC+Q_n\right]{\mathbf{b}}_q,\hfill \end{array}$$

(46)

and

$$\begin{array}{cc}\hfill D_{{\mathbf{b}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{n}}_q& =D_{{\mathbf{b}}_q}(-{\epsilon }_n{\mathbf{t}}_q-P{\mathbf{b}}_q)\hfill \\ \hfill & =\left[-{\left({\epsilon }_n\right)}_b+PD\right]{\mathbf{t}}_q+\left[C{\epsilon }_n+PQ\right]{\mathbf{n}}_q+\left[-D{\epsilon }_n-P_b\right]{\mathbf{b}}_q.\hfill \end{array}$$

(47)

The ${\mathbf{n}}_q$-components cancel. Hence,

$$P_b+Q_n=-AC-D{\epsilon }_n-P^2-Q^2-k_3(A+C),$$

which is precisely (33). For the $(b,s)$ pair, applying (29) componentwise to $f={\mathbf{t}}_q$ gives

$$D_{{\mathbf{t}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q=-k_2{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q+(C-k_3)D_{{\mathbf{n}}_q}{\mathbf{t}}_q-D{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q.$$

(48)

Using (25)–(27), the right-hand side becomes

$$\begin{array}{cc}\hfill & -k_2{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q+(C-k_3)D_{{\mathbf{n}}_q}{\mathbf{t}}_q-D{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q\hfill \\ \hfill & =\left[-k_1{k}_2+{\epsilon }_n(C-k_3)+CD\right]{\mathbf{n}}_q+\left[-k_2^2+A(C-k_3)-D^2\right]{\mathbf{b}}_q.\hfill \end{array}$$

(49)

On the other hand,

$$\begin{array}{cc}\hfill D_{{\mathbf{t}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q& =D_{{\mathbf{t}}_q}(-C{\mathbf{n}}_q+D{\mathbf{b}}_q)\hfill \\ \hfill & =(-C{k}_1-D{k}_2){\mathbf{t}}_q+\left[-C_s-D{k}_3\right]{\mathbf{n}}_q+\left[-C{k}_3+D_s\right]{\mathbf{b}}_q,\hfill \end{array}$$

(50)

whereas

$$\begin{array}{cc}\hfill D_{{\mathbf{b}}_q}{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q& =D_{{\mathbf{b}}_q}(k_1{\mathbf{n}}_q+k_2{\mathbf{b}}_q)\hfill \\ \hfill & =(-C{k}_1-D{k}_2){\mathbf{t}}_q+\left[{\left(k_1\right)}_b-Q{k}_2\right]{\mathbf{n}}_q+\left[Q{k}_1+{\left(k_2\right)}_b\right]{\mathbf{b}}_q.\hfill \end{array}$$

(51)

Therefore,

$$\begin{array}{cc}\hfill D_{{\mathbf{t}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{t}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q=& \left[-C_s-D{k}_3-{\left(k_1\right)}_b+Q{k}_2\right]{\mathbf{n}}_q\hfill \\ \hfill & +\left[-C{k}_3+D_s-Q{k}_1-{\left(k_2\right)}_b\right]{\mathbf{b}}_q.\hfill \end{array}$$

(52)

Comparing the ${\mathbf{n}}_q$- and ${\mathbf{b}}_q$-components of (49) and (52) gives (34) and (35), respectively. To obtain the remaining relation, we apply (29) componentwise to $f={\mathbf{n}}_q$. We have

$$D_{{\mathbf{t}}_q}{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q-D_{{\mathbf{b}}_q}{D}_{{\mathbf{t}}_q}{\mathbf{n}}_q=-k_2{D}_{{\mathbf{t}}_q}{\mathbf{n}}_q+(C-k_3)D_{{\mathbf{n}}_q}{\mathbf{n}}_q-D{D}_{{\mathbf{b}}_q}{\mathbf{n}}_q.$$

(53)

The ${\mathbf{b}}_q$-component of the left-hand side is

$$C{k}_2+Q_s-{\left(k_3\right)}_b+D{k}_1,$$

whereas the ${\mathbf{b}}_q$-component of the right-hand side is

$$-k_2{k}_3-P(C-k_3)-DQ.$$

Hence,

$$C{k}_2+Q_s-{\left(k_3\right)}_b+D{k}_1=-k_2{k}_3-P(C-k_3)-DQ,$$

which gives

$${\left(k_3\right)}_b-Q_s=C{k}_2+DQ+D{k}_1+P(C-k_3)+k_2{k}_3.$$

Thus, (36) follows. Finally, for the $(n,s)$ pair, applying (30) componentwise to $f={\mathbf{t}}_q$ gives

$$D_{{\mathbf{t}}_q}{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q-D_{{\mathbf{n}}_q}{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q=-k_1{D}_{{\mathbf{t}}_q}{\mathbf{t}}_q-{\epsilon }_n{D}_{{\mathbf{n}}_q}{\mathbf{t}}_q-(A-k_3)D_{{\mathbf{b}}_q}{\mathbf{t}}_q.$$

(54)

Substituting the directional derivative systems (25)–(27) and comparing the ${\mathbf{n}}_q$- and ${\mathbf{b}}_q$-components yields (37) and (38), respectively. Applying the same mixed-directional derivative relation componentwise to $f={\mathbf{b}}_q$ and comparing the remaining independent component gives (39). This completes the proof. □

3. Congruence Surfaces Associated with the Anholonomic $\mathit{q}$-Frame

In this section, we investigate three natural congruence surfaces associated with the extended anholonomic q-frame. For each case, the Frobenius theorem provides the necessary and sufficient condition for the integrability of the corresponding orthogonal distribution and, hence, for the local existence of its integral surfaces. Once the integrability condition is satisfied, the compatibility equations obtained in Theorem 3 reduce to the corresponding Gauss–Mainardi–Codazzi equations, which characterize the intrinsic and extrinsic geometry of the resulting congruence surfaces.

Definition 1. 

Let $\{{\mathbf{t}}_q,{\mathbf{n}}_q,{\mathbf{b}}_q\}$ be the extended orthonormal anholonomic q-frame defined on an open set $U\subset {\mathbb{E}}^3$.

(i) 

An integral surface of the distribution

$$span\{{\mathbf{n}}_q,{\mathbf{b}}_q\}$$

whose unit normal vector field is ${\mathbf{t}}_q$ is called a tangent congruence surface and is denoted by $X_T$.

(ii) 

An integral surface of the distribution

$$span\{{\mathbf{t}}_q,{\mathbf{b}}_q\}$$

whose unit normal vector field is ${\mathbf{n}}_q$ is called a normal congruence surface and is denoted by $X_N$.

(iii) 

An integral surface of the distribution

$$span\{{\mathbf{t}}_q,{\mathbf{n}}_q\}$$

whose unit normal vector field is ${\mathbf{b}}_q$ is called a binormal congruence surface and is denoted by $X_B$.

Theorem 4. 

The congruence surfaces introduced in Definition 1 exist if and only if the following integrability conditions are satisfied.

(i) 

The tangent congruence surface exists if and only if

$${\beta }_1={\alpha }_2.$$

(ii) 

The normal congruence surface exists if and only if

$${\beta }_1=-k_3.$$

(iii) 

The binormal congruence surface exists if and only if

$${\alpha }_2=k_3.$$

Proof. 

According to the Frobenius Integrability Theorem, each congruence surface exists locally precisely when the Lie bracket of the two vector fields spanning the corresponding distribution belongs to that distribution. Computing $[{\mathbf{n}}_q,{\mathbf{b}}_q],[{\mathbf{t}}_q,{\mathbf{b}}_q],[{\mathbf{t}}_q,{\mathbf{n}}_q]$ by means of the anholonomic directional derivative formulas gives the stated conditions. □

Theorem 5. 

Let $X_T$, $X_N$ and $X_B$ denote the tangent, normal and binormal congruence surfaces associated with the anholonomic q-frame.

(i) 

The tangent congruence surface satisfies

$$\begin{array}{cc}\hfill {\left({\epsilon }_n\right)}_b+C_n& =2AQ+(D-{\epsilon }_n)P,\hfill \\ \hfill A_b-D_n& =-2AP+(D-{\epsilon }_n)Q,\hfill \\ \hfill P_b+Q_n& =A^2-P^2-Q^2-D{\epsilon }_n.\hfill \end{array}$$

(ii) 

The normal congruence surface satisfies

$$\begin{array}{cc}\hfill {\left(k_1\right)}_b+{\left(k_3\right)}_s& =-2D{k}_3+Q{k}_2+k_1{k}_2,\hfill \\ \hfill {\left(k_2\right)}_b-D_s& =D^2-Q{k}_1+k_2^2-k_3^2,\hfill \\ \hfill {\left(k_3\right)}_b-Q_s& =2k_2{k}_3+D{k}_1+DQ.\hfill \end{array}$$

(iii) 

The binormal congruence surface satisfies

$$\begin{array}{cc}\hfill {\left(k_1\right)}_n-{\left({\epsilon }_n\right)}_s& ={\epsilon }_n^2+k_1^2-k_3^2-P{k}_2,\hfill \\ \hfill {\left(k_2\right)}_n-{\left(k_3\right)}_s& =2{\epsilon }_n{k}_3+P{k}_1+k_1{k}_2,\hfill \\ \hfill {\left(k_3\right)}_n+P_s& =2k_1{k}_3-P{\epsilon }_n-{\epsilon }_n{k}_2.\hfill \end{array}$$

These equations constitute the corresponding Gauss–Mainardi–Codazzi equations of the tangent, normal and binormal congruence surfaces, respectively.

Proof. 

The result follows directly from the compatibility equations established in Theorem 3. Applying the corresponding existence condition

$${\beta }_1={\alpha }_2,{\beta }_1=-k_3,{\alpha }_2=k_3$$

for the tangent, normal, and binormal congruence surfaces, respectively, the compatibility equations reduce to the corresponding Gauss–Mainardi–Codazzi equations. □

3.1. Geometric Invariants of the Congruence Surfaces

In this subsection, we determine the shape operators, principal curvatures, Gaussian curvatures, and mean curvatures of the tangent, normal, and binormal congruence surfaces associated with the anholonomic q-frame. We also derive the geodesic curvatures, normal curvatures, and geodesic torsions of the corresponding integral curves lying on these surfaces. Finally, we establish several geometric consequences of these results concerning minimality, flatness, and the characterization of asymptotic and principal lines.

Theorem 6. 

Let $X_T$, $X_N$, and $X_B$ denote the tangent, normal, and binormal congruence surfaces associated with the anholonomic q-frame, respectively. Then their shape operators, with respect to the orthonormal tangent bases $\{{\mathbf{n}}_q,{\mathbf{b}}_q\}$, $\{{\mathbf{t}}_q,{\mathbf{b}}_q\}$, and $\{{\mathbf{t}}_q,{\mathbf{n}}_q\}$, respectively, are

$$S_T=\left(\begin{array}{cc}-{\epsilon }_n& -\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)\\ -\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)& -div{\mathbf{t}}_q+{\epsilon }_n\end{array}\right),$$

$$S_N=\left(\begin{array}{cc}{k}_1& -k_3\\ -k_3& -div{\mathbf{n}}_q-k_1\end{array}\right),$$

and

$$S_B=\left(\begin{array}{cc}{k}_2& k_3\\ k_3& -div{\mathbf{b}}_q-k_2\end{array}\right).$$

Proof. 

Since the unit normal vector field of the tangent congruence surface $X_T$ is ${\mathbf{t}}_q$, its shape operator is given by

$$S_T\left(X\right)=-D_X{\mathbf{t}}_q,$$

for every tangent vector field X. Using the derivative formulas of Theorem 1, we have

$$S_T\left({\mathbf{n}}_q\right)=-{\epsilon }_n{\mathbf{n}}_q-\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right){\mathbf{b}}_q,$$

and

$$S_T\left({\mathbf{b}}_q\right)=\left(⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3\right){\mathbf{n}}_q-\left(div{\mathbf{t}}_q-{\epsilon }_n\right){\mathbf{b}}_q.$$

Hence, the matrix representation of the shape operator of $X_T$ with respect to the orthonormal basis $\{{\mathbf{n}}_q,{\mathbf{b}}_q\}$ is

$$S_T=\left(\begin{array}{cc}-{\epsilon }_n& \left(⟨curl{\mathbf{n}}_q,{\mathbf{n}}_q⟩+k_3\right)\\ -\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)& -div{\mathbf{t}}_q+{\epsilon }_n\end{array}\right).$$

Therefore, the above matrix reduces to the stated symmetric matrix representation of $S_T$. The shape operators of the normal and binormal congruence surfaces are obtained analogously by taking ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$ as the corresponding unit normal vector fields and using their respective existence conditions. □

Corollary 3. 

For each congruence surface, the sum of the principal curvatures equals the negative divergence of its corresponding unit normal vector field.

Proof. 

The principal curvatures are the eigenvalues of the corresponding shape operators. Hence, for each congruence surface, they are obtained from the characteristic equation

$$det(S-\kappa I)=0.$$

For the tangent congruence surface, using Theorem 6 yields

$$\left(-{\epsilon }_n-\kappa \right)\left(-div{\mathbf{t}}_q+{\epsilon }_n-\kappa \right)-{\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2=0.$$

Equivalently,

$${\kappa }^2+div{\mathbf{t}}_q\kappa +{\epsilon }_n\left(div{\mathbf{t}}_q-{\epsilon }_n\right)-{\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2=0.$$

Applying the quadratic formula, we obtain

$${\kappa }_{1,2}^T=-{\displaystyle \frac{div{\mathbf{t}}_q}{2}}\pm {\displaystyle \frac{1}{2}}\sqrt{{\left(div{\mathbf{t}}_q-2{\epsilon }_n\right)}^2+4{\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2}.$$

Similarly, applying the same procedure to the normal and binormal congruence surfaces gives

$${\kappa }_{1,2}^N=-{\displaystyle \frac{div{\mathbf{n}}_q}{2}}\pm {\displaystyle \frac{1}{2}}\sqrt{{\left(div{\mathbf{n}}_q+2k_1\right)}^2+4k_3^2},$$

$${\kappa }_{1,2}^B=-{\displaystyle \frac{div{\mathbf{b}}_q}{2}}\pm {\displaystyle \frac{1}{2}}\sqrt{{\left(div{\mathbf{b}}_q+2k_2\right)}^2+4k_3^2}.$$

This completes the proof. □

Theorem 7. 

The Gaussian curvatures of the tangent, normal and binormal congruence surfaces are

$$K_T={\epsilon }_n(div{\mathbf{t}}_q-{\epsilon }_n)-{\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2,$$

$$K_N=-k_1(div{\mathbf{n}}_q+k_1)-k_3^2,$$

and

$$K_B=-k_2(div{\mathbf{b}}_q+k_2)-k_3^2,$$

respectively.

Proof. 

Since the Gaussian curvature is the determinant of the corresponding shape operator, the stated formulas follow directly by computing the determinants of the matrices obtained in Theorem 6. □

Corollary 4. 

The tangent, normal and binormal congruence surfaces are flat if and only if

$${\epsilon }_n\left(div{\mathbf{t}}_q-{\epsilon }_n\right)={\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2,$$

$$k_1\left(div{\mathbf{n}}_q+k_1\right)=-k_3^2,$$

and

$$k_2\left(div{\mathbf{b}}_q+k_2\right)=-k_3^2,$$

respectively.

Proof. 

The result follows immediately by setting the corresponding Gaussian curvature equal to zero in Theorem 7. □

Remark 2. 

If the third q-curvature vanishes, then, among the q-curvatures, the flatness of the normal congruence surface depends only on $k_1$, whereas that of the binormal congruence surface depends only on $k_2$.

Theorem 8. 

The mean curvature of each congruence surface depends only on the divergence of its corresponding unit normal vector field.

Proof. 

Using Theorem 6, the mean curvatures of the tangent, normal and binormal congruence surfaces are easily obtained as

$$H_T=-{\displaystyle \frac{1}{2}}div{\mathbf{t}}_q,$$

$$H_N=-{\displaystyle \frac{1}{2}}div{\mathbf{n}}_q,$$

and

$$H_B=-{\displaystyle \frac{1}{2}}div{\mathbf{b}}_q,$$

respectively. □

Corollary 5. 

The tangent, normal and binormal congruence surfaces are minimal if and only if

$$div{\mathbf{t}}_q=0,div{\mathbf{n}}_q=0,div{\mathbf{b}}_q=0,$$

respectively.

In what follows, the s-, n-, and b-lines refer to the integral curves of ${\mathbf{t}}_q$, ${\mathbf{n}}_q$, and ${\mathbf{b}}_q$, respectively, whenever these curves lie on the corresponding congruence surface.

Theorem 9. 

The geodesic curvature, normal curvature, and geodesic torsion of the integral curves of the q-frame vector fields lying on the corresponding congruence surfaces are given as follows.

(i) 

For the tangent congruence surface $X_T$, the integral curves of ${\mathbf{n}}_q$, referred to as the n-lines, satisfy

$${\kappa }_g^n=-\left(div{\mathbf{b}}_q+k_2\right),{\kappa }_n^n=-{\epsilon }_n,{\tau }_g^n=-\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right),$$

while the integral curves of ${\mathbf{b}}_q$, referred to as the b-lines, satisfy

$${\kappa }_g^b=div{\mathbf{n}}_q+k_1,{\kappa }_n^b=-div{\mathbf{t}}_q+{\epsilon }_n,{\tau }_g^b=⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3.$$

(ii) 

For the normal congruence surface $X_N$, the s-lines satisfy

$${\kappa }_g^s=-k_2,{\kappa }_n^s=k_1,{\tau }_g^s=k_3,$$

while the b-lines satisfy

$${\kappa }_g^b=-div{\mathbf{t}}_q+{\epsilon }_n,{\kappa }_n^b=-\left(div{\mathbf{n}}_q+k_1\right),{\tau }_g^b=-k_3.$$

(iii) 

For the binormal congruence surface $X_B$, the s-lines satisfy

$${\kappa }_g^s=k_1,{\kappa }_n^s=k_2,{\tau }_g^s=k_3,$$

while the n-lines satisfy

$${\kappa }_g^n={\epsilon }_n,{\kappa }_n^n=-\left(div{\mathbf{b}}_q+k_2\right),{\tau }_g^n=-k_3.$$

Proof. 

For a unit-speed curve on an oriented surface with unit tangent vector T and unit normal vector N, we use

$${\kappa }_g=⟨D_{T}T,N\times T⟩,{\kappa }_n=⟨D_{T}T,N⟩,{\tau }_g=-⟨D_{T}N,N\times T⟩.$$

We first consider the tangent congruence surface $X_T$, whose unit normal vector field is ${\mathbf{t}}_q$. For an n-line, we have

$$T={\mathbf{n}}_q,{\mathbf{t}}_q\times {\mathbf{n}}_q={\mathbf{b}}_q.$$

Using the directional derivative formulas,

$$D_{{\mathbf{n}}_q}{\mathbf{n}}_q=-{\epsilon }_n{\mathbf{t}}_q-\left(div{\mathbf{b}}_q+k_2\right){\mathbf{b}}_q$$

and

$$D_{{\mathbf{n}}_q}{\mathbf{t}}_q={\epsilon }_n{\mathbf{n}}_q+\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right){\mathbf{b}}_q,$$

we obtain

$${\kappa }_g=-\left(div{\mathbf{b}}_q+k_2\right),{\kappa }_n=-{\epsilon }_n,{\tau }_g=-\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right).$$

For a b-line of $X_T$,

$$T={\mathbf{b}}_q,{\mathbf{t}}_q\times {\mathbf{b}}_q=-{\mathbf{n}}_q.$$

The corresponding directional derivative formulas, together with the existence condition of $X_T$, give

$${\kappa }_g=div{\mathbf{n}}_q+k_1,{\kappa }_n=-div{\mathbf{t}}_q+{\epsilon }_n,{\tau }_g=⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3.$$

The results for the corresponding integral curves of $X_N$ and $X_B$ follow in the same manner by taking ${\mathbf{n}}_q$ and ${\mathbf{b}}_q$ as their respective unit normal vector fields and applying the corresponding directional derivative formulas and existence conditions. This completes the proof. □

Corollary 6. 

If any of the congruence surfaces $X_T$, $X_N$, or $X_B$ is minimal, then its Gaussian curvature is non-positive. More precisely,

$$K_T=-{\epsilon }_n^2-{\left(⟨curl{\mathbf{b}}_q,{\mathbf{b}}_q⟩+k_3\right)}^2,K_N=-k_1^2-k_3^2,K_B=-k_2^2-k_3^2,$$

respectively. Hence, every non-flat point is hyperbolic. Moreover, $X_T$ is flat if and only if its n-lines are both asymptotic and principal, while $X_N$ and $X_B$ are flat if and only if their s-lines are both asymptotic and principal.

Remark 3. 

The geodesic torsion relations in Theorem 9 show that the two families of coordinate lines on each congruence surface are simultaneously principal. More precisely, the n-lines of $X_T$ are principal if and only if its b-lines are principal; the s-lines of $X_N$ are principal if and only if its b-lines are principal; and the s-lines of $X_B$ are principal if and only if its n-lines are principal.

3.2. An Illustrative Example

Example 1. 

Consider the surface

$$X_N(s,b)=\left(cos\theta ,sin\theta ,{\displaystyle \frac{s-b}{\sqrt{2}}}\right),\theta ={\displaystyle \frac{s+b}{\sqrt{2}}}.$$

Its partial derivatives are

$${\left(X_N\right)}_s={\displaystyle \frac{1}{\sqrt{2}}}(-sin\theta ,cos\theta ,1)={\mathbf{t}}_q$$

and

$${\left(X_N\right)}_b={\displaystyle \frac{1}{\sqrt{2}}}(-sin\theta ,cos\theta ,-1)={\mathbf{b}}_q.$$

Moreover,

$$\parallel {\mathbf{t}}_q\parallel =\parallel {\mathbf{b}}_q\parallel =1,⟨{\mathbf{t}}_q,{\mathbf{b}}_q⟩=0,$$

and the unit normal vector field is

$${\mathbf{n}}_q=(cos\theta ,sin\theta ,0).$$

Differentiating the q-frame along the s-direction gives

$${\displaystyle \frac{\partial }{\partial s}}\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right)=\left(\begin{array}{ccc}0& -{\displaystyle \frac{1}{2}}& 0\\ {\displaystyle \frac{1}{2}}& 0& {\displaystyle \frac{1}{2}}\\ 0& -{\displaystyle \frac{1}{2}}& 0\end{array}\right)\left(\begin{array}{c}{\mathbf{t}}_q\\ {\mathbf{n}}_q\\ {\mathbf{b}}_q\end{array}\right).$$

Hence,

$$k_1=-{\displaystyle \frac{1}{2}},k_2=0,k_3={\displaystyle \frac{1}{2}}.$$

On the other hand,

$${\left({\mathbf{t}}_q\right)}_b=-{\displaystyle \frac{1}{2}}{\mathbf{n}}_q,$$

which gives

$$C={\displaystyle \frac{1}{2}},{\beta }_1=-{\displaystyle \frac{1}{2}}=-k_3.$$

Therefore, the existence condition for the normal congruence surface is satisfied. Since $div{\mathbf{n}}_q=1,$ the shape operator of $X_N$ is

$$S_N=\left(\begin{array}{cc}{k}_1& -k_3\\ -k_3& -div{\mathbf{n}}_q-k_1\end{array}\right)=-{\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).$$

Consequently,

$$K_N=0,H_N=-{\displaystyle \frac{1}{2}}.$$

For a fixed $b=b_0$, the s-parameter curve $\gamma \left(s\right)=X_N(s,b_0)$ satisfies

$${\gamma }^{\prime }\left(s\right)={\mathbf{t}}_q,{\gamma }^{″}\left(s\right)=-{\displaystyle \frac{1}{2}}{\mathbf{n}}_q.$$

Similarly, for a fixed $s=s_0$, the b-parameter curve $\eta \left(b\right)=X_N(s_0,b)$ satisfies

$${\eta }^{\prime }\left(b\right)={\mathbf{b}}_q,{\eta }^{″}\left(b\right)=-{\displaystyle \frac{1}{2}}{\mathbf{n}}_q.$$

Therefore, both the s- and b-parameter curves are unit-speed circular helices with curvature $1/2$. Since their acceleration vectors are parallel to the surface normal ${\mathbf{n}}_q$, both families are geodesics on $X_N$.

Thus, this example illustrates how the invariants of the anholonomic q-frame can be used to characterize the geometry of an associated normal congruence surface.

4. Conclusions

We established an anholonomic coordinate formulation associated with the q-frame in Euclidean three-space and derived the corresponding directional derivative formulas, mixed-derivative relations, and compatibility equations. As the main geometric application, we constructed the tangent, normal, and binormal congruence surfaces and obtained their existence conditions, Gauss–Mainardi–Codazzi equations, shape operators, and curvature characterizations. These results reveal a direct connection between the differential invariants of the anholonomic q-frame and the intrinsic and extrinsic geometry of the associated congruence surfaces. The illustrative example further demonstrates how these invariants determine the geometry of a normal congruence surface and its associated parameter curves.

A significant feature of the proposed framework is that the compatibility conditions of the anholonomic q-frame are directly related to the Gauss–Mainardi–Codazzi equations of the corresponding congruence surfaces. Thus, the integrability of the moving-frame system and the geometry of the generated surfaces can be interpreted within the same differential-geometric framework. Moreover, the explicit expressions obtained for the shape operators, Gaussian curvatures, and mean curvatures make it possible to examine how conditions imposed on the q-curvature functions and the associated divergence and curl invariants are reflected in the geometry of the congruence surfaces. In particular, these formulas can be used to characterize special cases such as minimal or flat congruence surfaces.

The present formulation also provides several directions for further investigation. A natural extension is to Lorentzian geometry, where the causal characters of the frame vectors may lead to different anholonomic structures and associated congruence surfaces. Another possible direction is the study of time-dependent q-frames and their compatibility conditions in geometric evolution problems. In particular, the framework may be adapted to Hasimoto-type flows and their associated surface constructions, providing a basis for investigating the relationships among frame evolution, surface geometry, and the corresponding geometric and physical properties.