On generalised framed surfaces in the Euclidean space

1.   Introduction

We investigate differential geometric invariants of surfaces with singular points, that is, singular surfaces. The geometry of singular surfaces in the Euclidean space is a classical object (cf. ^{[1,2,3,4,5,9,11,18,19,20,21]}). For regular surfaces, the Gauss curvature and mean curvature are important invariants up to congruence. However, if we consider a deformation of a regular surface (for instance, parallel surfaces or caustics), it may have singular points. One of the idea is to consider the fronts or frontals as smooth surfaces with singular points (cf. ^{[1,2,10,13,16,19]}). The other idea is to consider one-parameter families of framed curves as smooth surfaces with singular points (cf. ^[7,15]). We generalise the consideration to treat the smooth surfaces with singular points. A more general notion of singular surfaces, called generalised framed surfaces, is introduced in this paper. The notion of generalised framed surfaces includes not only the notion of framed surfaces, but also the notion of one-parameter families of framed curves. It also includes surfaces with corank one singularities (cf. ^[12,14]).

We have introduced framed surfaces as surfaces with singular points in ^[6]. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals). In fact, if $f$ is a frontal, then the Jacobi ideal $J_f$ of $f$ is generated by one element ^[10]. On the other hand, we have also introduced one-parameter families of framed curves as surfaces with singular points in ^[8,15]. The relation between framed surfaces and one-parameter families of framed curves was investigated in ^[7]. In §2, we review the theories of framed surfaces and one-parameter families of framed curves. In §3, we introduce the basic invariants of generalised framed surfaces and give the existence and uniqueness theorems for the basic invariants of generalised framed surfaces. The properties of the generalised framed surfaces are investigated. We give conditions for a surface to become a generalised framed base surface (Theorem 3.10) and for a generalised framed surface to become a framed base surface (Theorem 3.11). In §4 and §5, we focus on surfaces with corank one singularities and corank two singularities, respectively. We prove that surfaces with corank one singularities can always be considered generalised framed surfaces at least locally (Theorem 4.1). Moreover, we find that a part of surfaces with corank two singularities can be considered as generalised framed surfaces. The conditions for special cases of surfaces with corank two singularities to become generalised framed surfaces and framed surfaces are given. As an application, we investigate two types of parallel surfaces of generalised framed surfaces and give concrete examples to illustrate our results in §6.

All maps and manifolds considered in this paper are differentiable of class $C^{\infty}$ unless stated otherwise.

2.   Preliminaries

Let $\mathbb{R}^3$ be the $3$-dimensional Euclidean space equipped with the inner product ${\boldsymbol{a}} \cdot {\boldsymbol{b}} = a_1 b_1 + a_2 b_2 + a_3 b_3$, where ${\boldsymbol{a}} = (a_1, a_2, a_3)$ and ${\boldsymbol{b}} = (b_1, b_2, b_3) \in \mathbb{R}^3$. The norm of ${\boldsymbol{a}}$ is given by $\vert {\boldsymbol{a}} \vert = \sqrt{ {\boldsymbol{a}} \cdot {\boldsymbol{a}}}$ and the vector product is given by

where ${\boldsymbol{e}}_1, {\boldsymbol{e}}_2$ and ${\boldsymbol{e}}_3$ are the canonical basis of $\mathbb{R}^3$. Let $U$ be a simply connected domain in $\mathbb{R}^2$ and $S^2$ be the unit sphere in $\mathbb{R}^3$, that is, $S^2 = \{ {\boldsymbol{a}} \in \mathbb{R}^3| | {\boldsymbol{a}}| = 1\}$. We denote a $3$-dimensional smooth manifold $\{({\boldsymbol{a}}, {\boldsymbol{b}}) \in S^2 \times S^2| {\boldsymbol{a}} \cdot {\boldsymbol{b}} = 0\}$ by $\Delta$.

2.1.   Framed surfaces

We quickly review the theory of framed surfaces in Euclidean 3-space; in detail, see ^[6,7]. Let $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}}):U \to \mathbb{R}^3 \times \Delta$ be a smooth mapping.

Definition 2.1. We say that $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}}):U \to \mathbb{R}^3 \times \Delta$ is a framed surface if ${\boldsymbol{x}}_u (u, v) \cdot {\boldsymbol{n}} (u, v) = {\boldsymbol{x}}_v(u, v) \cdot {\boldsymbol{n}}(u, v) = 0$ for all $(u, v) \in U$, where ${\boldsymbol{x}}_u(u, v) = (\partial {\boldsymbol{x}}/\partial u)(u, v)$ and ${\boldsymbol{x}}_v(u, v) = (\partial {\boldsymbol{x}}/\partial v)(u, v)$. We say that ${\boldsymbol{x}}:U \to \mathbb{R}^3$ is a framed base surface if there exists $({\boldsymbol{n}}, {\boldsymbol{s}}):U \to \Delta$ such that $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$ is a framed surface.

By definition, the framed base surface is a frontal. For the definition and properties of frontals see ^[1,2]. On the other hand, the frontal is a framed base surface at least locally.

We denote ${\boldsymbol{t}}(u, v) = {\boldsymbol{n}}(u, v) \times {\boldsymbol{s}}(u, v)$. Then $\{ {\boldsymbol{n}}(u, v), {\boldsymbol{s}}(u, v), {\boldsymbol{t}}(u, v)\}$ is a moving frame along ${\boldsymbol{x}}(u, v)$, and we have the following systems of differential equations:

where $a_i, b_i, e_i, f_i, g_i:U \to \mathbb{R}, i = 1, 2$ are smooth functions. We call these functions basic invariants of the framed surface. We denote the above matrices by $\mathcal{G}, \mathcal{F}_1$ and $\mathcal{F}_2$, respectively. We also call the matrices $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$ basic invariants of the framed surface $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$. Note that $(u, v)$ is a singular point of ${\boldsymbol{x}}$ if and only if ${\rm det}\ \mathcal{G}(u, v) = 0$.

Since the integrability conditions ${\boldsymbol{x}}_{uv} = {\boldsymbol{x}}_{vu}$ and $\mathcal{F}_{2, u}-\mathcal{F}_{1, v} = \mathcal{F}_1\mathcal{F}_2-\mathcal{F}_2\mathcal{F}_1$, the basic invariants should satisfy some conditions. Note that there are fundamental theorems for framed surfaces, namely, the existence and uniqueness theorems for the basic invariants of framed surfaces (cf. ^[6]).

2.2.   One-parameter families of framed curves

We also review the theory of one-parameter families of framed curves in the Euclidean 3-space, in detail, see ^[7,15]. Let $(\gamma, \nu_{1}, \nu_{2}):U \rightarrow \mathbb{R}^{3} \times \Delta$ be a smooth mapping.

Definition 2.2. We say that $(\gamma, \nu_1, \nu_2): U \to \mathbb{R}^3 \times \Delta$ is a one-parameter family of framed curves with respect to $u$ (respectively, with respect to $v$) if $(\gamma(\cdot, v), \nu_1(\cdot, v), \nu_2(\cdot, v))$ is a framed curve for each $v$ (respectively, $(\gamma(u, \cdot), \nu_1(u, \cdot), \nu_2(u, \cdot))$ is a framed curve for each $u$), that is, $\gamma_u(u, v) \cdot \nu_1(u, v) = \gamma_u(u, v) \cdot \nu_2(u, v) = 0$ (respectively, $\gamma_v(u, v) \cdot \nu_1(u, v) = \gamma_v(u, v) \cdot \nu_2(u, v) = 0$) for all $(u, v) \in U$. We say that $\gamma$ is a one-parameter family of framed base curves with respect to $u$ (respectively, with respect to $v$) if there exists $(\nu_1, \nu_2):U \to \Delta$ such that $(\gamma, \nu_1, \nu_2)$ is a one-parameter family of framed curves with respect to $u$ (respectively, with respect to $v$).

We denote $\mu(u, v) = \nu_{1}(u, v)\times\nu_{2}(u, v)$. Then $\{\nu_{1}(u, v), \nu_{2}(u, v), \mu(u, v)\}$ is a moving frame along $\gamma(u, v)$ and we have the Frenet-Serret type formula.

We call the mapping $(\ell, m, n, r, L, M, N, P, Q, R)$ the curvature of the one-parameter family of framed curves with respect to $u$ of $(\gamma, \nu_1, \nu_2)$.

Since the integrability conditions $\gamma_{uv}(u, v) = \gamma_{vu}(u, v), \; \nu_{1uv}(u, v) = \nu_{1vu}(u, v), \; \nu_{2uv}(u, v) = \nu_{2vu}(u, v)$ and $\mu_{uv}(u, v) = \mu_{vu}(u, v)$, the basic invariants should satisfy some conditions. Note that there are fundamental theorems for one-parameter families of framed curves, namely, the existence and uniqueness theorems for curvatures of one-parameter families of framed curves (cf. ^[7]).

3.   Generalised framed surfaces

We give a definition of a generalisation of framed surfaces and one-parameter families of framed curves. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a smooth mapping. We denote $\nu = {\boldsymbol{x}}_u \times {\boldsymbol{x}}_v$.

Definition 3.1. We say that $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface if there exist smooth functions $\alpha, \beta:U \to \mathbb{R}$ such that $\nu(u, v) = \alpha(u, v) \nu_1(u, v)+\beta(u, v) \nu_2(u, v)$ for all $(u, v) \in U$. We say that ${\boldsymbol{x}}:U \to \mathbb{R}^3$ is a generalised framed base surface if there exists $(\nu_1, \nu_2):U \to \Delta$ such that $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface.

Remark 3.2. Let $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}}):U \to \mathbb{R}^3 \times \Delta$ be a framed surface with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$. Then $\nu(u, v) = {\boldsymbol{x}}_{u}(u, v) \times {\boldsymbol{x}}_v(u, v) = (a_1(u, v)b_2(u, v)-a_2(u, v)b_1(u, v)) {\boldsymbol{n}}(u, v)$. If we take $\alpha(u, v) = a_1(u, v)b_2(u, v)-a_2(u, v)b_1(u, v)$ and $\beta(u, v) = 0$, then $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$ is also a generalised framed surface.

Remark 3.3. Let $(\gamma, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a one-parameter family of framed curves with respect to $u$ with curvature $(\ell, m, n, r, L, M, N, P, Q, R)$. Then $\nu(u, v) = \gamma_u(u, v) \times \gamma_v(u, v) = -r(u, v)Q(u, v)\nu_1(u, v)+ r(u, v)P(u, v)\nu_2(u, v)$. If we take $\alpha(u, v) = -r(u, v)Q(u, v)$ and $\beta(u, v) = r(u, v)P(u, v)$, then $(\gamma, \nu_1, \nu_2)$ is also a generalised framed surface.

We denote $\nu_3(u, v) = \nu_1(u, v) \times \nu_2(u, v)$. Then $\{\nu_1(u, v), \nu_2(u, v), \nu_3(u, v)\}$ is a moving frame along ${\boldsymbol{x}}(u, v)$, and we have the following systems of differential equations:

where $a_i, b_i, c_i, e_i, f_i, g_i:U \to \mathbb{R}, i = 1, 2$ are smooth functions with $a_1b_2-a_2b_1 = 0$. We call the functions basic invariants of the generalised framed surface. We denote the above matrices by $\mathcal{G}, \mathcal{F}_1$ and $\mathcal{F}_2$, respectively. We also call the matrices $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$ basic invariants of the generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2)$. By definition, we have

Since the integrability conditions ${\boldsymbol{x}}_{uv} = {\boldsymbol{x}}_{vu}$ and $\mathcal{F}_{2u}-\mathcal{F}_{1v} = \mathcal{F}_1\mathcal{F}_2-\mathcal{F}_2\mathcal{F}_1$, the basic invariants should satisfy the following conditions:

We give fundamental theorems for generalised framed surfaces, that is, the existence and uniqueness theorems for the basic invariants of generalised framed surfaces.

Theorem 3.4. (Existence Theorem for generalised framed surfaces). Let $(a_i, b_i, c_i, e_i, f_i, g_i):I \to \mathbb{R}^{12}, i = 1, 2$ be a smooth mapping satisfying $a_1b_2-a_2b_1 = 0$, satisfying the integrability conditions (3.1) and (3.2). Then there exists a generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ whose associated basic invariants are $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$.

Proof. Since the integrability conditions (3.1) and (3.2) exist, there exists a smooth mapping ${\boldsymbol{x}}: U \rightarrow \mathbb{R}^3$ and an orthonormal frame $\{\nu_1, \nu_2, \nu_3\}$ such that the condition holds. Therefore, there exists a generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2): U \rightarrow \Delta$ whose associated basic invariants are $(\mathcal{G}, \mathcal{F}_{1}, \mathcal{F}_{2})$. □

Definition 3.5. Let $({\boldsymbol{x}}, \nu_1, \nu_2), (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ be generalised framed surfaces. We say that $({\boldsymbol{x}}, \nu_1, \nu_2)$ and $(\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)$ are congruent as generalised framed surfaces if there exists a constant rotation $A \in SO(3)$ and a translation ${\boldsymbol{a}} \in \mathbb{R}^3$ such that $\widetilde{ {\boldsymbol{x}}}(u, v) = A({\boldsymbol{x}}(u, v)) + {\boldsymbol{a}}$, $\widetilde{\nu}_1(u, v) = A(\nu_1(u, v))$ and $\widetilde{\nu}_2(u, v) = A(\nu_2(u, v))$ for all $(u, v) \in U$.

Theorem 3.6. (Uniqueness Theorem for generalised framed surfaces). Let $({\boldsymbol{x}}, \nu_1, \nu_2), (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ be generalised framed surfaces with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2), (\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_1, \widetilde{\mathcal{F}}_2)$, respectively. Then $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$ and $(\widetilde{ {\boldsymbol{x}}}, \widetilde{ {\boldsymbol{n}}}, \widetilde{ {\boldsymbol{s}}})$ are congruent as generalised framed surfaces if and only if the basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$ and $(\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_1, \widetilde{\mathcal{F}}_2)$ coincide.

In order to prove the uniqueness theorem, we prepare the following two lemmas.

Lemma 3.7. Let $({\boldsymbol{x}}, \nu_1, \nu_2), (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ be generalised framed surfaces with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2), (\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_1, \widetilde{\mathcal{F}}_2)$, respectively. If $({\boldsymbol{x}}, \nu_1, \nu_2)$ and $(\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)$ are congruent as generalised framed surfaces, then $(\mathcal{G}, \mathcal{F}_{1}, \mathcal{F}_{2}) = (\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_{1}, \widetilde{\mathcal{F}}_{2})$.

Proof. By Definition 3.1 and a direct calculation, we obtain the lemma. □

Lemma 3.8. Let $({\boldsymbol{x}}, \nu_1, \nu_2), (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ be generalised framed surfaces with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2), (\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_1, \widetilde{\mathcal{F}}_2)$, respectively. If $(\mathcal{G}, \mathcal{F}_{1}, \mathcal{F}_{2}) = (\widetilde{\mathcal{G}}, \widetilde{\mathcal{F}}_{1}, \widetilde{\mathcal{F}}_{2})$ and $({\boldsymbol{x}}, \nu_1, \nu_2)(u_{0}, v_{0}) = (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)(u_{0}, v_{0})$ for some point $(u_{0}, v_{0})\in U$, then $({\boldsymbol{x}}, \nu_1, \nu_2) = (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)$.

Proof. Define a smooth function $f:U\rightarrow \mathbb{R}$ by

By the definition of the basic invariants, we have

By the assumption $(\mathcal{F}_{1}, \mathcal{F}_{2}) = (\widetilde{\mathcal{F}}_{1}, \widetilde{\mathcal{F}}_{2})$, we have $f_{u}(u, v) = 0$ for all $(u, v)\in U$. Similarly, we also have $f_{v}(u, v) = 0$ for all $(u, v)\in U$. Moreover, by the assumption $(\nu_1, \nu_2)(u_{0}, v_{0}) = (\widetilde{\nu}_1, \widetilde{\nu}_2)(u_{0}, v_{0})$, we have $f(u_0, v_0) = 3$. It concludes that $f(u, v) = 3$ for all $(u, v)\in U$. Hence, we have $\nu_1\cdot\widetilde{\nu}_1 = 1, \; \nu_2\cdot\widetilde{\nu}_2 = 1, \; \nu_3\cdot\widetilde{\nu}_3 = 1$. It follows that $\nu_1 = \widetilde{\nu}_1, \nu_2 = \widetilde{\nu}_2, \nu_3 = \widetilde{\nu}_3$. Next, we show ${\boldsymbol{x}} = \widetilde{ {\boldsymbol{x}}}$. By the assumption $\mathcal{G} = \widetilde{\mathcal{G}}$, we have ${\boldsymbol{x}}_u = a_1\nu_1+b_1\nu_2+ c_1\nu_3 = \widetilde{a}_1\widetilde{\nu}_1+\widetilde{b}_1\widetilde{\nu}_2+\widetilde{c}_1\widetilde{\nu}_3 = \widetilde{ {\boldsymbol{x}}}_u$ and ${\boldsymbol{x}}_v = a_2\nu_1+b_2\nu_2+ c_2\nu_3 = \widetilde{a}_2\widetilde{\nu}_1+\widetilde{b}_2\widetilde{\nu}_2+\widetilde{c}_2\widetilde{\nu}_3 = \widetilde{ {\boldsymbol{x}}}_v$. Then, we have $({\boldsymbol{x}}-\widetilde{ {\boldsymbol{x}}})_u = ({\boldsymbol{x}}-\widetilde{ {\boldsymbol{x}}})_v = 0$. Since ${\boldsymbol{x}}(u_0, v_0) = \widetilde{ {\boldsymbol{x}}}(u_0, v_0)$, we have ${\boldsymbol{x}}(u, v) = \widetilde{ {\boldsymbol{x}}}(u, v)$ for all $(u, v)\in U$. Therefore, we have $({\boldsymbol{x}}, \nu_1, \nu_2) = (\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)$. □

Proof of the Uniqueness Theorem. The necessary part of the theorem is Lemma 3.7. We prove the sufficient part of the theorem. For fix a point $(u_0, v_0)\in U$, there exist $A \in SO(3)$ and ${\boldsymbol{a}} \in \mathbb{R}^3$ such that $({\boldsymbol{x}}, \nu_1, \nu_2)(u_0, v_0) = (A\widetilde{ {\boldsymbol{x}}}+ {\boldsymbol{a}}, A\widetilde{\nu}_1, A\widetilde{\nu}_2)(u_0, v_0)$. By Lemmas 3.7 and 3.8, we have $({\boldsymbol{x}}, \nu_1, \nu_2) = (A\widetilde{ {\boldsymbol{x}}}+ {\boldsymbol{a}}, A\widetilde{\nu}_1, A\widetilde{\nu}_2)$, that is, $({\boldsymbol{x}}, \nu_1, \nu_2)$ and $(\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2)$ are congruent as generalised framed surfaces. □

By relations among basic invariants of generalised framed surfaces, basic invariants of framed surfaces, and curvatures of one-parameter families of framed curves, we have the following:

Proposition 3.9. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$.

(1) If $a_1(u, v) = a_2(u, v) = 0$ $($respectively, $b_1(u, v) = b_2(u, v) = 0)$ for all $(u, v) \in U$, then $({\boldsymbol{x}}, \nu_1, \nu_2)$ $($respectively, $({\boldsymbol{x}}, \nu_2, \nu_1))$ is a framed surface.

(2) If $a_1(u, v) = b_1(u, v) = 0$ $($respectively, $a_2(u, v) = b_2(u, v) = 0)$ for all $(u, v) \in U$, then $({\boldsymbol{x}}, \nu_1, \nu_2)$ $($respectively, $({\boldsymbol{x}}, \nu_2, \nu_1))$ is a one-parameter family of framed curves with respect to $u$ $($respectively, with respect to $v)$.

We give a condition for a surface to become a generalised framed base surface.

Theorem 3.10. Let ${\boldsymbol{x}}:U \to \mathbb{R}^3$ be a smooth mapping. We denote $\nu = {\boldsymbol{x}}_u\times {\boldsymbol{x}}_v = p_1 {\boldsymbol{e}}_1+p_2 {\boldsymbol{e}}_2+p_3 {\boldsymbol{e}}_3$, where ${\boldsymbol{e}}_1, {\boldsymbol{e}}_2$ and ${\boldsymbol{e}}_3$ are the canonical basis. Then ${\boldsymbol{x}}$ is a generalised framed base surface at least locally if and only if the functions $p_1, p_2$ and $p_3$ are linearly dependent.

Proof. If ${\boldsymbol{x}}$ is a generalised framed base surface at least locally, then there exist $(\nu_1, \nu_2):U\rightarrow \Delta$ and $\alpha, \beta:U\rightarrow \mathbb{R}$, such that $\nu = \alpha\nu_1+\beta\nu_2$. If we denote $\nu_3 = \nu_1\times\nu_2$, then there exists a rotation $A(u, v) \in SO(3)$, such that $^{T}(\nu_1, \nu_2, \nu_3) = A ^{T}({\boldsymbol{e}}_1, {\boldsymbol{e}}_2, {\boldsymbol{e}}_3)$, where $T$ is the transpose of the matrix. Therefore,

It follows that $(\alpha, \beta, 0)A = (p_1, p_2, p_3)$. Since the orthogonal transformation does not change the linearly relation of the set of functions, we have that the functions $p_1, p_2$ and $p_3$ are linearly dependent by $\alpha, \beta, 0$.

Conversely, if $p_1, p_2, p_3$ are linearly dependent, then there exist functions $k_1, k_2, k_3:U\rightarrow \mathbb{R}$ with $(k_1, k_2, k_3) \neq (0, 0, 0)$ such that $k_1p_1+k_2p_2+k_3p_3 = 0$. Without loss of generality, we may assume $k_1 \neq 0$, at least locally. Then we have

If we take

and

then $\nu = \alpha\nu_1+\beta\nu_2$. It follows that $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface, and hence ${\boldsymbol{x}}$ is a generalised framed base surface, at least locally. □

We also give a condition for a generalised framed surface to become a framed base surface.

Theorem 3.11. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with $\nu = \alpha\nu_1+\beta\nu_2$.

(1) If ${\boldsymbol{x}}$ is a framed base surface, then the functions $\alpha$ and $\beta$ are linearly dependent.

(2) Suppose that the set of regular points of ${\boldsymbol{x}}$ is dense in $U$. If the functions $\alpha$ and $\beta$ are linearly dependent, then ${\boldsymbol{x}}$ is a framed base surface, at least locally.

Proof. (1) If ${\boldsymbol{x}}$ is a framed base surface, then there exists $({\boldsymbol{n}}, {\boldsymbol{s}}):U\rightarrow \Delta$ such that $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$ is a framed surface. Therefore, there exists a smooth function $\ell:U\rightarrow \mathbb{R}$ such that $\nu = \alpha \nu_1 +\beta \nu_2 = \ell {\boldsymbol{n}}$. By $\alpha^2+\beta^2 = \ell^2$, there exists a smooth function $\theta:U \to \mathbb{R}$ such that $\alpha = \ell \cos \theta, \beta = \ell \sin \theta$. Since $\alpha \sin \theta-\beta \cos \theta = 0$, the functions $\alpha$ and $\beta$ are linearly dependent.

(2) If $\alpha$ and $\beta$ are linearly dependent, then there exist functions $k_1, k_2:U\rightarrow \mathbb{R}$ with $(k_1, k_2)\neq(0, 0)$ such that $k_1\alpha+k_2\beta = 0$. Without loss of generality, we may assume $k_1\neq0$ at least locally. Since the set of regular points of ${\boldsymbol{x}}$ is dense in $U$, we have

It follows that $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}}): U \to \mathbb{R}^3 \times \Delta$ is a framed surface, where

and hence ${\boldsymbol{x}}$ is a framed base surface at least locally. □

Corollary 3.12. Let $({\boldsymbol{x}}, \nu_1, \nu_2) : U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2).$ Suppose that the set of regular points of ${\boldsymbol{x}}$ is dense in $U$.

(1) If ${\boldsymbol{a}}(u, v) = (a_1, a_2)(u, v)$ and ${\boldsymbol{b}}(u, v) = (b_1, b_2)(u, v)$ are linearly dependent, then ${\boldsymbol{x}}$ is a framed base surface, at least locally.

(2) If ${\rm rank}({\boldsymbol{a}}, {\boldsymbol{b}}) = 1$ at $p \in U$, then ${\boldsymbol{x}}$ is a framed base surface around $p$.

Proof. (1) By assumption, there exist smooth functions $k_1, k_2:U \to \mathbb{R}$ with $(k_1, k_2)\not = (0, 0)$ such that $k_1 {\boldsymbol{a}}+k_2 {\boldsymbol{b}} = 0$. Since

$\alpha$ and $\beta$ are linearly dependent. By Theorem 3.11 (2), ${\boldsymbol{x}}$ is a framed base surface, at least locally.

(2) By assumption, ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ are linearly dependent around $p$. Therefore, we have the result by (1). □

Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with $\nu = \alpha \nu_1+\beta \nu_2$ and basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$. We consider other frames by using rotation and reflection.

Proposition 3.13. Under the above notations, we have the following:

(1) $({\boldsymbol{x}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface with

and the basic invariants

(2) $({\boldsymbol{x}}, \overline{\nu}_1, \overline{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface with $(\overline{\alpha}(u, v), \overline{\beta}(u, v)) = (\beta(u, v), \alpha(u, v))$ and the basic invariants

Proof. (1) Since

we have $\widetilde{\alpha} = \alpha\cos\theta-\beta\sin\theta, \; \widetilde{\beta} = \alpha\sin\theta+\beta\cos\theta.$ Moreover, $\widetilde{\nu}_1\cdot\widetilde{\nu}_2 = (\cos\theta\nu_1-\sin\theta\nu_2)\cdot(\sin\theta\nu_1+\cos\theta\nu_2) = 0.$ Therefore, $({\boldsymbol{x}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface. Since

we have the basic invariants

(2) Since $\nu = \alpha\nu_1+\beta\nu_2 = \beta\overline{\nu}_1+\alpha\overline{\nu}_2,$ we have $\overline{\alpha} = \beta, \; \overline{\beta} = \alpha.$ Moreover, $\overline{\nu}_1\cdot\overline{\nu}_2 = \nu_2\cdot\nu_1 = 0$. Therefore, $({\boldsymbol{x}}, \overline{\nu}_1, \overline{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface. Since $\overline{\nu}_3 = \overline{\nu}_1\times\overline{\nu}_2 = \nu_2\times\nu_1 = -\nu_3$, we have the basic invariants

□

Next, we consider a parameter change in the domain $U$ and a diffeomorphism in the target space $\mathbb{R}^3$.

Proposition 3.14. Let $({\boldsymbol{x}}, \nu_1, \nu_2) : U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2).$ Let $\phi: V \rightarrow U, (p, q) \mapsto \phi(p, q) = (u(p, q), v(p, q))$ be a parameter change, that is, a diffeomorphism of the domain. Then $(\widetilde{ {\boldsymbol{x}}}, \widetilde{\nu}_1, \widetilde{\nu}_2) = ({\boldsymbol{x}}, \nu_1, \nu_2)\circ \phi : V \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface with

and the basic invariants

Proof. Since

we have

Moreover, $\widetilde{\nu}_1(p, q)\cdot\widetilde{\nu}_2(p, q) = \nu_1(\phi(p, q))\cdot\nu_2(\phi(p, q)) = 0$. Therefore, $({\boldsymbol{x}}, \widetilde{\nu}_1, \widetilde{\nu}_2):U \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface. By the chain rule, we have

It follows that we have the first equation of the basic invariants. The second equation of the basic invariants can be proved similarly to the above by using the chain rule. □

Proposition 3.15. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface and $\Phi: \mathbb{R}^3 \to \mathbb{R}^3$ be a diffeomorphism. Then there exists a smooth mapping $(\nu_1^\Phi, \nu_2^\Phi) : U \to \Delta$ such that $(\Phi \circ {\boldsymbol{x}}, \nu_1^\Phi, \nu_2^\Phi):U \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface.

Proof. We denote the Jacobi matrix of $\Phi$ at $x$ by $D_\Phi(x)$, that is,

Since $\Phi$ is a diffeomorphism, $D_\Phi(x) \in GL(3, R)$. We define a mapping $(\nu_1^\Phi, \nu_2^\Phi) : U \rightarrow \Delta$ by

where $A = ^{T}((D_\Phi)^{-1}\circ {\boldsymbol{x}})$. Then we show that $(\Phi\circ {\boldsymbol{x}}, \nu_1^\Phi, \nu_2^\Phi):U\rightarrow \mathbb{R}^3\times\Delta$ is a generalised framed surface. In fact,

Thus, $\nu^\Phi = \alpha^\Phi\nu_1^\Phi+\beta^\Phi\nu_2^\Phi$, where

Moreover,

Therefore, $(\Phi\circ {\boldsymbol{x}}, \nu_1^\Phi, \nu_2^\Phi):U\rightarrow \mathbb{R}^3\times\Delta$ is a generalised framed surface. □

4.   Corank one singularities

Let ${\boldsymbol{x}}: U \to \mathbb{R}^3$ be a smooth mapping. Suppose that ${\rm corank}(d {\boldsymbol{x}}) = 1$ at a point $p \in U$. By using a parameter change of $U$, we may assume that ${\boldsymbol{x}}$ is given by ${\boldsymbol{x}}(u, v) = (u, f(u, v), g(u, v))$ at least locally, where $f, g: U \to \mathbb{R}$ are smooth functions. Then corank one singularities are always generalised framed base surfaces, at least locally.

Theorem 4.1. Let ${\boldsymbol{x}}:U \to \mathbb{R}^3$ be given by ${\boldsymbol{x}}(u, v) = (u, f(u, v), g(u, v))$. Then $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface, where

with

and the basic invariants

Proof. Since ${\boldsymbol{x}}_u(u, v) = (1, f_u(u, v), g_u(u, v)), \; {\boldsymbol{x}}_v(u, v) = (0, f_v(u, v), g_v(u, v)),$ we have

By ${\boldsymbol{x}}_u(u, v)\neq0$ and ${\boldsymbol{x}}_u(u, v) \cdot \nu(u, v) = 0$ for all $(u, v) \in U$, we have that the components of $\nu(u, v)$ are linearly dependent. Therefore, ${\boldsymbol{x}}$ is a generalised framed base surface by Theorem 3.10. By direct calculation, we have the basic invariants. □

Example 4.2 (Cross cap). Let $({\boldsymbol{x}}, \nu_1, \nu_2): (\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$,

Note that ${\boldsymbol{x}}$ at $0$ is a cross cap singular point (cf. ^[21]). Then $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface germ. By $f(u, v) = v^2, \ g(u, v) = uv$ in Theorem 4.1, we have $\alpha(u, v) = u, \ \beta(u, v) = -2v\sqrt{1+v^2}$ and the basic invariants

Example 4.3 ($S^\pm_1$ singular point). Let $({\boldsymbol{x}}, \nu_1, \nu_2): (\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$,

Note that ${\boldsymbol{x}}$ at $0$ is a $S^\pm_1$ singular point (cf. ^[14,17]). Then $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface germ. By $f(u, v) = v^2, \ g(u, v) = u^2v \pm v^3$ in Theorem 4.1, we have $\alpha(u, v) = u^2 \pm 3v^2, \ \beta(u, v) = -2v\sqrt{1+4u^2v^2}$ and the basic invariants

Remark 4.4. The $\mathcal{A}$-simple singularities of a map from a 2-dimensional manifold to a 3-dimensional one are also of corank one; see ^[14]. We can treat them as generalised framed surfaces.

We say that ${\boldsymbol{x}}$ at $p$ is a cross cap singular point (respectively, $S^\pm_1$ singular point) if ${\boldsymbol{x}}$ at $p$ is $\mathcal{A}$-equivalent (that is, right-left equivalent) to $(u, v) \mapsto (u, v^2, uv)$ (respectively, $(u, v) \mapsto (u, v^2, v(u^2 \pm v^2)))$. By using the criteria of cross cap and $S^{\pm}_1$ singular points, we have the following:

Proposition 4.5. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface which is given by the form of Theorem 4.1. Suppose that $p$ is a singular point of ${\boldsymbol{x}}$; that is, $a_2(p) = b_2(p) = c_2(p) = 0$. Then we have the following:

(1) ${\boldsymbol{x}}$ at $p$ is a cross cap singular point if and only if $-f_2 b_{2v}+g_2a_{2v}\not = 0$ at $p$.

(2) ${\boldsymbol{x}}$ at $p$ is a $S^+_1$ singular point if and only if $a_{2u}b_{2v}-b_{2u}a_{2v} = 0$, $(a_{2v}, b_{2v}) \not = (0, 0)$ and $H < 0$ at $p$, where

(3) ${\boldsymbol{x}}$ at $p$ is a $S^-_1$ singular point if and only if $a_{2u}b_{2v}-b_{2u}a_{2v} = 0$ and $H > 0$ at $p$.

Proof. Since $f_v(p) = g_v(p) = 0$, we have $a_2(p) = b_2(p) = c_2(p) = 0$. Note that $c_1(p) \not = 0$.

(1) By ${\boldsymbol{x}}_u = c_1 \nu_3, \ {\boldsymbol{x}}_v = a_2\nu_1+b_2\nu_2+c_2\nu_3$,

we have ${\rm det}({\boldsymbol{x}}_u, {\boldsymbol{x}}_{uv}, {\boldsymbol{x}}_{vv})(p) = c^2_1(p)(-f_2(p)b_{2v}(p)+g_2(p)a_{2v}(p))$. By the criterion of the cross cap singular point in ^[21], we have the result.

(2) We denote $\varphi = {\rm det}({\boldsymbol{x}}_u, {\boldsymbol{x}}_v, {\boldsymbol{x}}_{vv})$. By the calculation of (1) and

we have $\varphi = c_1(a_2(a_2e_2+b_{2v}-c_2g_2)-b_2(a_{2v}-b_2e_2-c_2f_2))$. It follows that

Therefore, $\varphi_u(p) = c_1(p)(a_{2u}(p)b_{2v}(p)-b_{2u}(p)a_{2v}(p))$ and $\varphi_v(p) = 0$. By the integrability condition (3.1) of the generalised framed surface, $-c_1(p)f_2(p) = a_{2u}(p), -c_1(p)g_2(p) = b_{2u}(p)$, and $c_{1v}(p) = c_{2u}(p)$. Hence $d\varphi(p) = 0$ if and only if $a_{2u}(p)b_{2v}(p)-b_{2u}(p)a_{2v}(p) = 0$ (equivalently, $f_2(p)b_{2v}(p)-g_2(p)a_{2v}(p) = 0$). By a direct calculation, we have

at $p$. By the criterion of the $S^{\pm}_1$ singular point in ^[17], ${\boldsymbol{x}}$ at $p$ is a $S^+_1$ singular point if and only if $d\varphi(p) = 0$, ${\boldsymbol{x}}_{u}$ and ${\boldsymbol{x}}_{vv}$ are linearly independent at $p$, and $H = \varphi_{uu}(p)\varphi_{vv}(p)-\varphi^2_{uv}(p) < 0$. We have the result.

(3) ${\boldsymbol{x}}$ at $p$ is a $S^-_1$ singular point if and only if $d\varphi(p) = 0$ and $H = \varphi_{uu}(p)\varphi_{vv}(p)-\varphi^2_{uv}(p) > 0$. We have the result. □

Remark 4.6. By the integrability condition (3.1), the condition $-f_2 b_{2v}+g_2a_{2v}\not = 0$ at $p$ in Proposition 4.5 (1) is equivalent to the condition $a_{2u}b_{2v}-b_{2u}a_{2v} \not = 0$ at $p$. That is, $d\varphi(p) \not = 0$ (cf. ^[17]).

5.   Corank two singularities

Let ${\boldsymbol{x}}:U \to \mathbb{R}^3$ be a smooth mapping. Suppose that corank$(d {\boldsymbol{x}}) = 2$ at a point $p \in U$. We consider one of the components of ${\boldsymbol{x}}(u, v)$ to be $2$-jet; that is, by using parameter change and up to sign, ${\boldsymbol{x}}(u, v)$ is given by

where $f, g:U \to \mathbb{R}$ are smooth functions. By direct calculation, $\nu(u, v)$ is given by

respectively. By Theorem 3.10, ${\boldsymbol{x}}$ is a generalised framed base surface at least locally if and only if the components of $\nu(u, v)$ are linearly dependent.

As special cases, we consider two of the components of ${\boldsymbol{x}}(u, v)$ to be $2$-jet.

Proposition 5.1. Let ${\boldsymbol{x}}: (\mathbb{R}^2, 0) \to \mathbb{R}^3$ be given by

and $j^2g(0) = 0$. Then we have the following:

(1) ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a generalised framed base surface germ if and only if there exists a function $h:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh$ or $g_v = vh$.

(2) Suppose that ${\boldsymbol{x}}$ is a generalised framed base surface germ. Then ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a framed base surface germ if and only if there exist functions $h_1, h_2:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh_1$ and $g_v = vh_2$.

Proof. (1) We show the sufficient part of the proposition. Since

we have $\nu(u, v) = \big(ug_v(u, v)+vg_u(u, v), vg_u(u, v)-ug_v(u, v), -2uv\big).$ If there exists a function $h:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh$ or $g_v = vh$, then we have

or

for all $(u, v) \in (\mathbb{R}^2, 0)$. It follows that the components of $\nu(u, v)$ are linearly dependent. By Theorem 3.10, ${\boldsymbol{x}}$ is a generalised framed base surface germ. In fact, we can take

with

$i = 1, 2$. Therefore, $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface germ.

Conversely, if ${\boldsymbol{x}}$ is a generalised framed surface germ, we have $ug_v+vg_u, vg_u-ug_v$ and $-2uv$ are linearly dependent. Then there exist functions $k_1, k_2, k_3:(\mathbb{R}^2, 0) \to \mathbb{R}$ with $(k_1, k_2, k_3) \neq 0$ at $0$, such that $k_1(ug_v+vg_u)+k_2(vg_u-ug_v)+k_3(-2uv) = 0$. By $j^2g(0) = 0$, we have $k_1(0) \neq 0$ or $k_2(0) \neq 0$. Without loss of generality, we assume $k_1(0)\neq0$. Then we have

for all $(u, v) \in (\mathbb{R}^2, 0)$. It follows that $v\left(1+{k_2(0, v)}/{k_1(0, v)} \right)g_u(0, v) = 0$ for all $v \in (\mathbb{R}, 0)$ and $u\left(1-{k_2(u, 0)}/{k_1(u, 0)}\right)g_v(u, 0) = 0$ for all $u \in (\mathbb{R}, 0)$. If $1+{k_2(0, 0)}/{k_1(0, 0)} \neq 0$, we have $vg_u(0, v) = 0$. By the continuous condition, $g_u(0, v) = 0$ for all $v \in (\mathbb{R}, 0)$. Thus, there exists a function $h$, such that $g_u(u, v) = uh(u, v)$. If $1-{k_2(0, 0)}/{k_1(0, 0)} \neq 0$, we have $ug_v(u, 0) = 0$. By the continuous condition, $g_v(u, 0) = 0$ for all $u \in (\mathbb{R}, 0)$. Thus, there exists a function $h:(\mathbb{R}^2, 0)\rightarrow \mathbb{R}$, such that $g_v(u, v) = vh(u, v)$.

(2) We show the sufficient part of the proposition. If there exist functions $h_1, h_2:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh_1$ and $g_v = vh_2$, then $vg_u-ug_v = uv(h_1-h_2)$. It follows that $\alpha$ and $\beta$ are linearly dependent from Eq (5.1). Since the set of regular points of ${\boldsymbol{x}}$ is dense, according to Theorem 3.11 (2), ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a framed base surface germ.

Conversely, since ${\boldsymbol{x}}$ is a generalised framed base surface germ, we have $g_u = uh_1$ or $g_v = vh_1$. Without loss of generality, we assume $g_u = uh_1$. By Theorem 3.11 (1), $\alpha$ and $\beta$ are linearly dependent. Then there exists a function $k:(\mathbb{R}^2, 0) \to \mathbb{R}$, such that $vg_u(u, v)-ug_v(u, v) = k(u, v) uv$. It follows that

Thus, $ug_v(u, 0) = 0$ for all $u \in (\mathbb{R}, 0)$. Then there exists a function $h_2:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_v(u, v) = vh_2(u, v)$ for all $(u, v) \in (\mathbb{R}^2, 0)$. □

Proposition 5.2. Let ${\boldsymbol{x}}: (\mathbb{R}^2, 0) \to \mathbb{R}^3$ be given by

and $j^2g(0) = 0$. Then we have the following:

(1) ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a generalised framed base surface germ if and only if there exists a function $h:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh$ or $g_v = vh$.

(2) Suppose that ${\boldsymbol{x}}$ is a generalised framed base surface germ. Then ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a framed base surface germ if and only if there exist functions $h_1, h_2:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh_1$ and $g_v = vh_2$.

Proof. (1) We show the sufficient part of the proposition. Since

we have $\nu(u, v) = \big(-vg_u(u, v), -ug_v(u, v), uv\big).$ If there exists a function $h:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh$, we have $-vg_u(u, v)+h(u, v)uv = 0.$ It follows that the components of $\nu(u, v)$ are linearly dependent. By Theorem 3.10, ${\boldsymbol{x}}$ is a generalised framed base surface germ. In fact, we can take

with

Therefore, $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface germ.

If there exists a function $h:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_v = vh$, we have $-ug_v(u, v)+h(u, v)uv = 0.$ It follows that the components of $\nu(u, v)$ are linearly dependent. By Theorem 3.10, ${\boldsymbol{x}}$ is a generalised framed base surface germ. In fact, we can take

with

Therefore, $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface germ.

Conversely, if ${\boldsymbol{x}}$ is a generalised framed surface germ, we have $-vg_u, -ug_v$ and $uv$ are linearly dependent. Then there exist functions $(k_1, k_2, k_3)\neq0$ at $0$, such that $k_1(-vg_u)+k_2(-ug_v)+k_3uv = 0$. By $j^2g(0) = 0$, we have $k_1(0)\neq0$ or $k_2(0)\neq0$. If $k_1(0)\neq0$, we have

for all $(u, v)\in(\mathbb{R}^2, 0)$. It follows that $vg_u(0, v) = 0$ for all $v\in(\mathbb{R}, 0)$. Thus, there exists a function $h:(\mathbb{R}^2, 0)\rightarrow \mathbb{R}$, such that $g_u(u, v) = uh(u, v)$. If $k_2(0)\neq0$, we have

for all $(u, v)\in(\mathbb{R}^2, 0)$. It follows that $ug_v(u, 0) = 0$ for all $u\in(\mathbb{R}, 0)$. Thus, there exists a function $h:(\mathbb{R}^2, 0)\rightarrow \mathbb{R}$, such that $g_v(u, v) = vh(u, v)$.

(2) We show the sufficient part of the proposition. If there exist functions $h_1, h_2:(\mathbb{R}^2, 0) \to \mathbb{R}$ such that $g_u = uh_1$ and $g_v = vh_2$, then $\alpha(u, v) = -uvh_2(u, v)$ or $\alpha(u, v) = -uv(h_1(u, v)+h_2(u, v))$. It follows that $\alpha$ and $\beta$ are linearly dependent by Eqs (5.2) and (5.3). Since the set of regular points of ${\boldsymbol{x}}$ is dense, according to Theorem 3.11 (2), ${\boldsymbol{x}}:(\mathbb{R}^2, 0) \to \mathbb{R}^3$ is a framed base surface germ.

Conversely, since ${\boldsymbol{x}}$ is a generalised framed base surface germ, we have $g_u = uh_1$ or $g_v = vh_1$. Without loss of generality, we assume $g_u = uh_1$. By Theorem 3.11 (1), $\alpha$ and $\beta$ are linearly dependent. Then there exists a function $k:(\mathbb{R}^2, 0)\rightarrow \mathbb{R}$, such that $ug_v(u, v) = k(u, v)uv$ by Eq (5.2). Thus, $ug_v(u, 0) = 0$ for all $u\in(\mathbb{R}, 0)$. Then there exists a function $h_2:(\mathbb{R}^2, 0)\rightarrow \mathbb{R}$, such that $g_v(u, v) = vh_2(u, v)$ for all $(u, v)\in(\mathbb{R}^2, 0)$. □

Example 5.3. Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be

where $k$ is a natural number. Note that $0$ is a corank two singular point of ${\boldsymbol{x}}$. By $g(u, v) = u^{k+2}v$ in Proposition 5.2 (1), $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface germ with

and the basic invariants

By Proposition 5.2 (2), ${\boldsymbol{x}}$ is not a framed base surface germ.

6.   Parallel surfaces of generalised framed surface

Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with $\nu = \alpha\nu_1+\beta\nu_2$ and basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$. We consider parallel surfaces of the generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2)$.

Definition 6.1. We say that ${\boldsymbol{x}}^\lambda:U \to \mathbb{R}^3$, ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda\nu$ is a parallel surface of the generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2)$, where $\lambda$ is a non-zero constant.

Remark 6.2. If there exist functions $\widetilde{\alpha}, \widetilde{\beta}, \ell:U \to \mathbb{R}$, such that $\alpha = \ell\widetilde{\alpha}, \beta = \ell\widetilde{\beta}$, then we take $\nu = \ell(\widetilde{\alpha}\nu_1+\widetilde{\beta}\nu_2) = \ell \widetilde{\nu}$. In this case, we consider the parallel surface as ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda \widetilde{\nu}$. Moreover, if the set of regular points of ${\boldsymbol{x}}$ is dense in $U$ and, $\alpha$ and $\beta$ are linearly dependent, then there exists a function $\widetilde{\ell}$ such that $\nu = \widetilde{\ell} {\boldsymbol{n}}$, where $({\boldsymbol{x}}, {\boldsymbol{n}}, {\boldsymbol{s}})$ is a framed surface (cf. Theorem 3.11). In this case, the parallel surface is given by ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda {\boldsymbol{n}}$.

Since ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda\nu$, we have

It follows that

By Theorem 3.10, ${\boldsymbol{x}}^\lambda$ is a generalised framed base surface at least locally if and only if the components of $\nu^\lambda$ are linearly dependent.

By Remark 6.2, if $k:U \to \mathbb{R}$ is a non-zero function, then $\alpha = \ell \widetilde{\alpha} = (\ell/k)k\widetilde{\alpha}, \beta = \ell \widetilde{\beta} = (\ell/k)k\widetilde{\beta}$. Therefore, $\widetilde{\nu}$ is not unique, that is, ${\boldsymbol{x}}^\lambda[k] = {\boldsymbol{x}}+\lambda k \widetilde{\nu}$.

Proposition 6.3. Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, p) \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with $k\widetilde{\nu} = k\widetilde{\alpha}\nu_1+k\widetilde{\beta}\nu_2$ and basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$. Suppose that $(\widetilde{\alpha}, \widetilde{\beta})(p) = 0$. Then we have the following:

(1) If $(a_1, a_2)(p) = (b_1, b_2)(p) = 0$ and $(\widetilde{\alpha}_u, \widetilde{\alpha}_v, \widetilde{\beta}_u, \widetilde{\beta}_v)(p)\neq0$, then ${\boldsymbol{x}}^\lambda[k]$ is also a generalised framed base surface around $p$.

(2) If $(c_1, c_2)(p)\neq0$, then ${\boldsymbol{x}}^\lambda[k]$ is also a generalised framed base surface around $p$.

Proof. Since ${\boldsymbol{x}}^\lambda[k] = {\boldsymbol{x}}+\lambda k \widetilde{\nu}$, we have

If $(\widetilde{\alpha}, \widetilde{\beta})(p) = 0$, $(a_1, a_2)(p) = (b_1, b_2)(p) = 0$ and $(\widetilde{\alpha}_u, \widetilde{\alpha}_v, \widetilde{\beta}_u, \widetilde{\beta}_v)(p)\neq0$, or $(\widetilde{\alpha}, \widetilde{\beta})(p) = 0$ and $(c_1, c_2)(p)\neq0$, then $({\boldsymbol{x}}^\lambda[k]_u, {\boldsymbol{x}}^\lambda[k]_v)(p)\neq0$ by the Eqs (6.3) and (6.4). It follows that the components of $\nu^\lambda[k]$ are linearly dependent by ${\boldsymbol{x}}^\lambda[k]_u(u, v)\cdot\nu^\lambda[k](u, v) = 0$ and ${\boldsymbol{x}}^\lambda[k]_v(u, v)\cdot\nu^\lambda[k](u, v) = 0$ for all $(u, v)\in(\mathbb{R}^2, p)$. By Theorem 3.10, ${\boldsymbol{x}}^\lambda[k]$ is a generalised framed base surface around $p$. □

Proposition 6.4. Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface germ, which is given by the form of Theorem 4.1. Suppose that $f_u(0) = f_v(0) = g_u(0) = g_v(0) = 0$. Then $({\boldsymbol{x}}^\lambda, \nu_1^\lambda, \nu_2^\lambda):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is also a generalised framed surface germ, where

with

Proof. Since ${\boldsymbol{x}}^\lambda = \big(u+\lambda(f_ug_v-f_vg_u), f-\lambda g_{v}, g+\lambda f_{v}\big)$, we have

Then

Since $f_u(0) = f_v(0) = g_u(0) = g_v(0) = 0$, we have ${\boldsymbol{x}}_u^\lambda(0) = (1, -\lambda g_{uv}(0), \lambda f_{uv}(0))$. It follows that the components of $\nu^\lambda$ are linearly dependent by ${\boldsymbol{x}}^\lambda_u(u, v)\cdot\nu^\lambda(u, v) = 0$ for all $(u, v)\in(\mathbb{R}^2, 0)$. By Theorem 3.10, ${\boldsymbol{x}}^\lambda$ is a generalised framed base surface germ. □

Corollary 6.5. Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface germ, which is given by the form of Theorem 4.1. Suppose that the set of regular points of ${\boldsymbol{x}}^\lambda$ is dense in $U$. Under the same assumptions in Proposition 6.4, we have the following:

(1) If $g_{vv}(0)\neq0$, then ${\boldsymbol{x}}^\lambda$ is a framed base surface germ.

(2) If $g_{vv}(0) = 0$ and $f_{vv}(0)\neq0$, then ${\boldsymbol{x}}^\lambda$ is a framed base surface germ.

Proof. By Proposition 6.4, we have

If $g_{vv}(0)\neq0$, then $\beta^\lambda(0)\neq0$. It follows that $\alpha^\lambda$ and $\beta^\lambda$ are linearly dependent around $0$. Also, if $g_{vv}(0) = 0$ and $f_{vv}(0)\neq0$, then $\alpha^\lambda(0)\neq0$. It follows that $\alpha^\lambda$ and $\beta^\lambda$ are linearly dependent around $0$. By Theorem 3.11 (2), ${\boldsymbol{x}}^\lambda$ is a framed base surface germ. □

We define the other type of parallel surfaces of the generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2)$.

Definition 6.6. We say that ${\boldsymbol{x}}^\lambda[\theta]:U \to \mathbb{R}^3$, ${\boldsymbol{x}}^\lambda[\theta] = {\boldsymbol{x}}+\lambda (\cos \theta \nu_1+\sin \theta \nu_2)$ is a $\theta$-parallel surface of the generalised framed surface $({\boldsymbol{x}}, \nu_1, \nu_2)$, where $\lambda$ is a non-zero constant and $\theta$ is a constant.

Remark 6.7. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$. If $a_1 = a_2 = 0$ (respectively, $b_1 = b_2 = 0$), then $({\boldsymbol{x}}, \nu_1, \nu_2)$ (respectively, $({\boldsymbol{x}}, \nu_2, \nu_1)$) is a framed surface by Proposition 3.9. That is ${\boldsymbol{n}} = \nu_1$ (respectively, ${\boldsymbol{n}} = \nu_2$). It follows that ${\boldsymbol{x}}^\lambda[0] = {\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda {\boldsymbol{n}}$ (respectively, ${\boldsymbol{x}}^\lambda[\pi/2] = {\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda {\boldsymbol{n}}$).

Since ${\boldsymbol{x}}^\lambda[\theta] = {\boldsymbol{x}}+\lambda (\cos \theta \nu_1+\sin \theta \nu_2)$, we have

It follows that

By Theorem 3.10, ${\boldsymbol{x}}^\lambda[\theta]$ is a generalised framed base surface at least locally if and only if the components of $\nu^\lambda[\theta]$ are linearly dependent.

Proposition 6.8. Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface germ, which is given by the form of Theorem 4.1. Suppose that $f_u(0) = f_v(0) = g_u(0) = g_v(0) = 0$. Then we have the following:

(1) If $f_{uu}(0)\cos\theta +g_{uu}(0)\sin \theta = 0$, then ${\boldsymbol{x}}^\lambda[\theta]$ is a generalised framed base surface germ.

(2) If $f_{uv}(0)\cos\theta +g_{uv}(0)\sin \theta \neq0$, then ${\boldsymbol{x}}^\lambda[\theta]$ is a generalised framed base surface germ.

Proof. Since ${\boldsymbol{x}}^\lambda[\theta] = {\boldsymbol{x}}+\lambda(\cos\theta\nu_1+\sin\theta\nu_2),$ we have

It follows that

If $f_{uu}(0)\cos\theta+ g_{uu}(0)\sin \theta = 0$ or $f_{uv}(0)\cos\theta+ g_{uv}(0)\sin \theta\neq0$, then 0 is a corank one singular point of ${\boldsymbol{x}}^\lambda[\theta]$. By Proposition 3.14 and Theorem 4.1, ${\boldsymbol{x}}^\lambda[\theta]$ is a generalised framed base surface germ. □

We consider special cases where $\nu_2$ or $\nu_1$ is a constant. If $\nu_2$ (respectively, $\nu_1$) is a constant, then we denote ${\boldsymbol{x}}^\lambda_1 = {\boldsymbol{x}}^\lambda[0] = {\boldsymbol{x}}+\lambda \nu_1$ (respectively, ${\boldsymbol{x}}^\lambda_2 = {\boldsymbol{x}}^\lambda[\pi/2] = {\boldsymbol{x}}+\lambda\nu_2$).

Proposition 6.9. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface with $\nu = \alpha\nu_1+\beta\nu_2$ and basic invariants $(\mathcal{G}, \mathcal{F}_1, \mathcal{F}_2)$.

(1) If $\nu_2$ is a constant, then $({\boldsymbol{x}}^\lambda_1, \nu_1, \nu_2)$ is also a generalised framed surface with $\widetilde{\alpha}^\lambda = \alpha+\lambda(b_1f_2-b_2f_1), \ \widetilde{\beta}^\lambda = \beta+\lambda(a_2f_1-a_1f_2)$ and the basic invariants

(2) If $\nu_1$ is a constant, then $({\boldsymbol{x}}^\lambda_2, \nu_1, \nu_2)$ is also a generalised framed surface with $\widetilde{\alpha}^\lambda = \alpha+\lambda(b_1g_2-b_2g_1), \ \widetilde{\beta}^\lambda = \beta+\lambda(a_2g_1-a_1g_2)$ and the basic invariants

Proof. (1) Since $\nu_2$ is a constant and ${\boldsymbol{x}}^\lambda_1 = {\boldsymbol{x}}+\lambda \nu_1$, we have

By a direct calculation, we have

It follows that $({\boldsymbol{x}}^\lambda_1, \nu_1, \nu_2)$ is also a generalised framed surface with $\widetilde{\alpha}^\lambda = \alpha+\lambda(b_1f_2-b_2f_1), \ \widetilde{\beta}^\lambda = \beta+\lambda(a_2f_1-a_1f_2)$. By a direct calculation, we have the basic invariants.

(2) Since $\nu_1$ is a constant and ${\boldsymbol{x}}^\lambda_2 = {\boldsymbol{x}}+\lambda \nu_2$, we have

By a direct calculation, we have

It follows that $({\boldsymbol{x}}^\lambda_2, \nu_1, \nu_2)$ is also a generalised framed surface with $\widetilde{\alpha}^\lambda = \alpha+\lambda(b_1g_2-b_2g_1), \ \widetilde{\beta}^\lambda = \beta+\lambda(a_2g_1-a_1g_2)$. By a direct calculation, we have the basic invariants. □

We give a relation between parallel surfaces ${\boldsymbol{x}}^\lambda$ and $\theta$-parallel surfaces ${\boldsymbol{x}}^\lambda[\theta]$.

Proposition 6.10. Let $({\boldsymbol{x}}, \nu_1, \nu_2):U \to \mathbb{R}^3 \times \Delta$ be a generalised framed surface, and $\nu = \alpha \nu_1+\beta \nu_2$.

(1) If ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}^\lambda[\theta]$, then ${\boldsymbol{x}}$ is a regular surface.

(2) Suppose that the set of regular points of ${\boldsymbol{x}}$ is dense in $U$ and $\alpha = \ell \widetilde{\alpha}, \beta = \ell \widetilde{\beta}$. If ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}^\lambda[\theta]$, where ${\boldsymbol{x}}^\lambda = {\boldsymbol{x}}+\lambda \widetilde{\nu}$, then ${\boldsymbol{x}}$ is a framed base surface at least locally.

Proof. (1) By assumption, $\alpha = \cos \theta$ and $\beta = \sin \theta$. Since $(\alpha, \beta) \not = (0, 0)$, ${\boldsymbol{x}}$ is a regular surface and hence a framed base surface.

(2) By assumption, $\widetilde{\alpha} = \cos \theta$ and $\widetilde{\beta} = \sin \theta$. Then $\alpha$ and $\beta$ are linearly dependent. By Theorem 3.11 (2), ${\boldsymbol{x}}$ is a framed base surface at least locally. □

By Proposition 6.10 (2), if $\ell(u_0, v_0) = 0$, then $(u_0, v_0)$ is a singular point of ${\boldsymbol{x}}$. The relation between two parallel surfaces ${\boldsymbol{x}}^\lambda$ and ${\boldsymbol{x}}^\lambda[\theta]$ measures not only regular surfaces, but also framed base surfaces.

Example 6.11 (Cross cap). Let $({\boldsymbol{x}}, \nu_1, \nu_2): (\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$,

Then $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface germ with $\alpha(u, v) = u, \beta(u, v) = -2v\sqrt{1+v^2}$, see Example 4.2.

We consider the parallel surfaces of $({\boldsymbol{x}}, \nu_1, \nu_2)$. Let $({\boldsymbol{x}}^\lambda, \nu_1^\lambda, \nu_2^\lambda):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be

Then $({\boldsymbol{x}}^\lambda, \nu_1^\lambda, \nu_2^\lambda)$ is a generalised framed surface germ with

In fact, $({\boldsymbol{x}}^\lambda, {\boldsymbol{n}}, {\boldsymbol{s}}):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a framed surface germ, where

The curvature of $({\boldsymbol{x}}^\lambda, {\boldsymbol{n}}, {\boldsymbol{s}})$ is given by

Moreover, ${\boldsymbol{x}}^\lambda$ is a regular surface germ, since ${J^F}^\lambda(0) = \sqrt{4\lambda^2(1+\lambda^2)}\neq0$.

Next, we consider $\theta$-parallel surface of $({\boldsymbol{x}}, \nu_1, \nu_2)$. Since $\nu_1$ is a constant, we consider

By a direct calculation, we have

It follows that $({\boldsymbol{x}}_2^\lambda, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface germ with

and the basic invariants

Moreover, we can see that the functions $\widetilde{\alpha}^\lambda$ and $\widetilde{\beta}^\lambda$ are not linearly dependent. By Theorem 3.11 (1), ${\boldsymbol{x}}_2^\lambda$ is not a framed base surface germ.

Example 6.12 ($H_k$ singular point). Let $({\boldsymbol{x}}, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be

where $k$ is a natural number with $k \ge 2$. Note that $0$ is a $H_k$ singular point of ${\boldsymbol{x}}$ (cf. ^[14]). Then $({\boldsymbol{x}}, \nu_1, \nu_2)$ is a generalised framed surface germ with

and the basic invariants

We consider the parallel surface of $({\boldsymbol{x}}, \nu_1, \nu_2)$. Let $({\boldsymbol{x}}^\lambda, \nu_1^\lambda, \nu_2^\lambda):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ be

Note that $0$ is a corank one singular point of ${\boldsymbol{x}}^\lambda$. Then $({\boldsymbol{x}}^\lambda, \nu_1^\lambda, \nu_2^\lambda)$ is a generalised framed surface germ with

and the basic invariants

Moreover, we can see that the functions $\alpha^\lambda$ and $\beta^\lambda$ are not linearly dependent. By Theorem 3.11 (1), ${\boldsymbol{x}}^\lambda$ is not a framed base surface germ.

Next, we consider $\theta$-parallel surface of $({\boldsymbol{x}}, \nu_1, \nu_2)$. Since $\nu_2$ is a constant, we consider

By a direct calculation, we have

It follows that $({\boldsymbol{x}}_1^\lambda, \nu_1, \nu_2):(\mathbb{R}^2, 0) \to \mathbb{R}^3 \times \Delta$ is a generalised framed surface germ with

and the basic invariants

Moreover, we can see that the functions $\widetilde{\alpha}^\lambda$ and $\widetilde{\beta}^\lambda$ are not linearly dependent. By Theorem 3.11 (1), ${\boldsymbol{x}}_1^\lambda$ is not a framed base surface germ.

Author contributions

Writing-original draft preparation, M.T. and H.Y.; writing-review and editing, M.T. and H.Y.; All authors equally contributed to this work. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank Professor Kentaro Saji for valuable discussions. The first author was partially supported by JSPS KAKENHI Grant Number JP 20K03573. The second author was partially supported by the Natural Science Foundation of Jilin Province, China, Grant Number JJKH20230181KJ.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.