Framed Surfaces in the Euclidean Space

Abstract

A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves.

Appendices

Framed Curves in the Euclidean Space

We quickly review on the theory of framed curves in the Euclidean space, see detail Honda and Takahashi (2016).

A framed curve in the Euclidean space is a smooth curve with a moving frame. We say that $(\gamma ,\nu _1,\nu _2):I \rightarrow \mathbb {R}^3 \times \varDelta$ is a framed curve if $\dot{\gamma }(t) \cdot \nu _1(t)=0$ and $\dot{\gamma }(t) \cdot \nu _2(t)=0$ for all $t \in I$. We say that $\gamma :I \rightarrow \mathbb {R}^3$ is a framed base curve if there exists $(\nu _1,\nu _2):I \rightarrow \varDelta$ such that $(\gamma ,\nu _1,\nu _2)$ is a framed curve.

We put $\mu (t) = \nu _1(t) \times \nu _2(t)$. Then $\{ \nu _1(t),\nu _2(t),\mu (t) \}$ is a moving frame along the framed base curve $\gamma (t)$ in $\mathbb {R}^3$ and we have the Frenet–Serret type formula,

$$\begin{aligned} \left( \begin{array}{c} \dot{\nu _1}(t)\\ \dot{\nu _2}(t)\\ \dot{\mu }(t) \end{array} \right) = \left( \begin{array}{ccc} 0 &{}\quad \ell (t) &{}\quad m(t)\\ -\ell (t) &{}\quad 0 &{}\quad n(t)\\ -m(t) &{}\quad -n(t) &{}\quad 0 \end{array}\right) \left( \begin{array}{c} \nu _1(t)\\ \nu _2(t)\\ \mu (t) \end{array}\right) , \ \dot{\gamma }(t)=\alpha (t)\mu (t) \end{aligned}$$

where $\ell (t) = \dot{\nu _1}(t) \cdot \nu _2(t)$, $m(t) = \dot{\nu _1}(t) \cdot \mu (t), n(t) = \dot{\nu _2}(t) \cdot \mu (t)$ and $\alpha (t)=\dot{\gamma }(t) \cdot \mu (t)$. We call the functions $(\ell ,m,n,\alpha )$the curvature of the framed curve. Note that $t_0$ is a singular point of $\gamma$ if and only if $\alpha (t_0) = 0$.

Definition 4

Let $(\gamma ,\nu _1,\nu _2)$ and $(\widetilde{\gamma },\widetilde{\nu }_1,\widetilde{\nu }_2):I \rightarrow \mathbb {R}^3 \times \varDelta$ be framed curves. We say that $(\gamma ,\nu _1,\nu _2)$ and $(\widetilde{\gamma },\widetilde{\nu }_1,\widetilde{\nu }_2)$ are congruent as framed curves if there exist a constant rotation $A \in SO(3)$ and a translation $a\in \mathbb {R}^3$ such that $\widetilde{\gamma }(t) = A(\gamma (t)) +a$, $\widetilde{\nu _1}(t) = A(\nu _1(t))$ and $\widetilde{\nu _2}(t) = A(\nu _2(t))$ for all $t \in I$.

Theorem 7

(The Existence Theorem for framed curves, Honda and Takahashi 2016) Let $(\ell ,m,n,\alpha ):I \rightarrow \mathbb {R}^4$ be a smooth mapping. There exists a framed curve $(\gamma ,\nu _1,\nu _2):I \rightarrow \mathbb {R}^3 \times \varDelta$ whose curvature of the framed curve is $(\ell ,m,n,\alpha )$.

Theorem 8

(The Uniqueness Theorem for framed curves, Honda and Takahashi 2016) Let $(\gamma ,\nu _1,\nu _2)$ and $(\widetilde{\gamma },\widetilde{\nu }_1,\widetilde{\nu }_2):I \rightarrow \mathbb {R}^3 \times \varDelta$ be framed curves with the curvature $(\ell ,m,n,\alpha )$ and $(\widetilde{\ell },\widetilde{m},\widetilde{n},\widetilde{\alpha })$, respectively. Then $(\gamma ,\nu _1,\nu _2)$ and $(\widetilde{\gamma },\widetilde{\nu }_1,\widetilde{\nu }_2)$ are congruent as framed curves if and only if the curvatures $(\ell , m, n, \alpha )$ and $(\widetilde{\ell }, \widetilde{m}, \widetilde{n}, \widetilde{\alpha })$ coincide.

Legendre Curves in the Euclidean Plane

We quickly review on the theory of Legendre curves in the unit tangent bundle over $\mathbb {R}^2$, see detail Fukunaga and Takahashi (2013).

We say that $(\gamma ,\nu ):I \rightarrow \mathbb {R}^2 \times S^1$ is a Legendre curve if $(\gamma ,\nu )^*\theta =0$ for all $t \in I$, where $\theta$ is a canonical contact form on the unit tangent bundle $T_1 \mathbb {R}^2=\mathbb {R}^2 \times S^1$ over $\mathbb {R}^2$ (cf. Arnol’d 1990; Arnol’d et al. 1986). This condition is equivalent to $\dot{\gamma }(t) \cdot \nu (t)=0$ for all $t \in I$. We say that $\gamma :I \rightarrow \mathbb {R}^2$ is a frontal if there exists $\nu :I \rightarrow S^1$ such that $(\gamma ,\nu )$ is a Legendre curve. Examples of Legendre curves see Ishikawa (2007), Ishikawa (2015). We denote $J(a) = (-a_2, a_1)$ the anticlockwise rotation by $\pi /2$ of a vector $a= (a_1, a_2) \in \mathbb {R}^2$. We put $\mu (t)=J(\nu (t))$. Then $\{\nu (t), \mu (t) \}$ is a moving frame of a frontal $\gamma (t)$ in $\mathbb {R}^2$ and we have the Frenet type formula,

$$\begin{aligned} \left( \begin{array}{c} \dot{\nu }(t)\\ \dot{\mu }(t) \end{array} \right) = \left( \begin{array}{cc} 0 &{}\quad \ell (t)\\ -\ell (t) &{}\quad 0 \end{array} \right) \left( \begin{array}{c} \nu (t)\\ \mu (t) \end{array} \right) , \ \dot{\gamma }(t) = \beta (t) \mu (t), \end{aligned}$$

where $\ell (t)=\dot{\nu }(t) \cdot \mu (t)$ and $\beta (t)=\dot{\gamma }(t) \cdot \mu (t)$. We call the pair $(\ell ,\beta )$the curvature of the Legendre curve.

Definition 5

Let $(\gamma ,\nu )$ and $(\widetilde{\gamma },\widetilde{\nu }):I \rightarrow \mathbb {R}^2 \times S^1$ be Legendre curves. We say that $(\gamma ,\nu )$ and $(\widetilde{\gamma },\widetilde{\nu })$ are congruent as Legendre curves if there exist a constant rotation $A \in SO(2)$ and a translation $a\in \mathbb {R}^2$ such that $\widetilde{\gamma }(t)=A(\gamma (t))+a$ and $\widetilde{\nu }(t)=A (\nu (t))$ for all $t \in I$.

Theorem 9

(The Existence Theorem for Legendre curves, Fukunaga and Takahashi 2013) Let $(\ell ,\beta ):I \rightarrow \mathbb {R}^2$ be a smooth mapping. There exists a Legendre curve $(\gamma ,\nu ):I \rightarrow \mathbb {R}^2 \times S^1$ whose curvature of the Legendre curve is $(\ell , \beta )$.

Theorem 10

(The Uniqueness Theorem for Legendre curves, Fukunaga and Takahashi 2013) Let $(\gamma ,\nu )$ and $(\widetilde{\gamma },\widetilde{\nu }):I \rightarrow \mathbb {R}^2 \times S^1$ be Legendre curves with the curvatures of Legendre curves $(\ell ,\beta )$ and $(\widetilde{\ell },\widetilde{\beta })$, respectively. Then $(\gamma ,\nu )$ and $(\widetilde{\gamma },\widetilde{\nu })$ are congruent as Legendre curves if and only if the curvatures $(\ell ,\beta )$ and $(\widetilde{\ell },\widetilde{\beta })$ coincide.