Research ContributionArnold Mathematical Journal

Received: 18 April 2015 / Revised: 28 July 2015 / Accepted: 24 August 2015

# Finite and Infinitesimal Flexibility of Semidiscrete Surfaces

Oleg Karpenkov University of Liverpool Liverpool UK

### Abstract

In this paper we study infinitesimal and finite flexibility for regular semidiscrete surfaces. We prove that regular 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of existence. Further we find a necessary condition of 3-ribbon infinitesimal flexibility. For an arbitrary $n\geq 3$ we prove that every regular $n$-ribbon surface has at most one degree of finite/infinitesimal flexibility. Finally, we discuss the relation between general semidiscrete surface flexibility and 3-ribbon subsurface flexibility. We conclude this paper with one surprising property of isometric deformations of developable semidiscrete surfaces.

###### Keywords
Semidiscrete surfaces, Flexibility, Infinitesimalflexibility

The work is partially supported by FWF Grant No. S09209

## 1 Introduction

A mapping $f:\mathbb{R}\times\mathbb{Z}\to\mathbb{R}^{3}$, where the dependence on the continuous parameter is smooth, is called a semidiscrete surface. Let us connect $f(t,z)$ with $f(t,z{+}1)$ by segments for all possible pairs $(t,z)$. The resulting surface is a piecewise ruled surface.

In this paper we study infinitesimal and finite flexibility for such semidiscrete surfaces. By isometric deformations of a semidiscrete surface $f$ we understand deformations that preserve inner geometry of the corresponding ruled surfaces and in addition that preserve all line segments connecting $f(t,z)$ with $f(t,z{+}1)$.

Many questions on discrete polyhedral surfaces have their origins in the classical theory of smooth surfaces. Flexibility is no exception from this rule. The general theory of flexibility of surfaces and polyhedra is discussed in the overview by [Sabitov1992].

[Bianchi1890] introduced a necessary and sufficient condition for the existence of isometric deformations of a surface preserving some conjugate system (i.e., two independent smooth fields of directions tangent to the surface), see also in [Eisenhart1960], etc. Such surfaces can be understood as certain limits of semidiscrete surfaces.

On the other hand, semidiscrete surfaces are themselves the limits of certain polygonal surfaces (or meshes). For the discrete case of flexible meshes much is now known. Basing on paper by [Stachel2010] made a significant contribution to discrete case classifying all flexible Kokotsakis quadrilateral meshes (see [Izmestiev2014]). We also refer the reader to [Bobenko et al.2008],  [Pottmann and Wallner2008],  [Kokotsakis1932], and [Karpenkov2010] for some other recent results in this area. For general relations to the classical case see a recent book by [Bobenko and Suris2008]. It is interesting to notice that the necessary flexibility conditions in the smooth case and the discrete case are of a different nature. Currently there is no clear description of relations between them in terms of limits.

The place of the study of semidiscrete surfaces is between the classical and the discrete cases. Main concepts of semidiscrete theory are described by  [Wallner2009]; [Wallner2012]. Some problems related to isothermic semidiscrete surfaces are studied by [Müller and Wallner2013]. Semidiscrete surfaces from the viewpoint of parallelity, offsets, and curvatures were studied by [Karpenkov and Wallner2014].

We investigate necessary condition for existence of isometric deformations of semidiscrete surfaces. To avoid pathological behavior related to noncompactness of semidiscrete surfaces we restrict ourselves to compact subsets of the following type. An $n$-ribbon surface is a mapping

 $\displaystyle f:[a,b]\times\{0,\ldots,n\}\to\mathbb{R}^{3},\qquad(t,i)\mapsto f _{i}(t).$

We also use the notion

 $\displaystyle\Delta f_{i}(t)=f_{i+1}(t)-f_{i}(t).$

While working with a rather abstract semidiscrete or $n$-ribbon surface $f$ we keep in mind the two-dimensional piecewise-ruled surface associated to it (see Fig. 1).

Note that, within this paper we traditionally consider $t$ as an argument of a semidiscrete surface $f$. The time parameter for deformations is $\lambda$.

In present paper we prove that every regular 2-ribbon surface (as a ruled surface) is flexible and has one degree of infinitesimal and finite flexibility in the regular case (Theorems 1, 2). This is quite surprising since regular 1-ribbon surfaces have infinitely many degrees of flexibility, see, for instance, in [Pottmann and Wallner2001], Theorem 5.3.10. We also find a system of differential equations for the deformation of 2-ribbon surfaces (Definition 18 and Proposition 9). In contrast to that, a regular $n$-ribbon surface is rigid for $n\geq 3$. For the case $n=3$ we prove the following statement (see Theorem 3 and Remark 10).

### 1.1 Infinitesimal Flexibility Condition

If a 3-ribbon surface is infinitesimally flexible then the following condition holds:

 $\displaystyle\dot{\Lambda}=(H_{2}-H_{1})\Lambda,$

where

 $\displaystyle\Lambda=\frac{\big(\dot{f}_{1},\ddot{f}_{1},\Delta f_{0}\big)}{ \big(\dot{f}_{2},\ddot{f}_{2},\Delta f_{2}\big)}\frac{\big(\dot{f}_{2},\Delta f _{1},\Delta f_{2}\big)^{2}}{\big(\dot{f}_{1},\Delta f_{0},\Delta f_{1}\big)^{2 }},$

and

 $\displaystyle H_{i}(t)=\frac{(\dot{f}_{i},\Delta\dot{f}_{i-1},\Delta f_{i})+( \dot{f}_{i},\Delta f_{i-1},\Delta\dot{f}_{i})}{(\dot{f}_{i},\Delta f_{i-1}, \Delta f_{i})},\quad i=1,2.$
###### Remark.

Throughout this paper we denote the derivative with respect to variable $t$ by the dot symbol.

Further in Theorem 4 we state that a regular $n$-ribbon surface ($n\geq 3$) has at most one degree of finite and infinitesimal flexibility. Finally, we show that a regular $n$-ribbon surface ($n\geq 4$) is infinitesimally or finitely flexible if and only if all its 3-ribbon subsurfaces are infinitesimally or finitely flexible (see Theorems 7 and 8). We say a few words in the case of developable semidiscrete surfaces whose finite isometric deformations have additional surprising properties. Let us mention that a similar flexibility question for smooth conjugate nets was solved by [Eisenhart1960] (see Section 141).

### 1.2 Organization of the Paper

We start in Sect. 2 with the introduction of necessary notions and definitions. In Sect. 3 we discuss flexibility of 2-ribbon surfaces. We study infinitesimal flexibility questions for 2-ribbon surfaces in Sects. 3.2 and 3.3. In Sect. 3.2 we give a system of differential equations for infinitesimal flexions, prove the existence of nonzero solutions, and show that all the solutions are proportional to each other. In Sect. 3.3 we define the operators of infinitesimal flexion which is studied further in the context of finite flexibility for 2-ribbon surfaces. In Sect. 3.4 we prove that a regular 2-ribbon surface is finitely flexible and has one degree of flexibility. In Sect. 4 we work with 3-ribbon surfaces. After some preliminary statements of Sect. 4.1 we gives a necessary infinitesimal flexibility condition for 3-ribbon surfaces in Sect. 4.2. In Sect. 5 we deal with general $n$-ribbon surfaces for $n\geq 3$. We prove that a regular $n$-ribbon surface has at most one degree of finite and infinitesimal flexibility in Sect. 5.1. Further after several preparatory statements of Sect. 5.2 we prove that finite or infinitesimal flexibility of regular $n$-ribbon surfaces is identified by finite or infinitesimal flexibility of all its 3-ribbon subsurfaces. We conclude the paper with flexibility of developable semidiscrete surfaces in Sect. 6. In this case isometric deformations have a remarkable geometric property.

## 2 Necessary Notions and Definitions

In this section we introduce central notions and definition of the article.

### 2.1 Differentiable Regular Semidiscrete Surfaces

###### Definition 1.

Let $M=(m_{0},\ldots,m_{n})$ be the $(n{+}1)$-tuple of non-negative integers. We say that an $n$-ribbon surface $f$ is a $M$-differentiable if for every $i\in\{0,\ldots,n\}$ and $j\in\{1,\ldots,m_{i}\}$ there exists a continuous derivative $f^{(j)}_{i}$.

Denote by $C^{m_{0},\ldots,m_{n}}([a,b],\mathbb{R}^{3})$ (or $C^{M}([a,b],\mathbb{R}^{3})$, for short) the Banach space of all $M$-differentiable $n$-ribbon surfaces (where $t\in[a,b]$) with the standard norm

 $\displaystyle\rho(f,g)=\max\limits_{i=\{0,\ldots,n\}}\max\limits_{j=\{1,\ldots ,m_{i}\}}\sup\limits_{[a,b]}(f^{(j)}_{i}-g^{(j)}_{i}).$
###### Remark 1.

Note that for two non-negative $(n{+}1)$-tuples $M=(m_{0},\ldots,m_{n})$ and $K=(k_{0},\ldots,k_{n})$ satisfying

 $\displaystyle m_{0}\geq k_{0},\quad\ldots,\quad m_{n}\geq k_{n}$

we have

 $\displaystyle C^{M}([a,b],\mathbb{R}^{3})\subset C^{K}([a,b],\mathbb{R}^{3}).$
###### Definition 2.

We say that an $n$-ribbon surface $f\in C^{1,2,2,\ldots,2,1}([a,b],\mathbb{R}^{3})$ is weakly regular if for every $t\in[a,b]$ and $i=1,\ldots,n-1$ we have

 $\displaystyle(\dot{f}_{i},\Delta f_{i-1},\Delta f_{i})\neq 0.$
###### Definition 3.

We say that an $n$-ribbon surface $f\in C^{1,2,2,\ldots,2,1}([a,b],\mathbb{R}^{3})$ is strongly regular if

• $f$ is weakly regular;

• for every $t\in[a,b]$ and $i=1,\ldots,n-1$ we have

 $\displaystyle\big(\dot{f}_{i}(t),\ddot{f}_{i}(t),\Delta f_{i-1}(t)\big)\neq 0 \qquad\hbox{and}\qquad\big(\dot{f}_{i}(t),\ddot{f}_{i}(t),\Delta f_{i}(t)\big) \neq 0.$

### 2.2 Isometric Semidiscrete Surfaces

Let us now study basic properties of the definition of isometric semidiscrete surfaces.

###### Definition 4.

Two $n$-ribbon surfaces $f$ and $g$ in the space $C^{1,1,\ldots,1}([a,b],\mathbb{R}^{3})$ are said to be isometric if

 $\displaystyle\left\{\begin{array}[]{l}|\dot{f}_{i}|=|\dot{g}_{i}|\\ |\Delta f_{i}|=|\Delta g_{i}|\\ \langle\dot{f}_{i},\Delta f_{i-1}\rangle=\langle\dot{g}_{i},\Delta g_{i-1} \rangle\\ \langle\dot{f}_{i},\Delta f_{i}\rangle=\langle\dot{g}_{i},\Delta g_{i}\rangle \\ \langle\dot{f}_{i},\dot{f}_{i+1}\rangle=\langle\dot{g}_{i},\dot{g}_{i+1} \rangle\end{array}\right.$

(for all admissible $i$ and $t$).

Before we continue let us show that the conditions of Definition 4 are precisely the isometric conditions for ruled surfaces. Let $f_{1}$ and $f_{2}$ be differentiable curves (denote by $\Delta_{1}f$ the curve $f_{2}{-}f_{1}$). Let us define a ruled surface $S(x,t)=xf_{1}(t)+(1{-}x)f_{2}(t)$. To show that the conditions of Definition 4 determine inner geometry we prove the following proposition.

###### Proposition 1.

The first fundamental form of the ruled surface $S(x,t)$ is uniquely defined by

 $\displaystyle|\dot{f}_{1}|,\quad|\dot{f}_{2}|,\quad|\Delta f_{1}|,\quad\langle \dot{f}_{1},\Delta f_{1}\rangle,\quad\langle\dot{f}_{2},\Delta f_{1}\rangle, \quad\langle\dot{f}_{1},\dot{f}_{2}\rangle$

and vice versa.

###### Proof.

Let us write all the coefficients of the first fundamental form of the surface in the coordinates $(x,t)$:

 $\displaystyle\Big\langle\frac{\partial S}{\partial x},\frac{\partial S}{ \partial x}\Big\rangle$ $\displaystyle=$ $\displaystyle\langle f_{1}{-}f_{2},f_{1}{-}f_{2}\rangle=|\Delta f_{1}|^{2};$ $\displaystyle\Big\langle\frac{\partial S}{\partial x},\frac{\partial S}{ \partial t}\Big\rangle$ $\displaystyle=$ $\displaystyle\big\langle f_{1}{-}f_{2},x\dot{f}_{1}{+}(1{-}x)\dot{f}_{2}\big \rangle=x\langle\Delta f_{1},\dot{f}_{1}\rangle+(1{-}x)\langle\Delta f_{1}, \dot{f}_{2}\rangle;$ $\displaystyle\Big\langle\frac{\partial S}{\partial t},\frac{\partial S}{ \partial t}\Big\rangle$ $\displaystyle=$ $\displaystyle\big\langle x\dot{f}_{1}{+}(1{-}x)\dot{f}_{2}(t),x\dot{f}_{1}{+}( 1{-}x)\dot{f}_{2}(t)\big\rangle$ $\displaystyle=$ $\displaystyle x^{2}|f_{1}|^{2}+2x(1{-}x)\langle\dot{f}_{1},\dot{f}_{2}\rangle+ (1{-}x)^{2}|f_{2}|^{2}.$

As we see, on the one hand the first fundamental form is defined by the above six functions. On the other hand the values of the first fundamental form at $x=0,1/2,1$ defines the values of the above six functions.

### 2.3 Deformations and Flexions of Semidiscrete Surfaces

###### Definition 5.

A deformation of a semidiscrete $n$-ribbon surface $f$ is a family of $n$-ribbon surfaces $\{f^{\lambda}\}$ with parameter $\lambda$ in the interval $[-\Lambda,\Lambda]$ for some positive $\Lambda$ such that $f^{0}=f$. In this paper we consider only deformations that are continuously differentiable in $\lambda$.

###### Remark 2.

In this paper $\lambda$ is the parameter of deformations, while $t$ is the first argument of semidiscrete surfaces.

Let us give a formal definition of deformations that do not change the inner geometry of a surface.

###### Definition 6.

We say that a deformation $\{f^{\lambda}\}$ of a semidiscrete $n$-ribbon surface $f$ is isometric if all the surfaces in the deformation are isometric to each other.

###### Definition 7.

Consider a family of functions, vector functions, or semidiscrete surfaces $\gamma=\{w^{\lambda}\}$ with parameter $\lambda\in[-\varepsilon,\varepsilon]$ for some positive $\varepsilon$, and let $w=w^{0}$. We say that the derivative

 $\displaystyle{\mathcal{D}_{\gamma}}w=\frac{\partial w^{\lambda}}{\partial \lambda}\Big|_{\lambda=0}$

is an infinitesimal deformation of $w$.

The infinitesimal deformation of an $n$-ribbon surface $f$ in $C^{M}([a,b],\mathbb{R}^{3})$ is an element of the tangent space $T_{f}C^{M}([a,b],\mathbb{R}^{3})$, which is naturally isomorphic to $C^{M}([a,b],\mathbb{R}^{3})$.

###### Definition 8.

Consider a deformation $\{f^{\lambda}\}$ of a semidiscrete $n$-ribbon surface $f$ in $C^{(1,2,2,\ldots,2,1)}([a,b],\mathbb{R}^{3})$. We say that the deformation $\{f^{\lambda}\}$ is infinitesimally flexible if

 $\displaystyle\begin{array}[]{c}{\mathcal{D}_{\gamma}}|\dot{f}_{i}^{\lambda}|=0 ,\qquad{\mathcal{D}_{\gamma}}|\Delta f_{i}^{\lambda}|=0,\qquad{\mathcal{D}_{ \gamma}}\langle\dot{f}_{i}^{\lambda},\Delta f_{i-1}^{\lambda}\rangle=0,\\ {\mathcal{D}_{\gamma}}\langle\dot{f}_{i}^{\lambda},\Delta f_{i}^{\lambda} \rangle=0,\quad\hbox{and}\quad{\mathcal{D}_{\gamma}}\langle\dot{f}_{i}^{ \lambda},\dot{f}_{i+1}^{\lambda}\rangle=0\end{array}$

(for all admissible $i$ and $t$).

In fact, infinitesimal flexibility is a property of tangent spaces rather than deformations.

###### Definition 9.

We say that a tangent vector $\mathcal{D}f$ at a semidiscrete surface $f$ is an infinitesimal flexion if the deformation ${\mathcal{D}_{\gamma}}f$ where

 $\displaystyle\gamma(\lambda)=f+\lambda\mathcal{D}f$

is infinitesimally isometric.

We say that an infinitesimal flexion $\mathcal{D}f$ is a finite flexion if there exists an isometric deformation $\gamma$ with $\gamma(0)=f$ such that $\mathcal{D}_{\gamma}f=\mathcal{D}f$.

Finally let us determine isometrically nontrivial infinitesimal flexions.

###### Definition 10.

An infinitesimal flexion of a weakly regular $n$-ribbon surface $f$ in $C^{0,1,0}([a,b],\mathbb{R}^{3})$ is said to be isometrically nontrivial (trivial) at point $(t,i)$ for some $t\in[a,b]$ and $n\in\{1,\ldots,n-1\}$ if the corresponding infinitesimal deformation of the angle between the planes spanned by $(\dot{f}_{i}(t)\Delta f_{i-1}(t))$ and $(\dot{f}_{i}(t)\Delta f_{i}(t))$ is nonzero (or zero, respectively).

We say that an infinitesimal flexion of $f$ is isometrically nontrivial if it is isometrically nontrivial at least at one point $(t,i)$. Otherwise an infinitesimal inflexion is said to be isometrically trivial.

We say that an infinitesimal flexion of $f$ is strongly isometrically nontrivial if it is isometrically nontrivial at every point $(t,i)$.

### 2.4 Spaces of Semidiscrete Surfaces with Fixed Initial Position

In order to calculate the degree of flexibility for a semidiscrete surface we should eliminate trivial Euclidean deformations of the surfaces. Let us do this as follows.

###### Definition 11.

Denote by

 $\displaystyle C_{0}^{M}([a,b],\mathbb{R}^{3})\subset C^{M}([a,b],\mathbb{R}^{3})$

the subset of all 2-ribbon surfaces with fixed initial position, namely an $n$-ribbon surface $f$ is in $C_{0}^{M}([a,b],\mathbb{R}^{3})$ if and only if

• $f_{1}(0)\in C^{M}([a,b],\mathbb{R}^{3})$;

• $f_{1}(0)=(0,0,0)$;

• the vector $\dot{f}_{1}(0)$ is proportional to $(1,0,0)$;

• the vector $\Delta f_{0}(0)$ has the coordinates $(p,q,0)$.

###### Remark 3.

Let $\Sigma$ denote all weakly non-regular semidiscrete surfaces. Notice that the set $C_{0}^{M}([a,b],\mathbb{R}^{3}){\setminus}\Sigma$ has a natural structure of an 8-fold covering of the quotient space of $C^{M}([a,b],\mathbb{R}^{3}){\setminus}\Sigma$ by the Euclidean congruence relation. In other words, for every weakly regular $M$-differentiable semidiscrete surface $f$ there exists exactly eight semidiscrete surfaces that are congruent to $f$. These 8 surfaces are obtained one from another by 8 symmetries of type

 $\displaystyle(e_{1},e_{2},e_{3})\to(\pm e_{1},\pm e_{2},\pm e_{3}).$

So, on the one hand one can consider any branch of the 8-fold for studying flexibility properties of the original $n$-ribbon curve. On the other hand the set $C_{0}^{M}([a,b],\mathbb{R}^{3})$ has a structure of a vector space. For these reasons from now on we prefer to consider the space $C_{0}^{M}([a,b],\mathbb{R}^{3})$, rather than the quotient space of $C^{M}([a,b],\mathbb{R}^{3}){\setminus}\Sigma$ by the group of all Euclidean transformation.

Since $C_{0}^{M}([a,b],\mathbb{R}^{3})$ is a subspace of $C^{M}([a,b],\mathbb{R}^{3})$ we have the induced metric and topology (in particular, $C_{0}^{M}([a,b],\mathbb{R}^{3})$ is a Banach space), definitions of deformations, isometric deformations, infinitesimal and finite flexions, isometrically trivial and nontrivial infinitesimal flexions in $C^{M}_{0}([a,b],\mathbb{R}^{3})$.

### 2.5 Rigid Semidiscrete Surfaces: Degrees of Flexibility

###### Definition 12.

The set of infinitesimal flexions in $C^{M}_{0}([a,b],\mathbb{R}^{3})$ is a linear space. We say that $f$ has $n$ degrees of infinitesimal flexibility if the dimension of the space of infinitesimal flexions is $n$. If $n=0$ we say that $f$ is infinitesimally rigid.

In the finite case we define only finitely regularly rigid semidiscrete surfaces and surfaces that has one degree of finite flexibility. In order to define finite regular rigidity we use the following definition.

###### Definition 13.

We say that an isometric deformation $\gamma$ of $f$ in $C^{M}_{0}([a,b],\mathbb{R}^{3})$ is regular at 0 if $\mathcal{D}_{\gamma}f\neq 0$.

###### Definition 14.

We say that an $n$-ribbon surface $f$ in $C^{M}_{0}([a,b],\mathbb{R}^{3})$ is finitely regularly rigid if the set of regular isometric deformations of $f$ is empty.

Let us finally give the definition of the property to have one degree of finite flexibility. As in infinitesimal case we consider only the space of semidiscrete surfaces with fixed initial position $C_{0}^{M}([a,b],\mathbb{R}^{3})$. This cancels excess trivial Euclidean rotations of the whole semidiscrete surface. Of course, every finite isometric deformations of a semidiscrete surface with fixed initial position still can be reparametrised, as a result one has another isometric deformation of the surface. So the best thing would be to try to normalize them.

In this paper we consider the following “natural parametrization” of an isometric deformation. It is clear that for every isometric deformation $\{f^{\lambda}\}$ in $C_{0}^{M}([a,b],\mathbb{R}^{3})$ we have

 $\displaystyle\mathcal{D}_{f^{\lambda}}\dot{f}(a)=0,\quad\mathcal{D}_{f^{ \lambda}}\Delta f_{0}(a)=0,\quad\hbox{and}\quad\mathcal{D}_{f^{\lambda}}\Delta f _{1}(a)=\alpha(\lambda)\dot{f}(a){\times}\Delta f_{1}(a)$

for some real valued function $\alpha$.

###### Definition 15.

We say that an isometric deformation $\{f^{\lambda}\}$ is normalized if and only if for every admissible values of parameter $\lambda$ we have $\alpha(\lambda)=1$, where $\alpha$ is the real-valued function defined in the last expression.

In our case by Corollary 1 below we have: if $\alpha(\lambda_{0})=0$ then $\mathcal{D}_{f^{\lambda}}f^{\lambda_{0}}=0$. Hence, there is no regular isometric deformation that preserves the frame $(\dot{f}_{1}(a),\Delta f_{0}(a),\Delta f_{1}(a))$ and the point $f_{1}(a)$. So we can give the following definition.

###### Definition 16.

We say that a weakly regular 2-ribbon surface $f$ has one degree of finite flexibility if

• $f$ has one degree of infinitesimal flexibility.

• for sufficiently small $\varepsilon>0$ there exists a unique normalized isometric deformation of $f$ defined on $[-\varepsilon,\varepsilon]$.

## 3 Finite and Infinitesimal Flexibility of 2-Ribbon Surfaces

In this section we describe flexions of 2-ribbon surfaces. Such surfaces are defined by three curves $f_{0}$, $f_{1}$, and $f_{2}$. Our main goal here is to prove under some natural genericity assumptions that every 2-ribbon surface is infinitesimally and finitely flexible and has one degree of infinitesimal and finite flexibility. Our first point is to describe the system of differential equations (System A) that determines infinitesimal flexions corresponding to finite flexions and find solutions to this system (see Sects. 3.2). We use it to derive finite flexibility in Theorem 1 (also in Sect. 3.2). Further via solutions of System A we define the operators of infinitesimal flexion ${\mathcal{V}}^{\pm}$ (in Sect. 3.3). Finally, to show finite flexibility of 2-ribbon surfaces we study Lipschitz properties for ${\mathcal{V}}^{\pm}$ and prove flexibility Theorem 2 (in Sect. 3.4).

### 3.1 Basic Relations for Infinitesimal Flexions

In this small subsection we collect some useful relations.

###### Proposition 2.

Let $f$ be a 2-ribbon surface in $C^{1,2,1}([a,b],\mathbb{R}^{3})$. Then for every infinitesimal flexion $\mathcal{D}f$ the following properties hold:

 $\displaystyle\langle\dot{f}_{1},\mathcal{D}\dot{f}_{1}\rangle=0;$ (1) $\displaystyle\langle\dot{f}_{1}-\Delta\dot{f}_{0},\mathcal{D}\dot{f}_{1}- \mathcal{D}\Delta\dot{f}_{0}\rangle=0;$ (2) $\displaystyle\langle\dot{f}_{1}+\Delta\dot{f}_{1},\mathcal{D}\dot{f}_{1}+ \mathcal{D}\Delta\dot{f}_{1}\rangle=0;$ (3) $\displaystyle\langle\Delta{f}_{0},\mathcal{D}\Delta\dot{f}_{0}\rangle+\langle \Delta\dot{f}_{0},\mathcal{D}\Delta f_{0}\rangle=0;$ (4) $\displaystyle\langle\Delta{f}_{1},\mathcal{D}\Delta\dot{f}_{1}\rangle+\langle \Delta\dot{f}_{1},\mathcal{D}\Delta f_{1}\rangle=0;$ (5) $\displaystyle\langle\dot{f}_{1},\mathcal{D}\Delta\dot{f}_{0}\rangle+\langle \mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle=0;$ (6) $\displaystyle\langle\dot{f}_{1},\mathcal{D}\Delta\dot{f}_{1}\rangle+\langle \mathcal{D}\dot{f}_{1},\Delta\dot{f}_{1}\rangle=0;$ (7) $\displaystyle\langle\mathcal{D}\ddot{f}_{1},\Delta{f}_{0}\rangle+\langle\ddot{ f}_{1},\mathcal{D}\Delta{f}_{0}\rangle=0;$ (8) $\displaystyle\langle\mathcal{D}\ddot{f}_{1},\Delta{f}_{1}\rangle+\langle\ddot{ f}_{1},\mathcal{D}\Delta{f}_{1}\rangle=0.$ (9)
###### Remark 4.

For a semidiscrete or $n$-ribbon surface $f$ the operations $\mathcal{D}$, $\Delta$, and $\frac{\partial}{\partial t}$ commute, so we do not pay attention to the order of these operations in compositions.

###### Proof.

Equations (1), (2), and (3) follow from the fact that infinitesimal flexions preserve the norms of $\dot{f}_{1}$, $\dot{f}_{0}=\dot{f}_{1}-\Delta\dot{f}_{0}$, and $\dot{f}_{2}=\dot{f}_{1}+\Delta\dot{f}_{1}$ respectively.

The invariance of the lengths of $\Delta f_{0}$ and $\Delta f_{1}$ imply Equations (4), and (5) respectively. They are equivalent to

 $\displaystyle\frac{\partial}{\partial t}\mathcal{D}\langle\Delta f_{0},\Delta f _{0}\rangle=0\quad\hbox{and}\quad\frac{\partial}{\partial t}\mathcal{D}\langle \Delta f_{1},\Delta f_{1}\rangle=0.$

Equations (6) and (7) follow from invariance of the angles between the vectors $\dot{f_{1}}$ and $\Delta\dot{f_{0}}$ and the vectors $\dot{f_{1}}$ and $\Delta\dot{f_{0}}$.

Let us prove Eq. (8). Since the angles between the vectors $\Delta f_{0}$ and $\dot{f_{1}}$ are preserved by infinitesimal flexions we have

 $\displaystyle\frac{\partial}{\partial t}\mathcal{D}\langle\dot{f}_{1},\Delta f _{0}\rangle=0.$

Therefore,

 $\displaystyle\langle\mathcal{D}\ddot{f}_{1},\Delta f_{0}\rangle+\langle\ddot{f }_{1},\mathcal{D}\Delta f_{0}\rangle+\langle\mathcal{D}\dot{f}_{1},\Delta\dot{ f}_{0}\rangle+\langle\dot{f}_{1},\mathcal{D}\Delta\dot{f}_{0}\rangle=0.$

By Eq. (6) we have $\langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle+\langle\dot{f}_{1}, \mathcal{D}\Delta\dot{f}_{0}\rangle=0$ and hence

 $\displaystyle\langle\mathcal{D}\ddot{f}_{1},\Delta f_{0}\rangle+\langle\ddot{f }_{1},\mathcal{D}\Delta f_{0}\rangle=0.$

We have arrived at Eq. (8).

Finally Eq. (9) is proved by analogy with Eq. (8).

### 3.2 Infinitesimal Flexibility of 2-Ribbon Surfaces

Our main goal for this subsection is to prove the following general theorem

###### Theorem 1.

Let $f\in C^{1,2,1}_{0}([a,b],\mathbb{R}^{3})$ be a weakly regular 2-ribbon surface with fixed initial position. Then $f$ has one degree of infinitesimal flexibility.

First we write down and investigate a supplementary system of differential equations (System A) which describes infinitesimal flexions of weakly regular 2-ribbon surfaces. We also show the uniqueness of the solution of System A for a given initial data (Proposition 3). The remaining part of this subsection is dedicated to the proof of Theorem 1 mentioned above. In Proposition 4 we show that every infinitesimal flexion satisfies System A. Then in Proposition 5 we prove that every solution of System A with certain initial data is an infinitesimal flexion. After that we prove Theorem 1.

#### 3.2.1 System A

Let

 $\displaystyle\begin{array}[]{lll}G_{11}=\langle\mathcal{D}\dot{f_{1}},\dot{f}_ {1}\rangle,&G_{12}=\langle\mathcal{D}\dot{f_{1}},\Delta f_{0}\rangle,&G_{13}= \langle\mathcal{D}\dot{f_{1}},\Delta f_{1}\rangle,\\ G_{21}=\langle\mathcal{D}\Delta f_{0},\dot{f}_{1}\rangle,&G_{22}=\langle \mathcal{D}\Delta f_{0},\Delta f_{0}\rangle,&G_{23}=\langle\mathcal{D}\Delta f _{0},\Delta f_{1}\rangle,\\ G_{31}=\langle\mathcal{D}\Delta f_{1},\dot{f}_{1}\rangle,&G_{32}=\langle \mathcal{D}\Delta f_{1},\Delta f_{0}\rangle,&G_{33}=\langle\mathcal{D}\Delta f _{1},\Delta f_{1}\rangle.\end{array}$ (10)

Denote by System A the following system of differential equations

$$\begin{cases} \dot G_{11}&=0,\\ \dot G_{12}&=\left( \frac{(\dot f_1,\Delta \dot f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} {+} \frac{(\ddot f_1,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right)G_{12}{+} \frac{(\dot f_1,\Delta f_0, \Delta \dot f_0)}{(\dot f_1,\Delta f_0, \Delta f_1)}G_{13}{-} \frac{(\dot f_1,\Delta f_0, \ddot f_1)}{(\dot f_1,\Delta f_0, \Delta f_1)}G_{23},\\ \dot G_{13}&=\frac{(\dot f_1,\Delta \dot f_1,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}G_{12}{+} \left( \frac{(\dot f_1,\Delta f_0,\Delta \dot f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}{+} \frac{(\ddot f_1,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right) G_{13}{-} \frac{(\dot f_1,\ddot f_1,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}G_{32},\\ \dot G_{21}&=-\left( \frac{(\dot f_1,\Delta \dot f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} + \frac{(\ddot f_1,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right)G_{12}- \frac{(\dot f_1,\Delta f_0, \Delta \dot f_0)}{(\dot f_1,\Delta f_0, \Delta f_1)}G_{13} \\ &+ \frac{(\dot f_1,\Delta f_0, \ddot f_1)}{(\dot f_1,\Delta f_0, \Delta f_1)}G_{23},\\ \dot G_{22}&=0,\\ \dot G_{23}&= -\left( \frac{(\Delta f_1,\Delta f_0,\dot f_1{\times} \Delta f_0)(\dot f_1,\Delta \dot f_0, \Delta f_1)}{|\dot f_1{\times} \Delta f_0|^2(\dot f_1,\Delta f_0, \Delta f_1)} -\frac{(\dot f_1,\Delta f_1,\dot f_1{\times} \Delta f_0)(\Delta \dot f_0,\Delta f_0,\Delta f_1)}{|\dot f_1{\times} \Delta f_0|^2(\dot f_1,\Delta f_0,\Delta f_1)}\right. \\ & \left.+ \frac{(\dot f_1,\Delta f_0{\times} \Delta \dot f_0,\Delta f_1)}{|\dot f_1{\times} \Delta f_0|^2}+ \frac{(\dot f_1{\times} \Delta \dot f_0,\Delta f_0,\Delta f_1)}{|\dot f_1{\times} \Delta f_0|^2}+ \frac{(\Delta \dot f_1,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right) G_{12} \\ & -\left( \frac{(\Delta f_1,\Delta f_0,\dot f_1{\times} \Delta f_0)(\dot f_1,\Delta f_0, \Delta \dot f_0)}{|\dot f_1{\times} \Delta f_0|^2(\dot f_1,\Delta f_0, \Delta f_1)} + \frac{(\dot f_1,\Delta f_0, \Delta f_0{\times} \Delta \dot f_0)}{|\dot f_1{\times} \Delta f_0|^2} \right)G_{13}\\ &- \left( \frac{(\dot f_1,\Delta f_1,\dot f_1{\times} \Delta f_0)(\dot f_1,\Delta f_0, \Delta \dot f_0)}{|\dot f_1{\times} \Delta f_0|^2(\dot f_1,\Delta f_0,\Delta f_1)} - \frac{(\dot f_1,\Delta f_0, \dot f_1{\times} \Delta \dot f_0)}{|\dot f_1{\times} \Delta f_0|^2} -\frac{(\dot f_1,\Delta f_0, \Delta \dot f_1)}{(\dot f_1,\Delta f_0, \Delta f_1)}\right)G_{23}, \\ \dot G_{31}&=-\frac{(\dot f_1,\Delta \dot f_1,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}G_{12}- \left( \frac{(\dot f_1,\Delta f_0,\Delta \dot f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}+ \frac{(\ddot f_1,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right) G_{13} \\ & +\frac{(\dot f_1,\ddot f_1,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)}G_{32},\\ \dot G_{32}&= -\left( \frac{(\Delta f_0,\Delta f_1,\dot f_1{\times} \Delta f_1)(\dot f_1,\Delta \dot f_1, \Delta f_1)}{|\dot f_1{\times} \Delta f_1|^2(\dot f_1,\Delta f_0, \Delta f_1)} +\frac{(\dot f_1,\Delta f_1, \Delta f_1{\times} \Delta \dot f_1)}{|\dot f_1{\times} \Delta f_1|^2} \right)G_{12}\\ &- \left( \frac{(\Delta f_0,\Delta f_1,\dot f_1{\times} \Delta f_1)(\dot f_1,\Delta f_0, \Delta \dot f_1)}{|\dot f_1{\times} \Delta f_1|^2(\dot f_1,\Delta f_0, \Delta f_1)} -\frac{(\dot f_1,\Delta f_0,\dot f_1{\times} \Delta f_1)(\Delta \dot f_1,\Delta f_0,\Delta f_1)}{|\dot f_1{\times} \Delta f_1|^2(\dot f_1,\Delta f_0,\Delta f_1)}\right. \\ & \left. +\frac{(\dot f_1,\Delta f_1{\times} \Delta \dot f_1,\Delta f_0)}{|\dot f_1{\times} \Delta f_1|^2}+ \frac{(\dot f_1{\times} \Delta \dot f_1,\Delta f_1,\Delta f_0)}{|\dot f_1{\times} \Delta f_1|^2}+ \frac{(\Delta \dot f_0,\Delta f_0,\Delta f_1)}{(\dot f_1,\Delta f_0,\Delta f_1)} \right) G_{13} \\ & -\left( \frac{(\dot f_1,\Delta f_0,\dot f_1{\times} \Delta f_1)(\dot f_1,\Delta \dot f_1, \Delta f_1)}{|\dot f_1{\times} \Delta f_1|^2(\dot f_1,\Delta f_0,\Delta f_1)} - \frac{(\dot f_1,\Delta f_1, \dot f_1{\times} \Delta \dot f_1)}{|\dot f_1{\times} \Delta f_1|^2} -\frac{(\dot f_1,\Delta \dot f_0, \Delta f_1)}{(\dot f_1,\Delta f_0, \Delta f_1)}\right)G_{32}, \\ \dot G_{33}&=0. \end{cases}$$
###### Remark 5.

In Proposition 10 below we show an explicit formula for the function $G_{23}{+}G_{32}$, it is $\Phi$ in our notation of Sect. 3.

Note also that $\dot{G}_{12}+\dot{G}_{21}=0$ and $\dot{G}_{13}+\dot{G}_{31}=0$ in System A.

###### Example 1.

Let us consider a simple example of a 2-ribbon curve where $\dot{f}$, $\Delta f_{0}$, and $\Delta f_{1}$ are all constants. Let us call these surfaces book-shaped surfaces. Direct calculations show that

 $\displaystyle\dot{G}_{11}=\dot{G}_{12}=\ldots=\dot{G}_{33}=0$

(this happens, since all the summands in the coefficients of System A contain either $\ddot{f}_{1}$, or $\Delta\dot{f}_{0}$, or $\Delta\dot{f}_{1}$ which are all zeroes in our case). Hence all the scalar products of the deformation with vectors $\dot{f}_{1},\Delta f_{0},\Delta f_{1}$ do not depend on $t$. Therefore, every element of every isometric deformations of a book-shaped surface is a book-shaped surface. Here is a typical example of isometric deformation in this class:

 $\displaystyle f_{1}^{\lambda}(t)=(t,0,0),\quad\Delta f_{0}^{\lambda}(t)=(0,1,0 ),\quad\Delta_{1}^{\lambda}(t)=(0,\sin\lambda,\cos\lambda).$

This deformation can be geometrically seen as an opening a museum book with two rigid plastic pages.

In the following proposition we prove that for every single 2-ribbon surface $f$ (not for a deformation) and initial data for $G_{ij}$ at one point $f(t_{0})$ System A has a unique solution. Recall that $t$ is an argument of $f$.

###### Proposition 3.

Let $f$ be a weakly regular 2-ribbon surface in $C^{1,2,1}([a,b],\mathbb{R}^{3})$. For every collection of initial data $G_{ij}(a)=c_{ij}$ there exists a unique solution of System A on $[a,b]$.

###### Proof.

System A is the system of homogeneous linear differential equations with smooth variable coefficients (since $(\dot{f}_{1},\Delta f_{0},\Delta f_{1})$ never vanishes on $[a,b]$) and hence for every collection of initial data it has a unique solution on the segment $[a,b]$. $\square$

#### 3.2.2 Every Infinitesimal Flexion Satisfies System A

Let us show the following statement.

###### Proposition 4.

Let $f$ be a weakly regular 2-ribbon surface in $C^{1,2,1}([a,b],\mathbb{R}^{3})$. Then for every infinitesimal flexion $\mathcal{D}f$ the functions $G_{11},G_{12},\ldots,G_{33}$ computed from the pair of semidiscrete surfaces $(f,\mathcal{D}f)$ satisfy system A.

We start the proof with the following general lemma.

###### Lemma 1.

For every infinitesimal flexion $\mathcal{D}f$ we have the equalities

 $\displaystyle G_{11}=G_{22}=G_{33}=0,\qquad G_{12}+G_{21}=0,\quad\hbox{and} \quad G_{13}+G_{31}=0.$
###### Proof.

The functions $|\dot{f}_{1}|$, $|\Delta f_{0}|$, and $|\Delta f_{1}|$ are infinitesimally preserved by infinitesimal flexions, hence $G_{11}$, $G_{22}$, and $G_{33}$ vanish.

The invariance of angles between $\dot{f}_{1}$ and $\Delta f_{0}$, and $\dot{f}_{1}$ and $\Delta f_{1}$ yield the equations $G_{12}+G_{21}=0$ and $G_{13}+G_{31}=0$, respectively. $\square$

###### Proof of Proposition 4..

From Lemma 1 the functions $G_{11}$, $G_{22}$, and $G_{33}$ are zero functions, thus $\dot{G}_{11}$, $\dot{G}_{22}$, and $\dot{G}_{33}$ are zero functions as well.

Let us prove the expression for $\dot{G}_{12}$ and $\dot{G}_{13}$. Note that

 $\displaystyle\dot{G}_{12}=\langle\mathcal{D}\ddot{f}_{1},\Delta f_{0}\rangle+ \langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle.$

Thus Eqs. (6) and (8) imply

 $\displaystyle\dot{G}_{12}=\langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0} \rangle-\langle\ddot{f}_{1},\mathcal{D}\Delta f_{0}\rangle.$

To obtain the expression for $\dot{G}_{12}$ rewrite $\Delta\dot{f}_{0}$ and $\ddot{f}_{1}$ in the basis consisting of vectors $\dot{f}_{1}$, $\Delta f_{0}$, and $\Delta f_{1}$.

 $\displaystyle\dot{G}_{12}$ $\displaystyle=\langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle-\langle \ddot{f}_{1},\mathcal{D}\Delta f_{0}\rangle$ $\displaystyle=\left(\frac{(\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot {f}_{1},\Delta f_{0},\Delta f_{1})}G_{11}+\frac{(\dot{f}_{1},\Delta\dot{f}_{0} ,\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{12}+\frac{(\dot{f}_ {1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G _{13}\right)$ $\displaystyle\quad-\left(\frac{(\ddot{f}_{1},\Delta f_{0},\Delta f_{1})}{(\dot {f}_{1},\Delta f_{0},\Delta f_{1})}G_{21}+\frac{(\dot{f}_{1},\ddot{f}_{1}, \Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{22}+\frac{(\dot{f}_{ 1},\Delta f_{0},\ddot{f}_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{23}\right)$ $\displaystyle=\left(\frac{(\dot{f}_{1},\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{ f}_{1},\Delta f_{0},\Delta f_{1})}+\frac{(\ddot{f}_{1},\Delta f_{0},\Delta f_{ 1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}\right)G_{12}$ $\displaystyle\quad+\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f }_{1},\Delta f_{0},\Delta f_{1})}G_{13}-\frac{(\dot{f}_{1},\Delta f_{0},\ddot{ f}_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{23}.$

The last equation holds since $G_{11}=0$, $G_{22}=0$, and $G_{21}=-G_{12}$.

The same strategy works for the function $\dot{G}_{13}$.

Now we study the expressions for $\dot{G}_{21}$ and $\dot{G}_{31}$. From Lemma 1 we know that $G_{21}=-G_{12}$ and $G_{31}=-G_{13}$ and hence $\dot{G}_{21}=-\dot{G}_{12}$ and $\dot{G}_{31}=-\dot{G}_{13}$. Therefore, the equations for $\dot{G}_{21}$ and $\dot{G}_{31}$ are satisfied.

In order to get the expression for $\dot{G}_{23}$, we first show that the function $(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})$ is an invariant of infinitesimal flexions. Indeed,

 $\displaystyle(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})=(\dot{f}_{1},\Delta f _{0},\dot{f}_{1}{-}\dot{f}_{0})=-(\dot{f}_{1},\Delta f_{0},\dot{f}_{0}).$

The vectors $\dot{f}_{0}$, $\dot{f}_{1}$, and $\Delta f_{0}$ form a rigid frame, hence their triple product is an invariant of infinitesimal flexions. Hence the function $(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})$ is an invariant as well.

The infinitesimal flexion invariance of $(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})$ implies that

 $\displaystyle\mathcal{D}(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})=0.$

So we get

 $\displaystyle(\mathcal{D}\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})+(\dot{f}_ {1},\mathcal{D}\Delta f_{0},\Delta\dot{f}_{0})+(\dot{f}_{1},\Delta f_{0}, \mathcal{D}\Delta\dot{f}_{0})=0.$

Rewrite

 $\displaystyle(\dot{f}_{1},\Delta f_{0},\mathcal{D}\Delta\dot{f}_{0})=$ $\displaystyle-(\mathcal{D}\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})-(\dot{f} _{1},\mathcal{D}\Delta f_{0},\Delta\dot{f}_{0})$ $\displaystyle=$ $\displaystyle-\langle\mathcal{D}\dot{f}_{1},\Delta f_{0}{\times}\Delta\dot{f}_ {0}\rangle+\langle\mathcal{D}\Delta f_{0},\dot{f}_{1}{\times}\Delta\dot{f}_{0}\rangle$ $\displaystyle=$ $\displaystyle-\frac{(\Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{0},\Delta f _{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{11}-\frac{(\dot{f}_{1}, \Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0} ,\Delta f_{1})}G_{12}$ $\displaystyle-\frac{(\dot{f}_{1},\Delta f_{0},\Delta f_{0}{\times}\Delta\dot{f }_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{13}+\quad\frac{(\dot{f}_{1} {\times}\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0 },\Delta f_{1})}G_{21}$ $\displaystyle+\frac{(\dot{f}_{1},\dot{f}_{1}{\times}\Delta\dot{f}_{0},\Delta f _{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{22}+\frac{(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}G_{23}.$

Secondly, we have

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle$ $\displaystyle=-\langle{\mathcal{D}\Delta f_{0}},\Delta\dot{f}_{0}\rangle$ $\displaystyle=-\frac{(\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{ 1},\Delta f_{0},\Delta f_{1})}G_{21}-\frac{(\dot{f}_{1},\Delta\dot{f}_{0}, \Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{22}-\frac{(\dot{f}_{ 1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_ {23}.$

Thirdly, we get

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle=-\langle \mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle=-\frac{(\dot{f}_{1},\Delta\dot {f}_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{12}-\frac{( \dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f _{1})}G_{13}.$

Fourthly,

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{1}\rangle=$ $\displaystyle\frac{(\Delta f_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0}) }{(\dot{f}_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}\langle\mathcal{D }\Delta\dot{f}_{0},\dot{f}_{1}\rangle+\frac{(\dot{f}_{1},\Delta f_{1},\dot{f}_ {1}{\times}\Delta f_{0})}{(\dot{f}_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f _{0})}\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle$ $\displaystyle+\frac{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}(\dot{f}_{1},\Delta f_{0}, \mathcal{D}\Delta\dot{f}_{0}).$

After the substitution of the four above expressions and simplifications we have

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{1}\rangle$ $\displaystyle\quad=-\left(\frac{(\Delta f_{1},\Delta f_{0},\dot{f}_{1}{\times} \Delta f_{0})(\dot{f}_{1},\Delta\dot{f}_{0},\Delta f_{1})}{|\dot{f}_{1}{\times }\Delta f_{0}|^{2}(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}-\frac{(\dot{f}_{1}, \Delta f_{1},\dot{f}_{1}{\times}\Delta f_{0})(\Delta\dot{f}_{0},\Delta f_{0}, \Delta f_{1})}{|\dot{f}_{1}{\times}\Delta f_{0}|^{2}(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}\right.$ $\displaystyle\qquad\left.+\frac{(\dot{f}_{1},\Delta f_{0}{\times}\Delta\dot{f} _{0},\Delta f_{1})}{|\dot{f}_{1}{\times}\Delta f_{0}|^{2}}+\frac{(\dot{f}_{1}{ \times}\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{|\dot{f}_{1}{\times} \Delta f_{0}|^{2}}\right)G_{12}$ $\displaystyle\qquad-\left(\frac{(\Delta f_{1},\Delta f_{0},\dot{f}_{1}{\times} \Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{|\dot{f}_{1}{\times }\Delta f_{0}|^{2}(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}+\frac{(\dot{f}_{1}, \Delta f_{0},\Delta f_{0}{\times}\Delta\dot{f}_{0})}{|\dot{f}_{1}{\times} \Delta f_{0}|^{2}}\right)G_{13}$ $\displaystyle\qquad-\left(\frac{(\dot{f}_{1},\Delta f_{1},\dot{f}_{1}{\times} \Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{|\dot{f}_{1}{\times }\Delta f_{0}|^{2}(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}-\frac{(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta\dot{f}_{0})}{|\dot{f}_{1}{\times}\Delta f _{0}|^{2}}\right)G_{23}.$

Further, decomposing the vector $\Delta\dot{f}_{1}$ into basis vectors $\dot{f}_{1}$, $\Delta f_{0}$, and $\Delta f_{1}$ we get

 $\displaystyle\langle\mathcal{D}\Delta f_{0},\Delta\dot{f}{}_{1}\rangle=\frac{( \Delta\dot{f}_{1},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f _{1})}G_{21}+\frac{(\dot{f}_{1},\Delta\dot{f}_{1},\Delta f_{1})}{(\dot{f}_{1}, \Delta f_{0},\Delta f_{1})}G_{22}+\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f }_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{23}.$

From the last two identities, by substituting $G_{22}=0$ and $G_{21}=-G_{12}$ (see Lemma 1), we obtain the expression for

 $\displaystyle\dot{G}_{23}=\frac{\partial}{\partial t}\langle\mathcal{D}\Delta f _{0},\Delta f_{1}\rangle=\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{1} \rangle+\langle\mathcal{D}\Delta f_{0},\Delta\dot{f}_{1}\rangle.$

The expression for $\dot{G}_{32}$ is calculated in a similar way. This concludes the proof. $\square$

#### 3.2.3 Existence of Infinitesimal Flexions

Let us prove that every solution of System A with certain initial data determines an infinitesimal flexion.

###### Proposition 5.

Let $f$ be a weakly regular 2-ribbon surface in $C^{1,2,1}([a,b],\mathbb{R}^{3})$. Then

1. (i)

For an arbitrary nonzero $\alpha$ there exists a unique tangent vector $\mathcal{D}f$ at $f$ satisfying System A and the boundary conditions

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\mathcal{D}\Delta f_{0}(a)=0, \quad\hbox{and}\quad\mathcal{D}\Delta f_{1}(a)=\alpha\dot{f}_{1}(a){\times} \Delta f_{1}(a).$
2. (ii)

This tangent vector is an infinitesimal flexion.

###### Remark 6.

Here and below, for a function $f$ defined on $[a,b]$ by $\dot{f}(a)$ we mean the one-sided derivative at $a$.

###### Proof.

We start with Proposition 5$(i)$. Consider three vectors

 $\displaystyle v_{1}=0,\quad v_{2}=0,\quad\hbox{and}\quad v_{3}=\alpha\dot{f}_{ 1}(a){\times}\Delta f_{1}(a).$

We introduce the notation

 $\displaystyle\begin{array}[]{lll}c_{11}=\langle v_{1},\dot{f}_{1}\rangle,&c_{1 2}=\langle v_{1},\Delta f_{0}\rangle,&c_{13}=\langle v_{1},\Delta f_{1}\rangle ,\\ c_{21}=\langle v_{2},\dot{f}_{1}\rangle,&c_{22}=\langle v_{2},\Delta f_{0} \rangle,&c_{23}=\langle v_{2},\Delta f_{1}\rangle,\\ c_{31}=\langle v_{3},\dot{f}_{1}\rangle,&c_{32}=\langle v_{3},\Delta f_{0} \rangle,&c_{33}=\langle v_{3},\Delta f_{1}\rangle.\end{array}$

By Proposition 3 there exists a unique solution $(G_{11},G_{12},\ldots,G_{33})$ satisfying the initial conditions $G_{ij}(a)=c_{ij}$. For every point $t\in[a,b]$ the values $\mathcal{D}\dot{f}_{1}$, $\mathcal{D}\Delta f_{0}$, and $\mathcal{D}\Delta f_{1}$ of the tangent vector $\mathcal{D}f$ are uniquely defined in the basis $(\dot{f}_{1},\Delta f_{0},\Delta f_{1})$ by Eq. (10): here we substitute the solution of System A with the initial conditions $G_{ij}(a)=c_{ij}$ to the right hand side of Eq. (10). Hence, there exists a unique tangent vector $\mathcal{D}f$ of $f$ satisfying System A and the boundary conditions

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\mathcal{D}\Delta f_{1}(a)=0, \quad\hbox{and}\quad\mathcal{D}\Delta f_{0}(a)=\alpha\dot{f}_{1}(a){\times} \Delta f_{0}(a).$

This concludes the proof of the first item of the proposition. $\square$

###### Proof of Proposition 5(ii).

By the definition of an infinitesimal flexion it is enough to check that the following 11 functions are preserved by the infinitesimal deformation:

 $\displaystyle|\dot{f}_{i}|,\qquad|\Delta f_{i}|,\qquad\langle\dot{f}_{i}, \Delta f_{i-1}\rangle,\qquad\langle\dot{f}_{i},\Delta f_{i}\rangle,\quad\hbox{ and}\quad\langle\dot{f}_{i},\dot{f}_{i+1}\rangle$

(for all possible admissible $i$).

Invariance of $|\dot{f}_{1}|$, $|\Delta f_{0}|$, $|\Delta f_{1}|$, $\langle\dot{f}_{1},\Delta f_{0}\rangle$, and $\langle\dot{f}_{1},\Delta f_{1}\rangle$.

From System A we have

 $\displaystyle\dot{G}_{11}=0,\quad\dot{G}_{22}=0,\quad\dot{G}_{33}=0,\quad\dot{ G}_{21}+\dot{G}_{12}=0,\quad\dot{G}_{31}+\dot{G}_{13}=0,$

and hence the functions

 $\displaystyle\begin{array}[]{c}\mathcal{D}(|\dot{f}_{1}|^{2})=2G_{11};\quad \mathcal{D}(|\Delta f_{0}|^{2})=2G_{22};\quad\mathcal{D}(|\Delta f_{1}|^{2})=2 G_{33};\\ \mathcal{D}\langle\dot{f}_{1},\Delta f_{0}\rangle=G_{12}+G_{21},\quad\hbox{and }\quad\mathcal{D}\langle\dot{f}_{1},\Delta f_{1}\rangle=G_{31}+G_{13}\end{array}$

are constant functions. So it is enough to show that they vanish at some point: we show this at point $a$.

 $\displaystyle\mathcal{D}\langle\dot{f}_{1}(a),\dot{f}_{1}(a)\rangle=2\langle \mathcal{D}\dot{f}_{1}(a),\dot{f}_{1}(a)\rangle=2\langle 0,\dot{f}_{1}(a) \rangle=0;$ $\displaystyle\mathcal{D}\langle\Delta f_{0}(a),\Delta f_{0}(a)\rangle=2\langle \mathcal{D}\Delta f_{0}(a),\Delta f_{0}(a)\rangle=2(0,\Delta f_{0}(a)\rangle=0;$ $\displaystyle\mathcal{D}\langle\Delta f_{1}(a),\Delta f_{1}(a)\rangle=2\langle \mathcal{D}\Delta f_{1}(a),\Delta f_{1}(a)\rangle=2\langle\alpha\dot{f}_{1}(a) {\times}\Delta f_{1}(a),\Delta f_{1}(a)\rangle=0;$ $\displaystyle\mathcal{D}\langle\dot{f}_{1}(a),\Delta f_{0}(a)\rangle=\langle{ \mathcal{D}_{\gamma}}\dot{f}_{1}(a),\Delta f_{0}(a)\rangle+\langle\dot{f}_{1}( a),\mathcal{D}\Delta f_{0}(a)\rangle=\langle 0,\Delta f_{0}(a)\rangle$ $\displaystyle\qquad+\langle\dot{f}_{1}(a),0\rangle=0.$ $\displaystyle\mathcal{D}\langle\dot{f}_{1}(a),\Delta f_{1}(a)\rangle=\langle \mathcal{D}\dot{f}_{1}(a),\Delta f_{1}(a)\rangle+\langle\dot{f}_{1}(a), \mathcal{D}\Delta f_{1}(a)\rangle=\langle 0,\Delta f_{0}(a)\rangle$ $\displaystyle\qquad+\langle\dot{f}_{1}(a),\alpha\dot{f}_{1}(a){\times}\Delta f _{1}(a)\rangle=0;$

Invariance of $\langle\dot{f}_{0},\Delta f_{0}\rangle$ and $\langle\dot{f}_{2},\Delta f_{1}\rangle$. Note that

 $\displaystyle\langle\dot{f}_{0},\Delta f_{0}\rangle=-\frac{1}{2}\frac{\partial }{\partial t}\langle\Delta f_{0},\Delta f_{0}\rangle+\langle\dot{f}_{1},\Delta f _{0}\rangle.$

Hence by the above item we have

 $\displaystyle\mathcal{D}\langle\dot{f}_{0},\Delta f_{0}\rangle=-\frac{1}{2} \frac{\partial}{\partial t}\mathcal{D}\langle\Delta f_{0},\Delta f_{0}\rangle+ \mathcal{D}\langle\dot{f}_{1},\Delta f_{0}\rangle=-\frac{1}{2}\frac{\partial}{ \partial t}(0)+0=0.$

Similar reasoning shows that $\mathcal{D}\langle\dot{f}_{2},\Delta f_{1}\rangle=0$.

Invariance of $\langle\dot{f}_{0},\dot{f}_{1}\rangle$ and $\langle\dot{f}_{1},\dot{f}_{2}\rangle$. Let us prove that $\mathcal{D}\langle\dot{f}_{0},\dot{f}_{1}\rangle=0$. First, note that

 $\displaystyle\langle\mathcal{D}\dot{f}_{0},\dot{f}_{1}\rangle$ $\displaystyle=\langle\mathcal{D}\dot{f}_{1},\dot{f}_{1}\rangle-\langle\mathcal {D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle=-\langle\mathcal{D}\Delta\dot{f}_{0}, \dot{f}_{1}\rangle$ $\displaystyle=\langle\mathcal{D}\Delta f_{0},\ddot{f}_{1}\rangle-\frac{ \partial}{\partial t}\langle\mathcal{D}\Delta f_{0},\dot{f}_{1}\rangle.$

Recall that $\frac{\partial}{\partial t}\langle\mathcal{D}\Delta f_{0},\dot{f}_{1}\rangle= \dot{G}_{21}=-\dot{G}_{12}$. Let us substitute the expression for $\dot{G}_{12}$ of System A and rewrite $\ddot{f}_{1}$ in the basis of vectors $\dot{f}_{1}$, $\Delta f_{0}$, and $\Delta f_{1}$. One obtains

 $\displaystyle\langle\mathcal{D}\dot{f}_{0},\dot{f}_{1}\rangle$ $\displaystyle=\langle\mathcal{D}\Delta f_{0},\ddot{f}_{1}\rangle+\dot{G}_{12}$ $\displaystyle=\frac{(\dot{f}_{1},\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1} ,\Delta f_{0},\Delta f_{1})}\langle\mathcal{D}\dot{f}_{1},\Delta f_{0}\rangle+ \frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}\langle\mathcal{D}\dot{f}_{1},\Delta f_{1}\rangle$ $\displaystyle=\langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle=-\langle \mathcal{D}\dot{f}_{1},\dot{f}_{0}\rangle.$

Hence

 $\displaystyle\mathcal{D}\langle\dot{f}_{0},\dot{f}_{1}\rangle=\langle\mathcal{ D}\dot{f}_{0},\dot{f}_{1}\rangle+\langle\mathcal{D}\dot{f}_{1},\dot{f}_{0} \rangle=-\langle\mathcal{D}\dot{f}_{1},\dot{f}_{0}\rangle+\langle\mathcal{D} \dot{f}_{1},\dot{f}_{0}\rangle=0.$

Therefore, $\langle\dot{f}_{0},\dot{f}_{1}\rangle$ is invariant under the infinitesimal deformation. The proof of the invariance of $\langle\dot{f}_{1},\dot{f}_{2}\rangle$ is analogous.

Invariance of $\langle\dot{f}_{0},\dot{f}_{0}\rangle$ and $\langle\dot{f}_{2},\dot{f}_{2}\rangle$. Let us prove that $\mathcal{D}\langle\dot{f}_{0},\dot{f}_{0}\rangle=0$.

 $\displaystyle\mathcal{D}\langle\dot{f}_{0},\dot{f}_{0}\rangle=2\langle\mathcal {D}\dot{f}_{0},\dot{f}_{0}\rangle=2\langle\mathcal{D}\Delta\dot{f}_{0},\Delta \dot{f}_{0}\rangle+2\mathcal{D}\langle\dot{f}_{1},\dot{f}_{0}\rangle-2\langle \mathcal{D}\dot{f}_{1},\dot{f}_{1}\rangle.$

We have already shown that $\mathcal{D}\langle\dot{f}_{1},\dot{f}_{0}\rangle=0$ and $\langle\mathcal{D}\dot{f}_{1},\dot{f}_{1}\rangle=0$. Hence

 $\displaystyle\mathcal{D}\langle\dot{f}_{0},\dot{f}_{0}\rangle=2\langle\mathcal {D}\Delta\dot{f}_{0},\Delta\dot{f}_{0}\rangle.$

We rewrite the last $\Delta\dot{f}_{0}$ in the last expression in the basis $\dot{f}_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0}$ and get

 $\displaystyle(\mathcal{D}\Delta\dot{f}_{0},\Delta\dot{f}_{0}\rangle=\frac{(\Delta\dot{f}_{0},\Delta f_{0},\dot{f}_{1}{\times}\Delta f _{0})}{(\dot{f}_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}\langle \mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle+\frac{(\dot{f}_{1},\Delta\dot{ f}_{0},\dot{f}_{1}{\times}\Delta f_{0})}{(\dot{f}_{1},\Delta f_{0},\dot{f}_{1} {\times}\Delta f_{0})}\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle$ (11) $\displaystyle+\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1} ,\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}(\mathcal{D}\Delta\dot{f}_{0}, \dot{f}_{1},\Delta f_{0}).$

Let us rewrite $\langle\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle$, $\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle$, and $(\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1},\Delta f_{0})$ in terms of $G_{11},\ldots,G_{33}$. First, we have:

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle=\langle \mathcal{D}\dot{f}_{0},\dot{f}_{1}\rangle=-\langle\mathcal{D}\dot{f}_{1},\dot{ f}_{0}\rangle=-\langle\mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle.$

The second equality holds since we have shown that $\mathcal{D}\langle\dot{f}_{0},\dot{f}_{1}\rangle=0$. If we rewrite $\Delta\dot{f}_{0}$ in the basis $\dot{f}_{1},\Delta f_{0},\Delta f_{1}$, we get the following:

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle=-\langle \mathcal{D}\dot{f}_{1},\Delta\dot{f}_{0}\rangle=-\frac{(\dot{f}_{1},\Delta\dot {f}_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{12}-\frac{( \dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f _{1})}G_{13}.$

Secondly, we have

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle=-\langle{ \mathcal{D}\Delta f_{0}},\Delta\dot{f}_{0}\rangle=\frac{(\Delta\dot{f}_{0}, \Delta f_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{12}- \frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}G_{23}.$

Thirdly, with

 $\displaystyle\dot{G}_{23}-\langle\mathcal{D}\Delta f_{0},\Delta\dot{f}_{1}\rangle$ $\displaystyle=\langle\mathcal{D}\Delta\dot{f}_{0},\Delta f_{1}\rangle=\frac{( \Delta f_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}{(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}\langle\mathcal{D}\Delta\dot{f}_ {0},\dot{f}_{1}\rangle$ $\displaystyle\quad+\frac{(\dot{f}_{1},\Delta f_{1},\dot{f}_{1}{\times}\Delta f _{0})}{(\dot{f}_{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}\langle \mathcal{D}\Delta\dot{f}_{0},\Delta f_{0}\rangle$ $\displaystyle\quad+\frac{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1} ,\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})}(\mathcal{D}\Delta\dot{f}_{0}, \dot{f}_{1},\Delta f_{0}).$

and the expression for $\dot{G}_{23}$ of System A we get:

 $\displaystyle(\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1},\Delta f_{0})=$ $\displaystyle-\left(\frac{(\dot{f}_{1}{\times}\Delta\dot{f}_{0},\Delta f_{0}, \Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}+\frac{(\dot{f}_{1}, \Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0} ,\Delta f_{1})}\right)G_{12}$ $\displaystyle-\frac{(\dot{f}_{1},\Delta f_{0},\Delta f_{0}{\times}\Delta\dot{f }_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{13}+\frac{(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}G_{23}.$

Finally, we substitute the obtained last three expressions for

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1}\rangle,\quad \langle\mathcal{D}\Delta\dot{f}{}_{0},\Delta f_{0}\rangle,\quad\hbox{and}\quad (\mathcal{D}\Delta\dot{f}_{0},\dot{f}_{1},\Delta f_{0})$

respectively to Expression (11) and arrive at

 $\displaystyle\langle\mathcal{D}\Delta\dot{f}_{0},\Delta\dot{f}_{0}\rangle$ $\displaystyle\quad=\left({-}\frac{(\Delta\dot{f}_{0},\Delta f_{0},\dot{f}_{1}{ \times}\Delta f_{0})(\dot{f}_{1},\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1} ,\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta f _{1})}+\frac{(\dot{f}_{1},\Delta\dot{f}_{0},\dot{f}_{1}{\times}\Delta f_{0})( \Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0},\dot{f }_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}\right.$ $\displaystyle\quad\qquad\left.-\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{ 0})(\dot{f}_{1}{\times}\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot{f}_ {1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}-\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})(\dot{f}_{1}, \Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1},\Delta f_{0} ,\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta f_{1})} \right)G_{12}$ $\displaystyle\quad\qquad+\left(-\frac{(\Delta\dot{f}_{0},\Delta f_{0},\dot{f}_ {1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f} _{1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}-\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})(\dot{f}_{1}, \Delta f_{0},\Delta f_{0}{\times}\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0} ,\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta f_{1})} \right)G_{13}$ $\displaystyle\quad\qquad+\left(-\frac{(\dot{f}_{1},\Delta\dot{f}_{0},\dot{f}_{ 1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_ {1},\Delta f_{0},\dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})}+\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})(\dot{f}_{1}, \Delta f_{0},\dot{f}_{1}{\times}\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0}, \dot{f}_{1}{\times}\Delta f_{0})(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}\right )G_{23}.$

It is clear that the coefficients of $G_{13}$ and $G_{23}$ vanish identically. Let us study the coefficient of $G_{12}$.

Consider the following mixed product $(\Delta\dot{f}_{0},\Delta\dot{f}_{0},\dot{f}_{1}{\times}\Delta f_{0})$, it is identical to zero. Let us rewrite $\Delta\dot{f}_{0}$ in the second position of the mixed product in the basis $\dot{f}_{0}$, $\Delta f_{0}$, $\Delta f_{1}$. We get the relation

 $\displaystyle\frac{(\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1} ,\Delta f_{0},\Delta f_{1})}(\Delta\dot{f}_{0},\dot{f}_{1},\dot{f}_{1}{\times} \Delta f_{0})+\frac{(\dot{f}_{1},\Delta\dot{f}_{0},\Delta f_{1})}{(\dot{f}_{1} ,\Delta f_{0},\Delta f_{1})}(\Delta\dot{f}_{0},\Delta f_{0},\dot{f}_{1}{\times }\Delta f_{0})$ $\displaystyle\quad=-\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{ f}_{1},\Delta f_{0},\Delta f_{1})}(\Delta\dot{f}_{0},\Delta f_{1},\dot{f}_{1}{ \times}\Delta f_{0}).$

We apply this identity to the first two summands of the coefficient of $G_{12}$ and get the following expression for the coefficient of $G_{12}$:

 $\displaystyle\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})(\Delta\dot{f}_ {0},\Delta f_{1},\dot{f}_{1}{\times}\Delta f_{0})}{(\dot{f}_{1},\Delta f_{0}, \Delta f_{1})|\dot{f}_{1}{\times}\Delta f_{0}|^{2}}-\frac{(\dot{f}_{1},\Delta f _{0}{\times}\Delta\dot{f}_{0},\Delta f_{1})(\dot{f}_{1},\Delta f_{0},\Delta \dot{f}_{0})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})|\dot{f}_{1}{\times} \Delta f_{0}|^{2}}$ $\displaystyle\quad-\frac{(\dot{f}_{1}{\times}\Delta\dot{f}_{0},\Delta f_{0}, \Delta f_{1})(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1},\Delta f _{0},\Delta f_{1})|\dot{f}_{1}{\times}\Delta f_{0}|^{2}}.$

We rewrite this as

 $\displaystyle\frac{(\dot{f}_{1},\Delta f_{0},\Delta\dot{f}_{0})}{(\dot{f}_{1}, \Delta f_{0},\Delta f_{1})|\dot{f}_{1}{\times}\Delta f_{0}|^{2}}\Big((\Delta \dot{f}{}_{0},\Delta f_{1},\dot{f}_{1}{\times}\Delta f_{0}){-}(\dot{f}_{1}, \Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{1})$ $\displaystyle\qquad{-}(\dot{f}_{1}{\times}\Delta\dot{f}_{0},\Delta f_{0}, \Delta f_{1})\Big).$

Let us study the expression in the brackets.

 $\displaystyle(\Delta\dot{f}_{0},\Delta f_{1},\dot{f}_{1}{\times}\Delta f_{0}){ -}(\dot{f}_{1},\Delta f_{0}{\times}\Delta\dot{f}_{0},\Delta f_{1}){-}(\dot{f}_ {1}{\times}\Delta\dot{f}_{0},\Delta f_{0},\Delta f_{1})$ $\displaystyle\quad=\!-\!\big(\Delta\dot{f}_{0}{\times}(\dot{f}_{1}{\times} \Delta f_{0}){+}\dot{f}_{1}{\times}(\Delta f_{0}{\times}\Delta\dot{f}_{0}){+} \Delta f_{0}{\times}(\Delta\dot{f}_{0}{\times}\dot{f}_{1}),\Delta f_{1}\big)\! =\!(0,\Delta f_{1})\!=\!0.$

The second equality holds by the Jacobi identity. Hence the coefficient of $G_{12}$ is zero. Therefore,

 $\displaystyle\mathcal{D}\langle\dot{f}_{0},\dot{f}_{0}\rangle=2\langle\mathcal {D}\Delta\dot{f}_{0},\Delta\dot{f}_{0}\rangle=0,$

and $\langle\dot{f}_{0},\dot{f}_{0}\rangle$ is invariant under the infinitesimal deformation.

The proof of the invariance of $\langle\dot{f}_{2},\dot{f}_{2}\rangle$ repeats the proof for $\langle\dot{f}_{0},\dot{f}_{0}\rangle$.

So we have checked the invariance of all the 11 functions in the definition of an infinitesimal flexion. Hence $\mathcal{D}f$ is an infinitesimal flexion. $\square$

Now we have all the ingredients to prove the main theorem of this subsection.

#### 3.2.4 Conclusion of the Proof of Theorem 1

Existence The existence of an infinitesimal flexion follows directly from Proposition 5$(i)$.

Uniqueness By Proposition 4 every infinitesimal flexion satisfies System A. Since we consider 2-ribbon surfaces with fixed initial position, for every non-zero infinitesimal flexion $\mathcal{D}f$ we have:

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\mathcal{D}\Delta f_{0}(a)=0, \quad\hbox{and}\quad\mathcal{D}\Delta f_{1}(a)=\alpha\dot{f}_{1}(a){\times} \Delta f_{1}(a)$

for some non-zero $\alpha$. Hence by Proposition 5 this is one of the flexions of Proposition 5$(i)$. So the set of infinitesimal flexions is one-dimensional. Since the set is a linear space, it is a line. Hence $f$ has one degree of infinitesimal flexibility. $\square$

Theorem 1 together with Proposition 5 imply the following.

###### Corollary 1.

Let $f\in C_{0}^{1,2,1}([a,b],\mathbb{R}^{3})$ be a weakly regular 2-ribbon surface with fixed initial position, and let $\mathcal{D}f$ be its infinitesimal flexion satisfying

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\mathcal{D}\Delta f_{1}(a)=0, \quad\hbox{and}\quad\mathcal{D}\Delta f_{0}(a)=0,$

Then $\mathcal{D}f=0$. $\square$

### 3.3 Operators Related to Infinitesimal Flexions

Let us fix an orthonormal basis $(e_{1},e_{2},e_{3})$ in $\mathbb{R}^{3}$. Denote by $\Omega^{1}_{3{\times}3}$ the Banach space

 $\displaystyle\big((C^{1}[a,b])^{3}\big)^{3}\cong(C^{1}[a,b])^{9}$

with the norm

 $\displaystyle\|(h_{11},h_{12},\ldots,h_{33})\|=\max\limits_{1\leq i,j\leq 3}( \max(\sup|h_{ij}|,\sup|\dot{h}_{ij}|)).$

Consider the following map

 $\displaystyle Z:C^{1,2,1}([a,b],\mathbb{R}^{3})\to\Omega^{1}_{3{\times}3},$

where for a 2-ribbon surface $f$ the image $Z(f)$ in the basis $(e_{1},e_{2},e_{3})$ is defined as

 $\displaystyle\dot{f}_{1}(t)=$ $\displaystyle\big(h_{11}(t),h_{12}(t),h_{13}(t)\big),$ $\displaystyle\Delta f_{0}(t)=$ $\displaystyle\big(h_{21}(t),h_{22}(t),h_{23}(t)\big),$ $\displaystyle\Delta f_{1}(t)=$ $\displaystyle\big(h_{31}(t),h_{32}(t),h_{33}(t)\big).$

Note that every 2-ribbon surface $f$ is defined by $\dot{f}_{1}$, $\Delta f_{0}$, and $\Delta f_{1}$ up to a translation. So after fixing, say, $f_{1}(a)=(0,0,0)$ one has a bijection.

We say that a point $h=(h_{11},h_{12},\ldots,h_{33})$ in $\Omega^{1}_{3{\times}3}$ is in general position if the determinant

 $\displaystyle\det\left(\begin{array}[]{c@{\quad}c@{\quad}c}h_{11}\quad&h_{12} \quad&h_{13}\\ h_{21}\quad&h_{22}\quad&h_{23}\\ h_{31}\quad&h_{32}\quad&h_{33}\\ \quad\end{array}\right)\neq 0$

for every $t\in[a,b]$. This condition obviously corresponds to the weak regularity condition, i.e., to

 $\displaystyle(\dot{f}_{1},\Delta f_{0},\Delta f_{1})\neq 0.$

Denote by $\Sigma_{\Omega}$ the set of all points $h$ that are not in general position.

###### Definition 17.

Denote by ${\mathcal{V}}^{\pm}:[0,\Lambda]\times(\Omega^{1}_{3{\times}3}{\setminus}\Sigma _{\Omega})\to\Omega^{1}_{3{\times}3}$ two operators of infinitesimal flexion in coordinates $(h_{11},h_{12},\ldots,h_{33})$:

 $\displaystyle{\mathcal{V}}^{\pm}_{l{-}1,m}(\lambda,h)$ $\displaystyle=$ $\displaystyle\displaystyle\frac{(e_{m},\Delta f_{0},\Delta f_{1})}{(\dot{f}_{1 },\Delta f_{0},\Delta f_{1})}G_{l{-}1,1}(h)+\frac{(\dot{f}_{1},e_{m},\Delta f_ {1})}{(\dot{f}_{1},\Delta f_{0},\Delta f_{1})}G_{l{-}1,2}(h)$ (12) $\displaystyle+\displaystyle\frac{(\dot{f}_{1},\Delta f_{0},e_{m})}{(\dot{f}_{1 },\Delta f_{0},\Delta f_{1})}G_{l{-}1,3}(h).$

for ($1\leq l,m\leq 3$). Here $G_{11}(h),G_{12}(h),\ldots,G_{33}(h)$ is a solution of System A at point $f$ with the initial conditions corresponding to

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\mathcal{D}\Delta f_{0}(a)=0, \quad\hbox{and}\quad\mathcal{D}\Delta f_{1}(a)=\pm\dot{f}_{1}(a){\times}\Delta f _{1}(a),$

i.e.,

 $\displaystyle\begin{array}[]{lll}G_{11}(a)=0,&G_{12}(a)=0,&G_{13}(a)=0,\\ G_{21}(a)=0,&G_{22}(a)=0,&G_{23}(a)=0,\\ G_{31}(a)=0,&G_{32}(a)=\pm(\dot{f}_{1}(a),\Delta f_{0}(a),\Delta f_{1}(a)),&G_ {33}(a)=0.\end{array}$ (13)

(Here we take “$+$” sign for ${\mathcal{V}}^{+}$ and “$-$” for ${\mathcal{V}}^{-}$.)

Note that both ${\mathcal{V}}^{+}$ and ${\mathcal{V}}^{-}$ are autonomous operators, they do not depend on time parameter $\lambda$.

It is important that the following statement holds.

###### Proposition 6.

Let $h$ be a point of $\Omega^{1}_{3{\times}3}$ in general position and $\lambda\in[0,\Lambda]$. Then we have

 $\displaystyle{\mathcal{V}}^{\pm}(\lambda,h)\in\Omega^{1}_{3{\times}3}.$
###### Proof.

The proof is straightforward, all functions involved in Expression (12) are continuously differentiable, and hence both ${\mathcal{V}}^{+}(\lambda,h)$ and ${\mathcal{V}}^{-}(\lambda,h)$ are continuously differentiable. $\square$

###### Remark 7.

Let us show in brief how to find the coordinates of the infinitesimal deformation $\mathcal{D}f$ in the basis $e_{1},e_{2},e_{3}$ satisfying

 $\displaystyle\begin{array}[]{c}\mathcal{D}f_{1}(a)=0,\qquad\mathcal{D}\dot{f}_ {1}(a)=0,\qquad\mathcal{D}\Delta f_{0}(a)=0,\\ \hbox{and}\quad\mathcal{D}\Delta f_{1}(a)=\dot{f}_{1}(a){\times}\Delta f_{1}(a ).\end{array}$

First, one should solve System A with the above initial data, then substitute the obtained solution $(G_{11},G_{12},\ldots,G_{33})$ to Eq. (12). Now we have the coordinates of $\mathcal{D}\dot{f}_{1}$, $\mathcal{D}\Delta f_{0}$, and $\mathcal{D}\Delta f_{1}$. Having the additional condition $\mathcal{D}f_{1}(a)=0$ one can construct $\mathcal{D}f_{1}$, $\mathcal{D}f_{0}$, and $\mathcal{D}f_{2}$:

 $\displaystyle\mathcal{D}f_{1}(t_{0})=\int\limits_{a}^{t_{0}}\mathcal{D}\dot{f} _{1}(t)d(t),\quad\mathcal{D}f_{0}=\mathcal{D}f_{1}-\mathcal{D}\Delta f_{0}, \quad\mathcal{D}f_{2}=\mathcal{D}f_{1}+\mathcal{D}\Delta f_{1}.$

Further we will work in the following subspace of $\Omega^{1}_{3{\times}3}$. Denote

 $\displaystyle\tilde{\Omega}^{1}_{3{\times}3}=\big\{h\in\Omega^{1}_{3{\times}3} \big|h_{12}(a)=h_{13}(a)=h_{23}(a)=0\big\}.$

It is clear that $\tilde{\Omega}^{1}_{3{\times}3}$ is a Banach space itself.

We have the following important property of $\tilde{\Omega}^{1}_{3{\times}3}$.

###### Proposition 7.

For every $\lambda\in[0,\Lambda]$ and $h\in\tilde{\Omega}^{1}_{3{\times}3}{\setminus}\Sigma_{\Omega}$ the subspace $\tilde{\Omega}^{1}_{3{\times}3}$ is an invariant space of the operators ${\mathcal{V}}^{+}(\lambda,h)$ and ${\mathcal{V}}^{-}(\lambda,h)$.

###### Proof.

From the conditions

 $\displaystyle\mathcal{D}\dot{f_{1}}(a)=0,\quad\hbox{and}\quad\mathcal{D}\Delta f _{0}(a)=0$

we have $G_{ij}(a)=0$ for all $i=1,2$, and $j=1,2,3$. Hence by Expression (12)

 $\displaystyle{\mathcal{V}}_{11}^{\pm}(\lambda,h)(a)={\mathcal{V}}_{12}^{\pm}( \lambda,h)(a)=\ldots={\mathcal{V}}^{\pm}_{23}(\lambda,h)(a)=0$

for all $\lambda\in[0,\Lambda]$ and $h\in\Omega^{1}_{3{\times}3}$. Therefore, for every $\lambda\in[0,\Lambda]$ and $h\in\tilde{\Omega}^{1}_{3{\times}3}{\setminus}\Sigma_{\Omega}$ we have ${\mathcal{V}}^{\pm}(\lambda,h)\in\tilde{\Omega}^{1}_{3{\times}3}$. $\square$

Finally we have the following important statement.

###### Proposition 8.

The map $Z$ is a bijection of $\tilde{\Omega}^{1}_{3{\times}3}$ and $C^{1,2,1}_{0}([a,b],\mathbb{R}^{3})$.

###### Proof.

The inverse map $Z^{-1}(h)=(f_{0},f_{1},f_{2})$ is defined as

 $\displaystyle f_{1}(t_{0})=\int\limits_{a}^{t_{0}} \begin{pmatrix}h_{11}(t)\\h_{12}(t)\\h_{13}(t)\end{pmatrix}dt$,    $f_{0}(t_{0})=f_{1}(t_{0})-\begin{pmatrix}h_{21}(t_{0})\\ h_{22}(t_{0})\\ h_{23}(t_{0})\end{pmatrix},$ $\displaystyle f_{2}(t_{0})=\displaystyle\begin{pmatrix}h_{31}(t_{0})\\ h_{32}(t_{0})\\ h_{33}(t_{0})\end{pmatrix}-f_{1}(t_{0}).$

at every $t_{0}\in[a,b]$. $\square$

### 3.4 Finite Flexibility of 2-Ribbon Surfaces

In Sect. 3.2 we showed that every 2-ribbon surface in general position is infinitesimally flexible and that the space of its infinitesimal flexions is one-dimensional. The aim of this subsection is to show that a weakly regular 2-ribbon surface is finitely flexible and has one degree of finite flexibility.

#### 3.4.1 Lipschitz Condition

We start with the discussion of the initial value problem for the following two differential equations on the set of all points $\tilde{\Omega}^{1}_{3{\times}3}$ in general position (here $\lambda$ is the time parameter):

 $\displaystyle\frac{\partial h}{\partial\lambda}={\mathcal{V}}^{+}(\lambda,h) \quad\hbox{and}\quad\frac{\partial h}{\partial\lambda}={\mathcal{V}}^{-}( \lambda,h).$ (14)

To solve the initial value problem we study local Lipschitz properties for ${\mathcal{V}}^{+}$ and ${\mathcal{V}}^{-}$.

###### Definition 18.

Consider a Banach space $E$ with a norm $|*|_{E}$, and a positive real number $\Lambda$. Let $U$ be a subset of $[0,\Lambda]\times E$. We say that a functional ${\mathcal{F}}:U\to E$ locally satisfies a Lipschitz condition if for every point $(\lambda_{0},p)$ in $U$ there exist a neighborhood $V$ of the point and a constant $K$ such that for every pair of points $(\lambda,p_{1})$ and $(\lambda,p_{2})$ in $V$ the inequality

 $\displaystyle|{\mathcal{F}}(\lambda,p_{1})-{\mathcal{F}}(\lambda,p_{2})|_{E} \leq K|p_{1}-p_{2}|_{E}$

holds.

First we verify a Lipschitz condition for the following operator. Define ${\mathcal{G}}:[0,\Lambda]\times\tilde{\Omega}^{1}_{3{\times}3}\to\tilde{\Omega }^{1}_{3{\times}3}$ by

 $\displaystyle\quad{\mathcal{G}}_{ij}(\lambda,h)=G_{ij}(h),\quad 1\leq i,j\leq 3,$

where $G_{ij}(h)$ are defined by Eq. (10).

###### Lemma 2.

For every point $h\in U$ in general position, there exists a neighborhood $V_{h}$ of $h$ such that the functional ${\mathcal{G}}$ locally satisfies a Lipschitz condition in $[0,\Lambda]\times V_{h}$.

###### Proof.

Consider a point $h\in U$. The element $(G_{11},G_{12},\ldots,G_{33})$ itself satisfies a system of linear differential equations (System A). The coefficients of this system depend only on a point of $\tilde{\Omega}_{3{\times}3}^{1}$. Since the point $h$ is in general position, there exists a positive real constant $K$ such that for a sufficiently small neighborhood $V_{h}$ of $h$ the dependence is $K$-Lipschitz, i.e., for $p$ and $q$ from $V_{h}$ every coefficient $c$ of System A satisfies the inequality

###### Acknowledgements
The author is grateful to J. Wallner for constant attention to this work, A. Weinmann for good remarks, and the unknown reviewer for excellent comments and suggestions. This work has been partially supported by the project “Computational Differential Geometry” (FWF Grant No. S09209).

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