Framed Curves, Ribbons, and Parallel Transport on the Sphere

Abstract

We consider curves $\gamma : [0, 1]\rightarrow {\mathbb {R}}^3$ endowed with an adapted orthonormal frame $r: [0, 1]\rightarrow SO(3)$. We wish to deform such framed curves $(\gamma , r)$ while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of $r'r^T$), and a global constraint prescribing the initial and terminal values of $\gamma$ and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve $\gamma$ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on $(\gamma , r)$ that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.

1 Introduction

Elastic inextensible rods can be modelled by framed curves, i.e., pairs $(\gamma , r)$ consisting of a curve $\gamma$ mapping some interval $I\subset {\mathbb {R}}$ into ${\mathbb {R}}^3$ and a frame field $r: I\rightarrow SO(3)$ which is adapted to $\gamma$ in the sense that $r^T e_1 = \gamma '$. In the context of the Cosserat theory of rods, the rows $r_2, r_3$ are called directors. For such rod theories and their generalizations we refer, e.g., to the monographs (Antman 2005; Ciarlet 1997). A concise exposition of Cosserat theory can be found, e.g., in Schuricht (2002). In this context Sobolev spaces of curves arise naturally; cf. also Maddocks (1984). Dynamical problems involving Kirchhoff rods (Kirchhoff 1859) were studied, e.g., in Lin and Schwetlick (2004), Dall’Acqua et al. (2017). Rods with fixed ends play a role in da Fonseca and de Aguiar (2003).

Framed curves also arise in differential geometry. There is a frame field – the Darboux frame – that is naturally associated with any curve $\gamma$ in an immersed surface. For this reason, framed curves play a role in asymptotic theories for nonlinearly elastic plates (Friesecke et al. 2002; Hornung 2011b) and, in particular, ribbons (Kirby and Fried 2015; Freddi et al. 2016; Dias and Audoly 2015; Paroni and Tomassetti 2019) such as the developable Möbius strip (Sadowsky 1930; Wunderlich 1962; Hinz and Fried 2015a, b; Starostin and van der Heijden 2007; Bartels and Hornung 2015). In Dias and Audoly (2015) elastic ribbons are described by framed curves endowed with a ‘material’ frame instead of the Frénet frame used, e.g., in Wunderlich (1962), Kirby and Fried (2015). For a general approach we refer to Hornung (2011b), Hornung (2011a).

In these situations a differential constraint of the form $a_{ij}:= (r^Te_i)'\cdot r^T e_j = 0$ arises naturally. More precisely, in Wunderlich (1962), Starostin and van der Heijden (2007), Kirby and Fried (2015), Randrup and Rogen (1996) the Frénet frame is used to model a narrow ribbon; it is characterized by the differential constraint $a_{13}\equiv 0$. In Dias and Audoly (2015), Freddi et al. (2016) plates and ribbons are described by framed curves satisfying the differential constraint $a_{12}\equiv 0$. They can also be described by framed curves arising from lines of curvature Hornung (2011b), Hornung (2011a), which therefore satisfy the differential constraint $a_{23}\equiv 0$ of vanishing torsion; this constraint also characterizes the relatively parallel frame in Bishop (1975).

Constrained framed curves also arise in DNA models (Chouaieb and Maddocks 2005; Chouaieb et al. 2006). They have also been studied for their own sake, see e.g. Honda and Takahashi (2020), Yı lmaz and Turgut (2010), Rogen (1999), Scherrer (1946), Solomon (1996), Hwang (1981). For a discussion of the various ways to frame a given space curve we refer to Bishop (1975), da Silva (2017).

For applications it is therefore useful to understand the possible shapes that can be attained by constrained framed curves and to be able to deform framed curves while preserving differential constraints of the form $a_{ij}\equiv 0$. Such results have proven to be useful in numerical analysis (Bartels and Reiter 2020; Bartels 2020) as well as the calculus of variations (Hornung 2011b; Freddi et al. 2016, 2022).

In fact, constrained framed curves arise in the context of nonlinearly elastic plates (Friesecke et al. 2002), because framed curves satisfying a suitable curvature constraint can be locally associated with $W^{2,2}$ isometric immersions, i.e., with finite energy deformations of such plates (Pakzad 2004; Hornung 2011a, 2023). The global constraint prescribing the endpoints of the framed curve arises in this context, e.g., when the framed curve models a ribbon clamped at its ends (Freddi et al. 2022). In a less immediate way it also arises in the context of nonlinearly elastic plates with finite width, even without regard to externally imposed boundary conditions: preserving the endpoints of a framed curve which describes the deformation locally allows for ‘local’ deformations of such plates (Hornung 2011a, b) in spite of their rigidity, as opposed to global deformations (Pakzad 2004).

Summarizing, we consider framed curves $(\gamma , r)$ satisfying two constraints: a differential constraint asserting the vanishing of a ‘curvature’, and a global constraint fixing the initial and terminal values of both the frame r and the curve $\gamma$. In the present article the deformation process is carried out in two successive stages. In the first stage we modify the frame $r: I\rightarrow SO(3)$ while preserving the differential constraint and possibly the value of r at its endpoints. In contrast to earlier work, this is not achieved by explicitly modifying the curvatures $a_{ij}$ and subsequently solving an ordinary differential equation. Instead, we interpret constraints such as $a_{ij}\equiv 0$ geometrically in terms of parallel transport on the sphere. This geometric viewpoint is explained in more detail in Sect. 2.3.1. It allows us to work directly with the frames, which is more intuitive than working with the curvatures. Moreover, it provides a link to differential geometry and therefore allows us to exploit some tools and notions from the differential geometry of surfaces. As a result, it also leads to a simple answer to a question left open in Hornung (2021), cf. Sect. 3 below.

In the second stage we reparametrize the frame in a suitable way in order to achieve the prescribed boundary conditions on $\gamma$. This stage, described in Sect. 2.3.2, is related to one-dimensional convex integration (Gromov 1986).

This article is organized as follows. In Sect. 2 we briefly introduce some basic notions and notation regarding framed curves, and then we proceed to describe in some detail the two stages outlined earlier. In Sects. 3 and 4 we provide two simple applications of this method:

In Sect. 3 we characterize the possible clamped boundary conditions on a framed curve that are compatible with the differential constraint. We show that a single constraint of the form $a_{ij}\equiv 0$ does not restrict the boundary conditions that can be accessed by a framed curve, therefore generalizing a corresponding local result in Hornung (2021); we also refer to Freddi et al. (2022).

In Sect. 4 we give a self-contained and short proof of one of the main results from Hornung (2021), Bartels and Reiter (2020), namely a smooth approximation result for framed curves with prescribed endpoints. The proof given here differs fundamentally from the proofs in those papers.

In Sect. 5 we consider ribbons with finite (as opposed to infinitesimal) width. Such objects are more rigid than framed curves, because the admissible deformations are isometric immersions of genuinely two-dimensional reference domains.

Notation. The set of rotation matrices is denoted by SO(n). The letter I will denote both the identity matrix and the open interval (0, 1). We denote by $(e_1, e_2, e_3)$ the canonical basis of ${\mathbb {R}}^3$ and by $B_r(a)$ the open ball of radius r centered at a. If $M\subset {\mathbb {R}}^n$ is a smooth manifold, then by $W^{1,p}(I, M)$ we mean the set of maps $a\in W^{1,p}(I, M)$ satisfying $a(t)\in M$ for almost every $t\in I$.

When $M\subset {\mathbb {R}}^k$ is a set then $M^{\perp }$ denotes its orthogonal complement in ${\mathbb {R}}^k$. If $x\in {\mathbb {R}}^2$ then $x^{\perp }$ denotes the vector obtained by rotating x counter-clockwise by $\frac{\pi }{2}$.

For two maps f and g at times we write f(g) instead of $f\circ g$ to denote their composition. The support of a continuous map f is denoted by ${{\,\textrm{spt}\,}}f$.

2 Parallel Transport and Naïve Convex Integration

2.1 Parallel Transport

For a given absolutely continuous curve $\beta : [a,b]\rightarrow {\mathbb {S}}^2$ let us denote by $\Pi _{\beta }$ the linear isomorphism from the tangent space $T_{\beta (a)}{\mathbb {S}}^2$ to ${\mathbb {S}}^2$ at the point $\beta (a)$ to the tangent space $T_{\beta (b)}{\mathbb {S}}^2$ that maps a tangent vector $v\in T_{\beta (a)}{\mathbb {S}}^2$ to the tangent vector $\Pi _{\beta }v\in T_{\beta (b)}{\mathbb {S}}^2$ obtained by parallel transporting v along $\beta$ from $\beta (a)$ to its endpoint $\beta (b)$. For details as well as geometric interpretations of parallel transport we refer, e.g., to do Carmo (1976, Section 4-4).

For the reader’s convenience here we briefly discuss parallel transport on ${\mathbb {S}}^2$. For a curve $\beta : [a,b]\rightarrow {\mathbb {S}}^2$ and a tangent vector $v\in T_{\beta (a)}{\mathbb {S}}^2$ the parallel transport $\Pi _{\beta }v$ is defined to be the terminal value V(b) of the unique vector field $V: I\rightarrow {\mathbb {R}}^3$ along $\beta$ with $V(a) = v$ which is everywhere tangent to ${\mathbb {S}}^2$ and whose covariant derivative vanishes at every point. In the present context, these last two conditions amount to the requirements

$$\begin{aligned} V\cdot \beta = 0\hbox { and }V'\parallel \beta \end{aligned}$$

(1)

everywhere on (a, b). If $V(a)\in T_{\beta (a)}{\mathbb {S}}^2$ then (1) are equivalent to the linear ordinary differential system

$$\begin{aligned} V' = -(\beta '\cdot V)\beta . \end{aligned}$$

(2)

In fact, differentiation of the first equation in (1) gives $V'\cdot \beta = -V\cdot \beta '$, which in view of the second condition in (1) implies (2). Conversely, if (2) is satisfied, then the second condition in (1) is satisfied and so is the derivative of the first one. Since $V(a)\cdot \beta (a) = 0$, the first condition in (1) is satisfied everywhere.

For $\beta \in W^{1,1}$ solutions V of (2) exist and are unique for any given initial value $V(a)\in T_{\beta (a)}{\mathbb {S}}^2$. Hence the parallel transport map $\Pi _{\beta }V(a):= V(b)$ along such curves $\beta$ is well-defined. For later use we note the following immediate consequences of (2).

Lemma 2.1

Let $p\in [1, \infty ]$, let $\beta \in W^{1,p}(I, {\mathbb {S}}^2)$, let $v\in T_{\beta (0)}{\mathbb {S}}^2$ and define $V(t) = \Pi _{\beta |_{(0, t)}}v$. Then the following are true:

• (i) We have $V\in W^{1,p}(I)$ and $|V(t)| = |v|$ for all $t\in I$.

• (ii) If $\beta _n\in W^{1,p}(I, {\mathbb {S}}^2)$ satisfy $\beta _n(0) = \beta (0)$ and $\beta _n\rightarrow \beta$ in $W^{1,p}$, then the vector fields $V_n$ defined by $V_n(t) = \Pi _{\beta _n|_{(0, t)}}v$ converge to V uniformly on I. In particular, if $\beta _n(1) = \beta (1)$ then $\Pi _{\beta _n}\rightarrow \Pi _{\beta }$.

Proof

Let $v\in T_{\beta (0)}{\mathbb {S}}^2$ and set $V(t) = \Pi _{\beta |_{(0, t)}}v$. Then (2) is satisfied. An immediate consequence of the definition of parallel transport is that it preserves the length of the transported vector field. In fact, (1) implies that $V\cdot V' = 0.$ Hence $|V(t)| = |v|$ for all $t\in I$. Since $|\beta (t)| = 1$ for almost every $t\in I$, we conclude that indeed $V'\in L^p(I)$.

To prove (ii), set

$$\begin{aligned} b_n = \beta \otimes \beta ' - \beta _n\otimes \beta _n'. \end{aligned}$$

Then $b_n\rightarrow 0$ in $L^p(I)$ and

$$\begin{aligned} (V_n - V)' = -\left( \beta '\cdot (V_n - V)\right) \beta + b_nV_n. \end{aligned}$$

Gronwall’s inequality implies that

$$\begin{aligned} |V_n(t) - V(t)| \le C\int _0^t|b_nV_n|\hbox { for all }t\in I. \end{aligned}$$

Since $|V_n| = |v|$ and $b_n\rightarrow 0$ in $L^1$, the right-hand side converges to zero uniformly. $\square$

The next lemma and its corollary are classical geometric facts. In the statement and proof of Lemma 2.2 we use the notation $DE_i$ to denote the differential of $E_i$, i.e., the linear map taking a tangent vector v to the directional derivative $D_vE_i$ of $E_i$ in the direction v.

Lemma 2.2

Let $U\subset {\mathbb {S}}^2$ be open and let $(E_1, E_2)$ be a smooth tangent orthonormal frame field on $\overline{U}$, and denote by $\omega = E_2\cdot DE_1$ the corresponding connection form. Let $\beta$, ${\widetilde{\beta }}: I\rightarrow U$ be absolutely continuous and such that $\beta (0) = {\widetilde{\beta }}(0)$ and $\beta (1) = {\widetilde{\beta }}(1)$. Then $\Pi _{\beta } = \Pi _{\widetilde{\beta }}$ if and only if $\int _{\beta }\omega = \int _{\widetilde{\beta }}\omega .$

Proof

Let $V_0$ be a unit vector in the tangent space to ${\mathbb {S}}^2$ at $\beta (0)$ and denote by $V(t) = \Pi _{\beta |_{(0, t)}}V_0$ its parallel transport along $\beta$. Since V is absolutely continuous, there exists an absolutely continuous function $\varphi : I\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} V(t) = E_1(\beta (t))\cos \varphi (t) + E_2(\beta (t))\sin \varphi (t)\hbox { for all }t\in I. \end{aligned}$$

We have

$$\begin{aligned} \partial _t V = \varphi '\cdot (-\sin \varphi E_1(\beta ) + \cos \varphi E_2(\beta )) + D_{\beta '}E_1(\beta )\cos \varphi + D_{\beta '}E_2(\beta )\sin \varphi . \end{aligned}$$

Since V is parallel along $\beta$, we have $E_1\cdot \partial _t V = E_2\cdot \partial _t V = 0$. We conclude that

$$\begin{aligned} \omega (\beta )(\beta ') + \varphi ' = 0\hbox { almost everywhere on }I, \end{aligned}$$

Integration over I yields

$$\begin{aligned} \int _{\beta }\omega = \varphi (0) - \varphi (1). \end{aligned}$$

(3)

The assertion follows from this formula and the same formula for ${\widetilde{\beta }}$. $\square$

The next result is a version of the Gauss-Bonnet formula relating holonomy and area of a domain. It follows from (3) by applying Stokes’ Theorem and using that $d\omega$ agrees with the area form on ${\mathbb {S}}^2$.

Proposition 2.3

Let $S\subset {\mathbb {S}}^2$ be a simply connected smoothly bounded domain with area $\theta$ and let $\beta \in C^1({\mathbb {S}}^1, {\mathbb {S}}^2)$ be a simple positively oriented parametrization of its boundary $\partial S$. Then $\Pi _{\beta }: T_{\beta (0)}{\mathbb {S}}^2\rightarrow T_{\beta (0)}{\mathbb {S}}^2$ is given by $\Pi _{\beta } = R_{\beta (0)}^{(\theta )}$.

2.2 Constrained Framed Curves

In this article, a framed curve $(\gamma , r)$ consists of an absolutely continuous frame field $r: I\rightarrow SO(3)$, whose rows will henceforth be denoted by

$$\begin{aligned} r_i = r^Te_i, \end{aligned}$$

and of a curve $\gamma : I\rightarrow {\mathbb {R}}^3$ satisfying $\gamma ' = r_1$. The frame r satisfies an ordinary differential equation of the form

$$\begin{aligned} \begin{pmatrix} r_1 \\ r_2 \\ r_3 \end{pmatrix}' = \begin{pmatrix} 0 &{}\quad a_{12} &{}\quad a_{13} \\ - a_{12} &{}\quad 0 &{}\quad a_{23} \\ - a_{13} &{}\quad - a_{23} &{}\quad 0 \end{pmatrix} \begin{pmatrix} r_1 \\ r_2 \\ r_3 \end{pmatrix}, \end{aligned}$$

(4)

for suitable ‘curvatures’ $a_{ij}: I\rightarrow {\mathbb {R}}$. For instance, if $\gamma$ is a curve in a surface and $r_3$ denotes the normal to the surface along $\gamma$, then r is the so-called Darboux frame. In this case, $a_{13}$ is the normal curvature of $\gamma$, whereas $a_{12}$ is its geodesic curvature and $a_{23}$ is its geodesic torsion.

As in Hornung (2021), let k, $l\in \{1, 2, 3\}$ be unequal and set

$$\begin{aligned} {\mathfrak {A}}_{kl} = \left\{ \begin{pmatrix} 0 &{}\quad a_{12} &{}\quad a_{13} \\ - a_{12} &{}\quad 0 &{}\quad a_{23} \\ - a_{13} &{}\quad - a_{23} &{}\quad 0 \end{pmatrix}\in {\mathbb {R}}^{3\times 3}: a_{kl} = 0 \right\} . \end{aligned}$$

(5)

This gives rise to three different two dimensional subspaces ${\mathfrak {A}}_{kl}\subset {\mathbb {R}}^{3\times 3}$, depending on the choice of k, l.

2.3 Deformation of Constrained Framed Curves in Two Stages

In this section we describe the deformation process for constrained framed curves which we will use in this article. It differs markedly from earlier approaches (Hornung 2021, 2011a, b; Pakzad 2004).

2.3.1 First Stage: Parallel Transport on the Sphere

In this short section we describe the viewpoint adopted in the first stage of the deformation process. The following simple observation links constrained framed curves to parallel transport on the sphere.

Proposition 2.4

Let (i, k, l) be a permutation of (1, 2, 3), let $r: I\rightarrow SO(3)$ be absolutely continuous and assume that $r_k'\cdot r_{l} \equiv 0$. Define $\beta : I\rightarrow {\mathbb {S}}^2$ by $\beta = r_i$. Then we have

$$\begin{aligned} r_j(t) = \Pi _{\beta |_{(0, t)}}r_j(0)\hbox { for all }t\in I \end{aligned}$$

(6)

for $j = k$, l.

Proof

Since $r_k$ is a unit vector field, we have $r_k'\cdot r_k \equiv 0$. Hence the constraint $r_k'\cdot r_{l} \equiv 0$ implies that $r_k'$ is normal to ${\mathbb {S}}^2$ along $\beta$. Hence $r_k$ is parallel transported along the curve $\beta$. The same is true for $r_{l}$. $\square$

In view of Proposition 2.4, prescribing the end value r(1) of the frame r is equivalent to prescribing the end value $\beta (1)$ of the curve $\beta : I\rightarrow {\mathbb {S}}^2$ as well as the associated parallel transport map $\Pi _{\beta }: T_{\beta (0)}{\mathbb {S}}^2\rightarrow T_{\beta (1)}{\mathbb {S}}^2$.

2.3.2 Second Stage: Convex Integration

A reparametrization of the curve $\beta$ clearly does not change its endpoints, nor does it affect the parallel transport map $\Pi _{\beta }$. It does, however, affect the base curve $\gamma : t\mapsto \int _0^t r_1$. This naturally leads us to consider reparametrizations of $\beta$ (or, equivalently, of the frame r) as a separate degree of freedom. This degree of freedom leads to different curves $\gamma$ corresponding to the same trace $r(I)\subset SO(3)$.

Let us be more precise. For a given continuous frame field $r: I\rightarrow SO(3)$ we define reparametrized frame fields as follows. Let $\eta : I\rightarrow {\mathbb {R}}$ be integrable, bounded from below by a positive constant and such that $\int _0^1\eta = 1$. For each such $\eta$ we define the reparametrized frame field ${\widetilde{r}}^{(\eta )}: I\rightarrow SO(3)$ by setting

$$\begin{aligned} {\widetilde{r}}^{(\eta )}\left( \int _0^t\eta \right) = r(t)\hbox { for all }t\in I. \end{aligned}$$

(7)

The following proposition is related to one-dimensional convex integration (Gromov 1986), but clearly the situation considered here is very basic.

Proposition 2.5

Let $r\in C^0(\overline{I}, SO(3))$ and let $\overline{\gamma }$ be a point in the interior of the convex hull of $r_1(\overline{I})\subset {\mathbb {R}}^3$. Then there exists an everywhere positive function $\eta \in C^{\infty }(\overline{I})$ with $\int _I \eta = 1$ such that the reparametrized frame field ${\widetilde{r}}^{(\eta )}: I\rightarrow SO(3)$ defined by (7) satisfies

$$\begin{aligned} \int _0^1 {\widetilde{r}}^{(\eta )}_1 = \overline{\gamma }. \end{aligned}$$

Remarks

• (i) Clearly, in the degenerate case when $r_1$ is constant, so is ${\widetilde{r}}_1^{(\eta )}$, for whichever choice of $\eta$. In this case we therefore have $\int _0^1 {\widetilde{r}}_1^{(\eta )} = \int _0^1 r_1$ for all $\eta$ as above. The generic case is covered by the proposition.

• (ii) In order to see the effect of the reparametrization (7) on the trace of the base curve, notice that the base curve determined by ${\widetilde{r}}^{(\eta )}$ is

  $$\begin{aligned} {\widetilde{\gamma }}^{(\eta )}(t) = \int _0^t {\widetilde{r}}^{(\eta )}_1. \end{aligned}$$

  A change of variables shows that

  $$\begin{aligned} {\widetilde{\gamma }}^{(\eta )}\left( \int _0^t\eta \right) = \int _0^t {\widetilde{r}}_1^{(\eta )}\left( \int _0^s\eta \right) \eta (s)\ ds. \end{aligned}$$

  Hence (7) implies

  $$\begin{aligned} {\widetilde{\gamma }}^{(\eta )}\left( \int _0^t\eta \right) = \int _0^t r_1\eta . \end{aligned}$$

  (8)

In the proof of Proposition 2.5 we will use the following lemma, the proof of which is left to the reader.

Lemma 2.6

Let $N\ge 4$, let x, $a_1$,..., $a_N\in {\mathbb {R}}^3$ and let $\varepsilon > 0$. Then there exists $\delta > 0$ such that the following is true: If $B_{\varepsilon }(x)$ is contained in the convex hull of $\{a_1,..., a_N\}$ and ${\widetilde{a}}_1,..., {\widetilde{a}}_N\in {\mathbb {R}}^3$ are such that $|{\widetilde{a}}_i - a_i| \le \delta$ for $i = 1,..., N$, then x is contained in the convex hull of $\{{\widetilde{a}}_1,..., {\widetilde{a}}_N\}$.

Proof of Proposition 2.5

There exist $N\in {\mathbb {N}}$, $\varepsilon > 0$ and $0 \le t_1< \cdots < t_N \le 1$ such that $B_{\varepsilon }(\overline{\gamma })$ is contained in the convex hull of $\{r_1(t_1),..., r_1(t_N)\}$. In fact, by Rockafellar (1997, Theorem 17.1) (cf. also Steinitz (1916)) the number N can be bounded in terms of the dimension, hence in the present case by an absolute constant.

Let $\delta$ be as furnished by Lemma 2.6 below, applied with $a_i = r_1(t_i)$ and $x = \overline{\gamma }$. For all $i = 1,..., N$ we can find a function $\eta _i\in C^{\infty }(\overline{I})$ with $\int _I\eta _i = 1$ which is positive everywhere on $\overline{I}$, and for which

$$\begin{aligned} {\widetilde{a}}_i = \int _I \eta _i r_1 \end{aligned}$$

(9)

is contained in $B_{\delta }(r_1(t_i))$. Such $\eta _i$ exist because $r_1$ is continuous and for each i there exists a sequence of positive functions in $C^{\infty }(\overline{I})$ with unit mass converging weakly-$*$ as measures to the Dirac measure concentrated at $t_i$.

By Lemma 2.6 the point $\overline{\gamma }$ is contained in the convex hull of the ${\widetilde{a}}_i$. Hence by Rockafellar (1997, Chapter I, Theorem 2.3) there exist nonnegative numbers $\lambda _1$,..., $\lambda _N$ with

$$\begin{aligned} \sum _{i = 1}^N\lambda _i = 1 \end{aligned}$$

(10)

and such that

$$\begin{aligned} \overline{\gamma }= \sum _{i = 1}^N\lambda _i{\widetilde{a}}_i. \end{aligned}$$

(11)

Set $\eta = \sum _{i = 1}^N \lambda _i\eta _i$. Then $\eta \in C^{\infty }(\overline{I})$ is positive because all $\eta _i$ are positive and at least one $\lambda _i$ is nonzero. Moreover, $\int _I\eta = 1$ due to (10). By the definition (9) of ${\widetilde{a}}_i$,

$$\begin{aligned} \int _0^1 {\widetilde{r}}^{(\eta )}_1 = \int _0^1 \eta r_1 = \sum _{i = 1}^N\lambda _i{\widetilde{a}}_i. \end{aligned}$$

By (11) the right-hand side agrees with $\overline{\gamma }$. $\square$

2.4 Frames with Prescribed Curvature

In Sect. 2.3.1 we only addressed homogeneous constraints of the form $a_{ij}\equiv 0$. As an immediate consequence, one can also handle inhomogeneous constraints of the form $a_{ij} = \kappa$ with a prescribed function $\kappa \in L^1(I)$. In order to be specific, we will only consider the constraint $a_{12} = \kappa$. The following proposition is included here for completeness; it will not be relevant elsewhere in this article.

Proposition 2.7

Let $\kappa \in L^1(I)$ and let $r\in L^{\infty }(I, SO(3))$ be such that $r_3\in W^{1,1}(I)$. Then the following are equivalent:

• (i) For all $t\in I$ we have

  $$\begin{aligned} \begin{aligned} r_1(t)&= \cos \left( \int _0^t\kappa \right) \Pi _{r_3|_{(0,t)}}r_1(0) + \sin \left( \int _0^t\kappa \right) \Pi _{r_3|_{(0,t)}}r_2(0). \end{aligned} \end{aligned}$$

  (12)

• (ii) $r\in W^{1,1}(I)$ and $r_1'\cdot r_2 = \kappa$ almost everywhere.

• (iii) $r\in W^{1,1}(I)$ and there exist $\tau$, $\mu \in L^1(I)$ such that

  $$\begin{aligned} r' = \begin{pmatrix} 0 &{}\quad \kappa &{}\quad \mu \\ -\kappa &{}\quad 0 &{}\quad \tau \\ -\mu &{}\quad -\tau &{}\quad 0 \end{pmatrix} r \end{aligned}$$

  (13)

Proof

The equivalence of (iii) and (ii) is clear.

Notice that if (i) is satisfied, then Lemma 2.1 implies that $r\in W^{1,1}(I)$. So in order to prove the equivalence of (i) and (ii) we must show the following: if r, ${\widetilde{r}}\in W^{1,1}(I, SO(3))$ satisfy $r_3 = {\widetilde{r}}_3$ and (with $K(t) = \int _0^t\kappa$)

$$\begin{aligned} r_1 = {\widetilde{r}}_1\cos K + {\widetilde{r}}_2\sin K, \end{aligned}$$

(14)

then

$$\begin{aligned} {\widetilde{r}}_1(t) = \Pi _{r_3|_{(0, t)}} r_1(0) \end{aligned}$$

(15)

if and only if $r_1'\cdot r_2 = \kappa$.

To verify this it is enough to take derivatives on both sides of (14) to see that

$$\begin{aligned} r_1'\cdot r_2 = {\widetilde{r}}_1'\cdot {\widetilde{r}}_2 + \kappa . \end{aligned}$$

So $r_1'\cdot r_2 = \kappa$ if and only if ${\widetilde{r}}_1'\cdot {\widetilde{r}}_2 = 0$, which in turn is equivalent to (15). $\square$

Remarks

• (i) In what follows, for $z\in {\mathbb {S}}^2$, an angle $\theta \in {\mathbb {R}}$ and $v\in T_z{\mathbb {S}}^2$, we will use the notation

  $$\begin{aligned} R_z^{(\theta )}v = v\cos \theta + (z\times v)\sin \theta . \end{aligned}$$

  This expression, which can be viewed as a particular case of the Rodrigues rotation formula, describes a counter-clockwise rotation by an angle $\theta$ of the tangent vector v within the tangent plane $T_z{\mathbb {S}}^2$.

  With this notation, (12) can be written as

  $$\begin{aligned} r_1(t) = R_{r_3(t)}^{\left( \int _0^t\kappa \right) }\Pi _{r_3|_{(0,t)}}r_1(0). \end{aligned}$$

• (ii) Let $\alpha \in (\frac{1}{2},1)$. Then the parallel transport $\Pi _{\beta }$ along a curve $\beta \in C^{0, \alpha }(I, {\mathbb {S}}^2)$ is well-defined (Borisov 1971). Therefore, one can use Proposition 2.7 to make the following definition: A frame $r\in L^{\infty }(I, SO(3))$ with $r_3\in C^{0, \alpha }(I)$ is said to have geodesic curvature $\kappa \in L^1(I)$ if it satisfies (12).

2.5 Curves with Bounded Geodesic Curvature

We recall that the curves $\beta : I\rightarrow {\mathbb {S}}^2$ considered in this article are not immersed in general.

Definition 2.8

An adapted frame for a curve $\beta \in W^{1,1}(I, {\mathbb {S}}^2)$ is a map $r\in L^{\infty }(I, SO(3))$ satisfying

$$\begin{aligned} r_3&= \beta \hbox { almost everywhere on }I \end{aligned}$$

(16)

$$\begin{aligned} \beta '\times r_1&= 0\hbox { almost everywhere on }I. \end{aligned}$$

(17)

For $\beta \in W^{1,1}(I, {\mathbb {S}}^2)$ define the open set

$$\begin{aligned} C_{\beta } = \{t\in I: \beta \hbox { is constant in a neighbourhood of }t\}. \end{aligned}$$

For v, $w\in {\mathbb {S}}^2$ we will write

$$\begin{aligned} v = \pm w \end{aligned}$$

to mean that $v\in \{w, -w\}$, i.e., that v and w agree as elements of the projective space ${\mathbb {P}}^2$.

Lemma 2.9

Let $\beta \in W^{1,1}(I, {\mathbb {S}}^2)$ and let $r\in L^{\infty }(I, SO(3))$ be an adapted frame for $\beta$. Then the following are true:

• (i) We have $\beta ' = (\beta '\cdot r_1)\ r_1$ almost everywhere on I and

  $$\begin{aligned} r_1 = \pm \frac{\beta '}{|\beta '|}\hbox { almost everywhere on }\{\beta '\ne 0\}. \end{aligned}$$

  (18)

• (ii) We have $r_3\in W^{1,1}(I)$ and

  $$\begin{aligned} r_3'\cdot r_2 = 0\hbox { almost everywhere on }I. \end{aligned}$$

  (19)

• (iii) If $r\in C^0(I, SO(3))$ and if ${\widetilde{r}}\in C^0(I, SO(3))$ is another adapted frame, then

  $$\begin{aligned} {\widetilde{r}}_1 = \pm r_1\hbox { on }I\setminus C_{\beta }. \end{aligned}$$

  (20)

Remarks

• (i) An adapted frame $r\in L^{\infty }(I, SO(3))$ always exists: set $r_3 = \beta$ and on the set where $\beta '\ne 0$ define $r_1 = \frac{\beta '}{|\beta '|}$. Elsewhere set $r_1 = e_1$, say.

• (ii) Since $r_3 = \beta$, Eq. (20) implies that ${\widetilde{r}}_2 = \pm r_2$ on $I\setminus C_{\beta }$ as well.

Proof of Lemma 2.9

Equation (18) is obvious. Equation (19) follows from (16), (17) and the fact that $|\beta |\equiv 1$.

Now assume that r, ${\widetilde{r}}$ are continuous adapted frames for $\beta$. Let $t_0\in I\setminus C_{\beta }$. Then for all $\varepsilon > 0$ the set $I_{\varepsilon } = (t_0-\varepsilon , t_0+\varepsilon )\cap \{\beta ' \ne 0\}$ has positive measure. Hence by (18) we have $r_1 = \pm {\widetilde{r}}_1$ almost everywhere on $I_{\varepsilon }$, hence by the arbitrariness of $\varepsilon$ and by continuity $r_1(t_0) = \pm {\widetilde{r}}_1(t_0)$. $\square$

Definition 2.10

A curve $\beta \in W^{1,1}(I, {\mathbb {S}}^2)$ is said to have geodesic curvature $\kappa \in L^1(I)$ if there exists an adapted frame $r\in W^{1,1}(I, SO(3))$ for $\beta$ satisfying

$$\begin{aligned} r_1'\cdot r_2 = \kappa \hbox { almost everywhere on }I. \end{aligned}$$

(21)

We will say that $\beta$ has bounded geodesic curvature provided that there exists an adapted frame $r\in W^{1,1}(I, SO(3))$ for $\beta$ satisfying $r_1'\cdot r_2\in L^{\infty }(I)$.

Remarks

• (i) An apparently more general definition would only require $r\in L^{\infty }(I)$ and replace (21) by the condition that r satisfy (12). However, when $\beta \in W^{1,1}$ then it is equivalent to Definition 2.10. This follows from Proposition 2.7.

• (ii) In view of Lemma 2.9 the geodesic curvature $\kappa$ is uniquely determined by $\beta$ almost everywhere on $I\setminus C_{\beta }$.

The next lemma shows that having bounded geodesic curvature is a local property.

Lemma 2.11

Let $\beta \in W^{1,1}(I, {\mathbb {S}}^2)$ and assume that for every $t_0\in \overline{I}$ there exists $\varepsilon > 0$ such that the restriction of $\beta$ to $\overline{I}\cap (t_0-\varepsilon , t_0+\varepsilon )$ has bounded geodesic curvature. Then $\beta$ has bounded geodesic curvature.

Proof

We cover $\overline{I}$ by finitely many relatively open intervals $J_1$,..., $J_N\subset \overline{I}$ on each of which $\beta$ has bounded geodesic curvature. So for each $i = 1,..., N$ there is an adapted frame $r^{(i)}\in W^{1,1}(J_i, SO(3))$ for $\beta$ satisfying $(r^{(i)}_1)'\cdot r^{(i)}_2\in L^{\infty }(J_i)$.

By iteration we may assume without loss of generality that $N = 2$ and that $0\in J_1$ and $1\in J_2$. To construct an adapted frame defined on all of $J_1\cup J_2$, we observe that two cases can occur:

If $J_1\cap J_2\setminus C_{\beta }$ is nonempty, then let $t_0$ be a point in this set and define

$$\begin{aligned} r = {\left\{ \begin{array}{ll} r^{(1)}&{}\hbox { on }(0, t_0] \\ r^{(2)}&{}\hbox { on }(t_0, 1). \end{array}\right. } \end{aligned}$$

The map r is continuous because (after possibly replacing $r^{(2)}$ by $-r^{(2)}$) we have $r^{(1)}(t_0) = r^{(2)}(t_0)$ by Lemma 2.9 (iii). Hence $r\in W^{1,1}(I)$ and, moreover, $r_1'\cdot r_2$ is bounded. So indeed $\beta$ has bounded geodesic curvature on I.

If $J_1\cap J_2\subset C_{\beta }$ then $J_1\cap J_2$ is contained in a maximal interval $(t_0, t_1)\subset C_{\beta }$. If $(t_0, t_1) = I$, then $\beta$ is constant and there is nothing to prove. Otherwise, we denote by ${\widetilde{r}}_1: [t_0, t_1]\rightarrow {\mathbb {S}}^2$ a smooth interpolation from $r_1^{(1)}(t_0)$ to $r_1^{(2)}(t_1)$ within the unit circle in the plane $T_{\beta (t_0)}{\mathbb {S}}^2$. Then we define ${\widetilde{r}}: [t_0, t_1]\rightarrow SO(3)$ by setting ${\widetilde{r}}_3 = \beta$, and we define

$$\begin{aligned} r = {\left\{ \begin{array}{ll} r^{(1)} &{}\hbox { on }[0, t_0] \\ {\widetilde{r}} &{}\hbox { on }(t_0, t_1) \\ r^{(2)} &{}\hbox { on }[t_1, 1]. \end{array}\right. } \end{aligned}$$

It is easy to see that $r\in W^{1,1}$ with $r_1'\cdot r_2\in L^{\infty }$. $\square$

3 Accessible Boundary Conditions

A (midline of a) ribbon with clamped lateral boundaries is described by a framed curve for which the initial value $(\gamma (0), r(0))$ and its terminal value $(\gamma (1), r(1))$ are prescribed, see Freddi et al. (2022). However, a constraint of the form $a_{ij}\equiv 0$ clearly restricts the possible shapes of framed curves $(\gamma , r)$. Therefore, it is a priori not obvious which clamped boundary conditions can be achieved by such constrained framed curves.

By applying a rigid motion we may assume without loss of generality that $\gamma (0)$ is the origin and r(0) is the identity matrix. Then the problem reduces to the question which values $\gamma (1)$ and r(1) can be attained for the solution r of $r' = Ar$ with inital value $r(0) = I$ and $\gamma (t) = \int _0^t r_1$, where A is a map taking values only in one of the three sets ${\mathfrak {A}}_{kl}$ defined by (5).

This question is answered in this section. Exploiting the connection to parallel transport on the sphere elicited by the first stage, the possible values of r(1) will be readily seen to be arbitrary, because the holonomy group of ${\mathbb {S}}^2$ is SO(2). The arbitrariness of $\gamma (1)$ then follows by carrying out the second stage.

More precisely, in this section we will prove the following result. It asserts that all boundary conditions can be attained by a suitable choice of (even a smooth) $A: I\rightarrow {\mathfrak {A}}_{kl}$.

Proposition 3.1

Let k, $l\in \{1,2,3\}$ satisfy $k\ne l$. For all $\overline{r}\in SO(3)$ and all $\overline{\gamma }\in B_1(0)$ there is an $A\in C^{\infty }(\overline{I}, {\mathfrak {A}}_{kl})$ such that the solution r of $r' = Ar$ with inital value $r(0) = I$ satisfies $r(1) = \overline{r}$ and $\int _0^1 r_1 = \overline{\gamma }$.

Remarks

• (i) Proposition 3.1 has been used in Freddi et al. (2022) to prove a Gamma convergence result for ribbons with clamped ends.

• (ii) The proof will show that there is much freedom in the choice of A. It could also be chosen to be piecewise constant.

• (iii) In view of the second stage, we will see that it is enough to satisfy only the condition $r(1) = \overline{r}$. Below, we prove the existence of such an r by means of the first stage, but one could apply intead, e.g., the techniques from Hornung (2021) or use properties of the exponential map for SO(3).

• (iv) An endpoint $\overline{\gamma }$ with $|\overline{\gamma }| = 1$ can clearly only be achieved if $\overline{\gamma }= e_1$, and in this case we must have $r_1\equiv e_1$.

  • In the case $a_{23} \equiv 0$ this implies that $\beta \equiv e_1$ and so necessarily $\overline{r} = I$.

  • In the case $a_{13} \equiv 0$ we have $\beta = r_2$, so it implies that $\Pi _{\beta |_{(0, t)}}e_1 = e_1$ for all t. This is the case if, and only if, $r_2$ takes its values in the great circle ${\mathbb {S}}^2\cap \{e_1\}^{\perp }$. (The fact that $r_2$ must be contained in a great circle of course follows as well from the observation that $r_2$ must take values in the orthogonal complement of $e_1$.) So $\overline{r}$ can be attained precisely if $\overline{r}_2$ lies in the circle ${\mathbb {S}}^2\cap \{e_1\}^{\perp }$.

  • The case $a_{12}\equiv 0$ is similar to the case $a_{13}\equiv 0$.

In order to prove Proposition 3.1, we will first show that for all $\overline{r}\in SO(3)$ there is some $A\in C^{\infty }(\overline{I}, {\mathfrak {A}}_{kl})$ such that the solution r of $r' = Ar$ with $r(0) = I$ satisfies $r(1) = \overline{r}$. This will follow from the connectedness of ${\mathbb {S}}^2$ and from the fact that the holonomy group of ${\mathbb {S}}^2$ is SO(2).

3.1 Modifying Curves

In order to provide explicit constructions, we will perform some simple operations on curves, such as introducing a loop into a given curve $\beta$. This allows to modify the parallel transport along $\beta$.

3.1.1 Concatenation of Continuous Curves

In what follows we will consider reparametrized concatenations of continuous curves

$$\begin{aligned} \beta ^{(i)}: [t_0^{(i)}, t_1^{(i)}]\rightarrow X\hbox { for }i = 1,..., N \end{aligned}$$

into some manifold X satisfying

$$\begin{aligned} \beta ^{(i)}(t_1^{(i)}) = \beta ^{(i+1)}\left( t_0^{(i + 1)}\right) \hbox { for }i = 1,..., N-1. \end{aligned}$$

The reparametrized concatenation $\beta = \beta ^{(N)}\bullet \cdots \bullet \beta ^{(1)}$ is the continuous curve $I\rightarrow X$ obtained by concatenation of the $\beta ^{(i)}$ and subsequent reparametrization by a constant factor.

To make this precise, set $T^{(0)} = 0$ and for $i = 1,..., N$ define $T^{(i)} = \sum _{k = 1}^i (t_1^{(k)} - t_0^{(k)})$. Then set

$$\begin{aligned} I^{(i)} = \left[ \frac{T^{(i-1)}}{T^{(n)}}, \frac{T^{(i)}}{T^{(n)}}\right] . \end{aligned}$$

We define $\beta ^{(N)} \bullet \cdots \bullet \beta ^{(1)}: I\rightarrow X$ by setting

$$\begin{aligned} (\beta ^{(N)} \bullet \cdots \bullet \beta ^{(1)})(t) = \beta ^{(i)}\left( t_0^{(i)} + T^{(n)} t - T^{(i-1)}\right) \hbox { whenever }t\in I^{(i)}. \end{aligned}$$

In what follows we will use the notation $I^{(i)}$ introduced above, i.e., $I^{(i)}$ is the subinterval of I on which $\beta ^{(N)} \bullet \cdots \bullet \beta ^{(1)}$ agrees with $\beta ^{(i)}$ up to a reparametrization.

3.1.2 Inserting a Loop While Preserving the Parametrization Near the Boundary

We will later wish to insert a loop ${\widehat{\beta }}$ into a given parametrized curve $\beta$ without affecting $\beta$ near the boundaries of its interval of definition. The basic construction is as follows.

Let $a\in [0, 1)$, let $\ell > 0$ and let $\varepsilon$ be a positive number satisfying

$$\begin{aligned} \varepsilon < \frac{1-a}{4\ell }. \end{aligned}$$

(22)

Let $X = {\mathbb {S}}^2$ or $X = {\mathbb {R}}^2$ and let $\beta : \overline{I}\rightarrow X$ and ${\widehat{\beta }}: [0, \ell ]\rightarrow X$ be continuous and such that ${\widehat{\beta }}(0) = {\widehat{\beta }}(\ell ) = \beta (a)$.

Let $\varphi \in C^{\infty }({\mathbb {R}})$ be a nonnegative function satisfying

$$\begin{aligned} {{\,\textrm{spt}\,}}\varphi \subset \left( a, a + (1+\varepsilon )\ell \right) \end{aligned}$$

and $\int \varphi = 1$ as well as

$$\begin{aligned} \sup \varphi \le \frac{1}{(1+\varepsilon /2)\ell }. \end{aligned}$$

(23)

Define $\tau : [0, 1+\ell ]\rightarrow \overline{I}$ by

$$\begin{aligned} \tau (t) = t - \ell \int _0^t\varphi . \end{aligned}$$

(24)

In view of (23) we have

$$\begin{aligned} \tau ' = 1 - \ell \varphi \ge \frac{\varepsilon }{2+\varepsilon }. \end{aligned}$$

(25)

Hence we can define the curve $\overline{\beta }: \overline{I}\rightarrow X$ by

$$\begin{aligned} \overline{\beta }(\tau (t)) = {\left\{ \begin{array}{ll} \beta (t) &{}\hbox { if }t\in [0, a] \\ {\widehat{\beta }}(t - a) &{}\hbox { if }t\in [a, a+\ell ] \\ \beta (t - \ell ) &{}\hbox { if }t\in [a + \ell , 1 + \ell ]. \end{array}\right. } \end{aligned}$$

(26)

The hypotheses on $\beta$ and ${\widehat{\beta }}$ ensure that $\overline{\beta }$ is continuous. Moreover,

$$\begin{aligned} \overline{\beta }= \beta \hbox { on }\overline{I}\setminus (a, a + \varepsilon \ell ). \end{aligned}$$

(27)

This follows from the definition because

$$\begin{aligned} \tau (t) = t\hbox { on } [0, a] \hbox { and } \tau (t) = t - \ell \hbox { on }[a + (1+\varepsilon )\ell , 1 + \ell ]; \end{aligned}$$

(28)

so in particular

$$\begin{aligned} \tau \left( [a + (1+\varepsilon )\ell , 1 + \ell ]\right) = [a + \varepsilon \ell , 1]. \end{aligned}$$

Condition (22) ensures that the interval $I{\setminus } (a, a + \varepsilon \ell )$ has positive length.

Finally, notice that $\tau$ is a $C^{\infty }$ diffeomorphism. Therefore, if $\beta \in C^{\infty }(\overline{I})$ and ${\widehat{\beta }}\in C^{\infty }([0, \ell ])$ and if, moreover, there is $r > 0$ such that

$$\begin{aligned} \begin{aligned} {\widehat{\beta }}(t)&= \beta (t + a) \hbox { for all }t\in [0, r] \\ {\widehat{\beta }}(t + \ell )&= \beta (t + a) \hbox { for all }t\in [- r, 0], \end{aligned} \end{aligned}$$

(29)

then $\overline{\beta }\in C^{\infty }(\overline{I})$. Conditions (29) are clearly satisfied if $\beta = \beta (a)$ near a and ${\widehat{\beta }} = \beta (a)$ near $\{0, \ell \}$,

3.2 Modifying the Parallel Transport

The following lemma provides a way to modify the parallel transport along a given spherical curve by inserting a small loop.

Fig. 1

A schematic depiction of the proof of Lemma 3.2. On the left: the orginal curve $\beta$. On the right: the modified curve ${\widetilde{\beta }}$ to which a loop has been added in order to achieve (30)

Lemma 3.2

Let $\beta \in C^{\infty }([a, b], {\mathbb {S}}^2)$ be nonconstant and let J be an open interval with $\overline{J}\subset (a, b)$ and such that $\beta ' \ne 0$ everywhere on J. Let $\theta \in [-\pi , \pi ]$ and let $V\subset {\mathbb {S}}^2$ be a neighbourhood of $\beta ([a, b])$.

Then there exists ${\widetilde{\beta }}\in C^{\infty }([a, b], {\mathbb {S}}^2)$ with the following properties:

• (i) ${\widetilde{\beta }}([a, b])\subset V$.

• (ii) ${\widetilde{\beta }} = \beta$ on $[a,b]\setminus J$ and ${\widetilde{\beta '}}\ne 0$ on J.

• (iii) The parallel tansport maps along $\beta$ and ${\widetilde{\beta }}$ are related by

  $$\begin{aligned} \Pi _{{\widetilde{\beta }}} = R_{\beta (b)}^{(\theta )}\circ \Pi _{\beta }. \end{aligned}$$

  (30)

Remarks

• (i) The construction provided here is elementary geometric in that it modifies the parallel transport along a curve simply by adding a loop; see Fig. 1. Since the construction is quite explicit, on can easily arrange (i), simply by making the additional loop small enough.

• (ii) If the original curve $\beta$ was immersed, then ${\widetilde{\beta }}$ provided by Lemma 3.2 is immersed as well. We have included this feature because, in the context of ribbons, an immersed curve $\beta$ corresponds to a surface without affine regions, which is a nondegeneracy condition which one may wish to preserve.

• (iii) As in Proposition 4.2 below, one can also show that ${\widetilde{\beta }}$ is arbitrarily close to $\beta$ in $W^{1,p}$ if $\theta$ is small enough.

In the proof of Lemma 3.2 we will insert a particular loop described in Sect. 3.2.1 into the curve $\beta$.

3.2.1 A Building Block

Let us give an explicit construction of an immersed smooth parametrization ${\widehat{\beta }}: {\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$ of a square with rounded corners. The construction provided here provides a loop whose addition to a given smooth immersed curve $\beta$ will preserve both the smoothness of $\beta$ and its immersedness.

Let ${\widetilde{\rho }}: {\mathbb {R}}\rightarrow [1, \sqrt{2}]$ be the $\frac{\pi }{2}$-periodic function determined by

$$\begin{aligned} {\widetilde{\rho }}(t) = \frac{1}{\cos t}\hbox { for }t\in \left[ -\frac{\pi }{4}, \frac{\pi }{4}\right] . \end{aligned}$$

Then

$$\begin{aligned} {\widetilde{\rho }}\in C^{\infty }\left( {\mathbb {R}}\setminus \left( \frac{\pi }{4} + \frac{\pi }{2}{\mathbb {Z}}\right) \right) . \end{aligned}$$

(31)

Let $\Theta _i$ be such that

$$\begin{aligned} 1< \Theta _1< \Theta _2< \Theta _3 < \sqrt{2}. \end{aligned}$$

Choose $\Theta \in C^{\infty }({\mathbb {R}})$ such that $\Theta '$ is nonincreasing and such that

$$\begin{aligned} \Theta (t) = {\left\{ \begin{array}{ll} t &{}\hbox { if }t \le \Theta _1 \\ \Theta _2 &{}\hbox { if }t \ge \Theta _3. \end{array}\right. } \end{aligned}$$

Observe that $\Theta '\ge 0$ and that $\Theta$ is just a smoothened version of the function

$$\begin{aligned} \chi _{(-\infty , \Theta _2)}\cdot id + \chi _{[\Theta _2, \infty )}\cdot \Theta _2. \end{aligned}$$

Now define $\rho : {\mathbb {R}}\rightarrow [1, \Theta _2]$ by $\rho = \Theta \circ {\widetilde{\rho }}.$ Choosing

$$\begin{aligned} \Theta _1 = {\widetilde{\rho }}\left( \frac{\pi }{8}\right) \end{aligned}$$

we have $\rho = {\widetilde{\rho }}$ on $(-\pi /8, \pi /8)$.

Now define the immersed smooth curve $\Xi : {\mathbb {R}}\rightarrow {\mathbb {R}}^2$ by setting

$$\begin{aligned} \Xi (t) = \rho (t) e^{it}. \end{aligned}$$

It is clearly $2\pi$-periodic and it parametrizes the boundary of a square with rounded off corners, whose perimeter we denote by ${\widehat{\ell }} = \int _0^{2\pi }|\Xi '|$.

We define ${\widehat{\beta }}: {\mathbb {R}}\rightarrow {\mathbb {R}}^2$ by setting

$$\begin{aligned} {\widehat{\beta }}\left( \int _0^t |\Xi '| \right) = \Xi \left( t - \frac{\pi }{2}\right) . \end{aligned}$$

Then $|{\widehat{\beta '}}|\equiv 1$ and ${\widehat{\beta }}(0) = -e_2$, and ${\widehat{\beta }}$ is ${\widehat{\ell }}$-periodic.

3.2.2 Proof of Lemma 3.2

Proof of Lemma 3.2

We assume that $\theta > 0$; the case $\theta < 0$ is similar. After possibly shrinking J we may assume that $\beta$ is injective on $\overline{J}$. After an affine reparametrization we may moreover assume that $\overline{I}\subset J$.

The map $\Phi : J\times {\mathbb {R}}\rightarrow {\mathbb {S}}^2$ given by

$$\begin{aligned} \Phi (t,s) = \frac{\beta (t) + s\beta (t)\times \beta '(t)}{|\beta (t) + s\beta (t)\times \beta '(t)|} \end{aligned}$$

clearly satisfies

$$\begin{aligned} \beta (t) = \Phi (t, 0)\hbox { for all }t\in \overline{I}. \end{aligned}$$

Moreover, since $\beta$ is smooth with uniformly bounded curvature, since $\beta |_{\overline{J}}$ is injective and since $\beta '\ne 0$ on $\overline{J}$, there is $\varepsilon _1 > 0$ such that $\Phi$ is a $C^{\infty }$ diffeomorphism from $U = J\times (-\varepsilon _1, \varepsilon _1)$ onto $\Phi (U)\subset {\mathbb {S}}^2$. After possibly shrinking $\varepsilon _1$ we may moreover assume that $\Phi (U)\subset V$.

Adding a loop. Let $\delta > 0$, $k\in {\mathbb {N}}$ and let ${\widehat{\beta }}$ be the building block introduced in Sect. 3.2.1. Let ${\widehat{\ell }}$ be as in Sect. 3.2.1 and define the immersed smooth curve ${\widehat{\beta }}_{\delta , k}: [0, k\delta {\widehat{\ell }}] \rightarrow {\mathbb {R}}^2$ by setting

$$\begin{aligned} {\widehat{\beta }}_{\delta , k}(t) = \left( \frac{1}{2}, \delta \right) + \delta {\widehat{\beta }}\left( \frac{t}{\delta }\right) . \end{aligned}$$

This is a k-fold parametrization of the boundary of a smoothened version of the square

$$\begin{aligned} (1/2-\delta , 1/2+ \delta )\times (0, 2\delta )\subset {\mathbb {R}}^2; \end{aligned}$$

compare Fig. 1. We denote the domain enclosed by ${\widehat{\beta }}_{\delta , 1}$ by $S_{\delta }$. The perimeter of $S_{\delta }$ is $\delta {\widehat{\ell }}$.

Now we define $\overline{\beta }: I\rightarrow {\mathbb {R}}^2$ by inserting, as described in Sect. 3.1.2, the loop ${\widehat{\beta }}_{\delta ,k}$ at $a = \frac{1}{2}$ into the curve $t\mapsto te_1$; so in the notation of Sect. 3.1.2 we have $\ell = k\delta {\widehat{\ell }}$, and we choose $\varepsilon = \left( 10 k\delta {\widehat{\ell }}\right) ^{-1},$ so that (22) is satisfied. The resulting curve $\overline{\beta }$ is in $C^{\infty }(\overline{I})$ because the definition of ${\widehat{\beta }}_{\delta , k}$ ensures that conditions (29) are satisfied. In addition, (27) and our choice of $\varepsilon$ show that

$$\begin{aligned} \overline{\beta }(t) = (t, 0) \hbox { for all }t\in \overline{I}\setminus \left( \frac{1}{2}, \frac{3}{5}\right) . \end{aligned}$$

(32)

Definition of ${\widetilde{\beta }}$ and computation of $\Pi _{{\widetilde{\beta }}}$. Clearly, there is $\delta _0 > 0$ such that $\overline{\beta }(\overline{I})$ is contained in U whenever $\delta \le \delta _0$. Hence for such $\delta$ we can define ${\widetilde{\beta }}: [a,b]\rightarrow {\mathbb {S}}^2$ by setting

$$\begin{aligned} {\widetilde{\beta }}(t) = {\left\{ \begin{array}{ll} \Phi \left( \overline{\beta }(t)\right) &{}\hbox { if }t\in I \\ \beta (t) &{}\hbox { if }t\in [a, b]\setminus I. \end{array}\right. } \end{aligned}$$

Then ${\widetilde{\beta }}\in C^{\infty }([a,b])$ because both $\overline{\beta }$ and $\beta$ are smooth and $\Phi \circ \overline{\beta }= \beta$ in a neighbourhood of $\partial I$, due to (32). Conditions (i) and (ii) are satisfied as well.

Now define $\Theta : [0, \delta _0)\rightarrow [0, \infty )$ by setting $\Theta (0) = 0$ and

$$\begin{aligned} \Theta (a) = \int _{{\widehat{S}}_a} \left( \det (\nabla \Phi ^T\nabla \Phi )\right) ^{1/2} \end{aligned}$$

for $a > 0$. Since $\det (\nabla \Phi ^T\nabla \Phi )$ is bounded from below and above by positive constants, $\Theta$ is continuous and strictly increasing on $[0, \delta _0)$. Moreover, denoting the area form on ${\mathbb {S}}^2$ by $\eta _{{\mathbb {S}}^2}$, we have

$$\begin{aligned} \int _{\Phi ({\widehat{S}}_{\delta })}\eta _{{\mathbb {S}}^2} = \Theta (\delta ). \end{aligned}$$

Hence Proposition 2.3 shows that the parallel transport along ${\widetilde{\beta }}$ satisfies (30) with $k\Theta (\delta )$ instead of $\theta$. Since $\Theta$ is continuous and increasing, we can choose $k\in {\mathbb {N}}$ and $\delta \in (0, \delta _0)$ such that $k\Theta (\delta ) = \theta$. $\square$

3.3 Proof of Proposition 3.1

Set $\beta ^{(0)} = \beta (0)$. Since $\overline{\gamma }$ is contained in the interior of the convex hull of ${\mathbb {S}}^2\subset {\mathbb {R}}^3$, there exists an $N\in {\mathbb {N}}$ and $\overline{r}_1^{(1)},..., \overline{r}_1^{(N)}\in {\mathbb {S}}^2$ such that $\overline{\gamma }$ lies in the interior of the convex hull of $\{\overline{r}_1^{(1)},..., \overline{r}_1^{(N)}\}$. Set $\overline{r}^{(0)}_1 = e_1$ and $\overline{r}_1^{(N+1)} = \overline{r}_1$. Choose a partition $0 = t_0< t_1< \cdots< t_N < t_{N+1} = 1$ of I.

Let us first consider the constraint $a_{12} = 0$. Choose, for example, a constant speed geodesic ${\widetilde{\beta }}: \overline{I}\rightarrow {\mathbb {S}}^2$ with endpoints ${\widetilde{\beta }}(0) = e_3$ and ${\widetilde{\beta }}(1) = \overline{r}_3$. By Lemma 3.2 we can replace ${\widetilde{\beta }}$ on each interval $[t_i, t_{i+1}]$ with a curve $\beta \in C^{\infty }([t_i, t_{i+1}])$ in such a way that

$$\begin{aligned} \Pi _{\beta |_{(t_i, t_{i+1})}}\overline{r}_1^{(i)} = \overline{r}_1^{(i+1)}\hbox { for all }i = 0,..., N. \end{aligned}$$

By Lemma 3.2 we also have $\beta = {\widetilde{\beta }}$ near each $t_i$. Hence $\beta$ is smooth on all of [0, 1]. Define the frame $r: I\rightarrow SO(3)$ by setting $r_3 = \beta$ and, for $j = 1, 2$,

$$\begin{aligned} r_j(t) = \Pi _{\beta |_{(0, t)}}e_j\hbox { for all }t\in I. \end{aligned}$$

(33)

Then r is smooth and satisfies $r_1'\cdot r_2 \equiv 0$; moreover, $r(1) = \overline{r}$ and $r_1(t_i) = \overline{r}_1^{(i)}$ for $i = 0,..., N$.

In particular, $\overline{\gamma }$ is contained in the interior of the convex hull of $r_1(I)$. So by Proposition 2.5, after reparametrizing r smoothly, we can arrange that $\overline{\gamma }= \int _0^1 r_1$.

The case $a_{13} = 0$ is similar to the case $a_{12} = 0$, so it remains to consider the case $a_{23} = 0$. Choose a smooth curve ${\widetilde{\beta }}: \overline{I}\rightarrow {\mathbb {S}}^2$ with constant speed which satisfies ${\widetilde{\beta }}(t_i) = \overline{r}_1^{(i)}$ for all $i = 0,..., N+1$. By Lemma 3.2 there exists a smooth curve $\beta : I\rightarrow {\mathbb {S}}^2$ whose trace agrees with ${\widetilde{\beta }}$ except for an additional small loop and such that, moreover, the frame $r: I\rightarrow SO(3)$ defined by setting $r_1 = \beta$ and by (33) for $j = 2, 3$ satisfies $r(1) = \overline{r}$. Since, in particular, ${\widetilde{\beta }}(I)\subset \beta (I)$ and $r_1 = \beta$, we see that $\overline{\gamma }$ is contained in the interior of the convex hull of $r_1(\overline{I})$. Hence we can use Proposition 2.5 as before.

4 Smooth Approximation of Framed Curves

As another application of the two stages, we will now prove the following result.

Theorem 4.1

Let $p\in [1, \infty )$ and let k, $l\in \{1,2,3\}$ be unequal. For all $A\in L^p(I, {\mathfrak {A}}_{kl})$ there exist $A_n\in C^{\infty }(\overline{I}, {\mathfrak {A}}_{kl})$ such that $A_n\rightarrow A$ strongly in $L^p(I, {\mathbb {R}}^{3\times 3})$ and, moreover, the solutions r, $r^{(n)}$ of $r' = Ar$ and of $(r^{(n)})' = A_nr^{(n)}$ with $r(0) = r^{(n)}(0) = I$ satisfy

$$\begin{aligned} r^{(n)}(1) = r(1)\hbox { and }\int _0^1 r^{(n)}_1 = \int _0^1 r_1 \hbox { for all }n\in {\mathbb {N}}. \end{aligned}$$

Remarks

• (i) We will obtain Theorem 4.1 as an immediate corollary of Proposition 4.2 below, which is the main result of this section.

• (ii) A more detailed result was obtained in Hornung (2021) by a different approach; compare also Bartels and Reiter (2020).

4.1 Smooth Approximation Preserving Parallel Transport

Using the viewpoint of Sect. 2.3.1, we see that the essential point in the proof of Theorem 4.1 will be to smoothly approximate a given $W^{1,p}$ curve on ${\mathbb {S}}^2$ while preserving its endpoint and its parallel transport map. The correct endpoint for $\gamma$ is then achieved by carrying out stage two.

Proposition 4.2

Let $p\in [1,\infty )$ and $\beta \in W^{1,p}(I, {\mathbb {S}}^2)$. Then there exist $\beta _n\in C^{\infty }(\overline{I}, {\mathbb {S}}^2)$ such that $\beta _n$ converges to $\beta$ strongly in $W^{1,p}(I)$ and, moreover, for all $n\in {\mathbb {N}}$ we have

• (i) $\beta _n(t) = \beta (0)$ for all $t\in I$ near 0 and $\beta _n(t) = \beta (1)$ for all $t\in I$ near 1,

• (ii) $\Pi _{\beta _n} = \Pi _{\beta }$.

4.1.1 First Proof of Proposition 4.2

The following proof of Proposition 4.2 applies the same basic idea as in Sect. 3.2, namely to add a loop. The situation here is different because the initial curve $\beta$ is in general neither smooth nor immersed on any subinterval. In addition, we need to estimate the $W^{1,p}$ distance between the modified curves and the original one.

Proof of Proposition 4.2

Let $\beta _n\in C^{\infty }(\overline{I}, {\mathbb {S}}^2)$ converge to $\beta$ in $W^{1,p}(I)$ and assume that for each n we have $\beta _n(t) = \beta (0)$ for all $t\in I$ near 0 and $\beta _n(t) = \beta (1)$ for all $t\in I$ near 1. Such $\beta _n$ are easy to construct.

Let $\ell _n\in \left( 0, \frac{1}{8}\right)$ be such that $\ell _n\rightarrow 0$. Let ${\widehat{\beta }}_n\in C^{\infty }([0, \ell _n], {\mathbb {S}}^2)$ satisfy

$$\begin{aligned} {\widehat{\beta }}_n(t) = \beta (0)\hbox { for all } t\in [0, \ell _n] \hbox { near }\{0, \ell _n\}, \end{aligned}$$

(34)

as well as

$$\begin{aligned} |{\widehat{\beta }}_n'(t)|\le C\hbox { for all } n \hbox { and all }t\in [0, \ell _n]. \end{aligned}$$

(35)

Define $\overline{\beta }_n$ by adding the loop ${\widehat{\beta }}_n$ to $\beta _n$ at $a_n = 0$, as decribed in Sect. 3.1.2. Choosing $\varepsilon _n = 1$ for all n, we ensure that condition (22) is satisfied.

In view of (34), since $\beta _n = \beta (0)$ near 0 and since each $\tau _n$ is smooth, we have $\overline{\beta }_n\in C^{\infty }(\overline{I})$. In view of (27) we also have

$$\begin{aligned} \overline{\beta }_n = \beta _n\hbox { on }(\ell _n, 1). \end{aligned}$$

(36)

Since ${\widehat{\beta }}_n = \beta (0)$ near 0, we conclude that, like $\beta _n$, each $\overline{\beta }_n$ satisfies

$$\begin{aligned} \overline{\beta }_n = \beta (0) \hbox { near }0 \hbox { and } \overline{\beta }_n = \beta (1)\hbox { near }1. \end{aligned}$$

(37)

We claim that $\overline{\beta }_n\rightarrow \beta$ strongly in $W^{1,p}(I)$. In view of (37) this follows provided that $\overline{\beta }_n'\rightarrow \beta '$ strongly in $L^p(I)$. Therefore, by (36) it is enough to show that

$$\begin{aligned} \int _0^{\ell _n} |\overline{\beta }_n' - \beta '|^p \rightarrow 0\hbox { as }n\rightarrow \infty . \end{aligned}$$

(38)

After a change of variables, the left-hand side of (38) equals

$$\begin{aligned} \int _{\tau _n^{-1}(0, \ell _n)} |\overline{\beta }_n'(\tau _n) - \beta '(\tau _n)|^p\tau _n'&= \int _0^{2\ell _n} |(\overline{\beta }_n(\tau _n))' - (\beta (\tau _n))'|^p(\tau _n')^{1-p}. \end{aligned}$$

We have used that $\tau _n(0) = 0$ and $\tau _n(2\ell _n) = \ell _n$ by (28). By definition, on $(0, \ell _n)$ we have $\overline{\beta }_n(\tau _n) = {\widehat{\beta }}_n$. So

$$\begin{aligned} \int _0^{\ell _n} |(\overline{\beta }_n(\tau _n))' - (\beta (\tau _n))'|^p(\tau _n')^{1-p}&\le C\int _0^{\ell _n} |{\widehat{\beta }}_n'|^p + C\int _0^{\tau (\ell _n)} |\beta '|^p. \end{aligned}$$

In the first term we used that $\tau _n'\ge 1/2$ due to (25) and our choice $\varepsilon _n = 1$. In the second term we undid the change of variables. The second term on the right-hand side converges to zero because $\beta '\in L^p$ and $\tau _n(\ell _n)\rightarrow 0$. The first one converges to zero because the integrands are uniformly bounded, due to (35).

By (26), on $(\ell _n, 2\ell _n)$ we have $\overline{\beta }_n(\tau _n) = \beta _n(\cdot -\ell _n)$. Hence

$$\begin{aligned} \int _{\ell _n}^{2\ell _n} |(\overline{\beta }_n(\tau _n))' - (\beta (\tau _n))'|^p(\tau _n')^{1-p}&= \int _{\ell _n}^{2\ell _n} |\beta _n'(\cdot -\ell _n) - (\beta (\tau _n))'|^p(\tau _n')^{1-p} \\&\le C \int _{\ell _n}^{2\ell _n} |\beta _n'(\cdot -\ell _n) - \beta '(\cdot -\ell _n)|^p \\&\quad + C \int _{\ell _n}^{2\ell _n} |\beta '(\cdot -\ell _n) - \beta '(\tau _n)\tau _n'|^p(\tau _n')^{1-p}. \end{aligned}$$

Again we have used that $\tau _n'\ge 1/2$. The first integral on the right-hand side equals $\int _0^{\ell _n} |\beta _n' - \beta '|^p$ and therefore converges to zero. The second integral does not exceed a constant times

$$\begin{aligned} \int _0^{\ell _n} |\beta '|^p + \int _{\tau _n(\ell _n)}^{\tau _n(2\ell _n)} |\beta '|^p. \end{aligned}$$

Both terms converge to zero because $\beta '\in L^p$ and the length of each integration interval converges to zero. This concludes the proof of (38). Hence we have shown that $\overline{\beta }_n\rightarrow \beta$ in $W^{1,p}(I)$.

It remains to specify ${\widehat{\beta }}_n$. For each n there is a unique $\theta _n\in (-\pi , \pi ]$ such that

$$\begin{aligned} \Pi _{\beta _n} = R^{(-\theta _n)}_{\beta (1)}\Pi _{\beta }. \end{aligned}$$

(39)

Assume that $\theta _n > 0$; the other nontrivial case is obtained by reversing the orientation of ${\widehat{\beta }}_n$. Let $D_n\subset {\mathbb {S}}^2$ be a geodesic disc containing $\beta (0)$ in its boundary and whose area equals $\theta _n$. Let ${\widehat{\alpha }}_n\in C^{\infty }({\mathbb {S}}^1, {\mathbb {S}}^2)$ be a positively oriented constant speed parametrization of $\partial D_n$ with ${\widehat{\alpha }}(0) = \beta (0)$. So

$$\begin{aligned} |{\widehat{\alpha }}_n'| = \frac{\ell _n}{2\pi }, \end{aligned}$$

(40)

where $\ell _n$ is now defined to be the perimeter of $D_n$. Since $\beta _n\rightarrow \beta$ in $W^{1,p}$, Lemma 2.1 (ii) implies that $\theta _n\rightarrow 0$. Hence $\ell _n \rightarrow 0$.

Let $\psi \in C^{\infty }_0(0, 1)$ satisfy $\int \psi = 1$ and $0\le \psi \le 2$. Define ${\widehat{\beta }}_n: [0, \ell _n]\rightarrow {\mathbb {S}}^2$ by

$$\begin{aligned} {\widehat{\beta }}_n(t) = {\widehat{\alpha }}_n \left( e^{2\pi i \int _0^{t/\ell _n}\psi } \right) . \end{aligned}$$

Then for all $t\in (0, \ell _n)$ we have

$$\begin{aligned} {\widehat{\beta }}_n'(t) = {\widehat{\alpha }}_n' \left( e^{2\pi i \int _0^{t/\ell _n}\psi } \right) \cdot \frac{2\pi i}{\ell _n}\psi \left( \frac{t}{\ell _n}\right) . \end{aligned}$$

(41)

Hence $|{\widehat{\beta }}_n'| \le 2$ by (40), so (35) is satisfied. Since $\psi$ has compact support, Eq. (41) also shows that ${\widehat{\beta }}_n$ is constant in a neighbourhood of $\{0, \ell _n\}$, namely equal to $\beta (0)$.

Since the loop ${\widehat{\beta }}_n$ is a simple positively oriented parametrization of the boundary of the geodesic disc $D_n$ with area $\theta _n$, Proposition 2.3 implies that

$$\begin{aligned} \Pi _{\overline{\beta }_n} = R^{(\theta _n)}_{\beta (1)}\Pi _{\beta _n}. \end{aligned}$$

In view of (39) this concludes the proof of (ii). $\square$

4.1.2 Alternative Proof of Proposition 4.2

Here we provide an alternative, more analytic, proof of Proposition 4.2. It less explicit than the previous one, but nevertheless very basic.

Lemma 4.3

For every nonconstant $h\in L^{\infty }(I)$ and all $\delta \in {\mathbb {R}}$ there exists a ${\widehat{\mu }}\in C_0^{\infty }(I)$ such that $\int {\widehat{\mu }} = 0$ and $\int h{\widehat{\mu }} = \delta$, as well as

$$\begin{aligned} \Vert {\widehat{\mu }}\Vert _{L^{\infty }}\le |\delta |\cdot \frac{4\Vert h\Vert _{L^{\infty }(I)}}{\sigma ^2_h}, \end{aligned}$$

(42)

where $\sigma _h^2 = \int _I h^2 - \left( \int _I h\right) ^2$.

Proof

For the readers’ convenience we include the elementary construction. We only need to consider the case $\delta = 1$; for arbitrary $\delta \in {\mathbb {R}}$ we take $\delta {\widehat{\mu }}$ instead of ${\widehat{\mu }}$.

Let $\varepsilon > 0$. By cutting off and mollifying we can find $\mu _1$, $\mu _2\in C_0^{\infty }(I)$ satisfying

$$\begin{aligned} \Vert \mu _1\Vert _{L^{\infty }(I)}\le 1\hbox { and } \Vert \mu _2\Vert _{L^{\infty }(I)}\le \Vert h\Vert _{L^{\infty }(I)}, \end{aligned}$$

(43)

as well as

$$\begin{aligned} \Vert \mu _1 - 1\Vert _{L^2(I)} + \Vert \mu _2 - h\Vert _{L^2(I)} \le \varepsilon . \end{aligned}$$

(44)

For $i = 1, 2$ define $\lambda _i = \int _I\mu _i$ and $\zeta _i = \int _I h\mu _i$. Let us now choose $\varepsilon$ so small that

$$\begin{aligned} \begin{aligned} |\lambda _2|&\le \frac{3\left| \int _I h\right| }{2} \\ |\lambda _1|&\le \frac{3}{2} \\ |\lambda _1\zeta _2 - \lambda _2\zeta _1|&\ge \frac{3\sigma _h^2}{4} \end{aligned} \end{aligned}$$

(45)

In fact, these conditions are satisfied whenever $\varepsilon$ is small enough, because (44) implies that $\lambda _1\rightarrow 1$ and $\lambda _2\rightarrow \int _I h$, as well as $\zeta _1\rightarrow \int h$ and $\zeta _2\rightarrow \int _I h^2$, as $\varepsilon \rightarrow 0$. We claim that

$$\begin{aligned} {\widehat{\mu }} = \frac{\lambda _1\mu _2 - \lambda _2\mu _1}{\lambda _1\zeta _2 - \lambda _2\zeta _1} \end{aligned}$$

has the desired properties. In fact, ${\widehat{\mu }}\in C_0^{\infty }(I)$, and one can readily verify that $\int _I {\widehat{\mu }} = 0$ and $\int _I h{\widehat{\mu }} = 1$. It remains to estimate:

$$\begin{aligned} \Vert {\widehat{\mu }}\Vert _{L^{\infty }(I)}&\le \frac{|\lambda _1|\Vert \mu _2\Vert _{L^{\infty }(I)} + |\lambda _2|\Vert \mu _1\Vert _{L^{\infty }(I)}}{ |\lambda _1\zeta _2 - \lambda _2\zeta _1|}. \end{aligned}$$

In view of (43) and (45) the right-hand side does not exceed the right-hand side of (42). $\square$

Proof of Proposition 4.2

We may subdivide the interval I into finitely many subintervals and restrict the construction to each subinterval. In fact, due to (i) we will be able to smoothly glue together the smooth approximations obtained separately on each subinterval. Therefore, since $\beta$ is continuous, we may assume without loss of generality that every point in $p\in \beta (\overline{I})$ satisfies $p\cdot e_1 > 0$ and $p\cdot e_3 > 0$. In particular, $\beta (\overline{I})$ does not contain the north pole $e_3$.

We parametrize the relevant portion of ${\mathbb {S}}^2$ by

$$\begin{aligned} \Psi : (0, 1)\times (-\pi , \pi )&\rightarrow {\mathbb {S}}^2 \\ (\rho , \varphi )&\mapsto (\rho \cos \varphi , \rho \sin \varphi , h(\rho )), \end{aligned}$$

where $h(\rho ) = \sqrt{1 - \rho ^2}$. We define the orthonormal frame field $(E_1, E_2)$ by

$$\begin{aligned} E_1(\rho , \varphi )&= (h(\rho )\cos \varphi , h(\rho )\sin \varphi , -\rho ) \\ E_2(\rho , \varphi )&= (-\sin \varphi , \cos \varphi , 0). \end{aligned}$$

The connection form $\omega = E_2\cdot DE_1$ is $\omega = h(\rho )\ d\varphi .$ In view of our hypotheses on the range of $\beta$, there exist $\rho$, $\varphi \in W^{1,p}(I)$ such that $\beta (t) = \Psi (\rho (t), \varphi (t))$ for all $t\in I$.

Let $\varepsilon > 0$ be small and let ${\widetilde{\rho }}\in C^{\infty }(\overline{I})$ be such that $\Vert {\widetilde{\rho }} - \rho \Vert _{W^{1,p}(I)} < \varepsilon$ and such that ${\widetilde{\rho }} = \rho (0)$ near 0 and ${\widetilde{\rho }} = \rho (1)$ near 1. Let us assume for the moment that $\rho$ is not constant. Then, with the notation from Lemma 4.3, we have $\sigma _{h\circ \rho } > 0$ and

$$\begin{aligned} \sigma _{h\circ {\widetilde{\rho }}} \ge \frac{1}{2}\sigma _{h\circ \rho }\hbox { for all } \varepsilon > 0 \hbox { small enough.} \end{aligned}$$

(46)

Set $\mu = \varphi '$. Let $\overline{\mu }\in C_0^{\infty }(I)$ be such that $\Vert \overline{\mu }- \mu \Vert _{L^p}<\varepsilon$ and $\int _I\overline{\mu }= \int _I\mu$. Define

$$\begin{aligned} \delta = \int h({\widetilde{\rho }})\overline{\mu }- \int h(\rho )\mu . \end{aligned}$$

By Lemma 4.3 there exists ${\widehat{\mu }}\in C_0^{\infty }(I)$ satisfying $\int _I {\widehat{\mu }} = 0$ and $\int h({\widetilde{\rho }}){\widehat{\mu }} = - \delta$ as well as

$$\begin{aligned} \Vert {\widehat{\mu }}\Vert _{L^{\infty }}\le C|\delta |. \end{aligned}$$

(47)

Here, the constant C can be chosen to be independent of $\varepsilon$ because by (46) the variance $\sigma _{h\circ {\widetilde{\rho }}}$ is bounded away from zero, uniformly in $\varepsilon$ for all $\varepsilon$ small enough.

The function ${\widetilde{\mu }} = \overline{\mu }+ {\widehat{\mu }}$ satisfies

$$\begin{aligned} \int {\widetilde{\mu }} = \int \mu \end{aligned}$$

(48)

as well as

$$\begin{aligned} \int h({\widetilde{\rho }}){\widetilde{\mu }} = \int h(\rho )\mu . \end{aligned}$$

(49)

In view of (48) and (49) and since $\delta$ (and therefore ${\widehat{\mu }}$, by (47)) can be made arbitrarily small by choosing $\varepsilon$ small enough, the functions ${\widetilde{\varphi }}(t) = \varphi (0) + \int _0^t{\widetilde{\mu }}$ and ${\widetilde{\rho }}$ define a smooth curve ${\widetilde{\beta }} = \Psi ({\widetilde{\rho }}, {\widetilde{\varphi }})$ with the desired properties (here ${\widetilde{\beta }}$ plays the role of $\beta _n$ in the statement). In fact, (48) implies that ${\widetilde{\varphi }}(1) = \varphi (1)$; since ${\widetilde{\rho }}$ agrees with $\rho$ in 0 and in 1 this implies that ${\widetilde{\beta }}$ and $\beta$ have the same endpoints. The compact support of ${\widetilde{\mu }}$ ensures that ${\widetilde{\varphi }}$ is constant near 0 and 1, and since the same is true for ${\widetilde{\rho }}$, we see that (i) is satisfied. Finally, (49) implies that $\int _{{\widetilde{\beta }}}\omega = \int _{\beta }\omega$ and therefore $\Pi _{{\widetilde{\beta }}} = \Pi _{\beta }$, by Lemma 2.2.

It remains to consider the degenerate case when $\rho$ is constant. Then $h(\rho )$ is constant as well. We set ${\widetilde{\rho }} = \rho$ and observe that in this situation (49) follows from (48). Therefore, we can simply take ${\widehat{\mu }} = 0$ in the earlier argument. $\square$

4.2 Proof of Theorem 4.1

We only consider the case $a_{12} = 0$; the others are similar. Set $\beta = r_3$ and let $\beta _n\in C^{\infty }(\overline{I}, {\mathbb {S}}^2)$ as provided by Proposition 4.2. Define $r^{(n)}\in C^{\infty }(\overline{I}, SO(3))$ by setting $r^{(n)}_3 = \beta _n$ and, for $i = 1, 2$,

$$\begin{aligned} r^{(n)}_i(t) = \Pi _{\beta _n|_{(0, t)}}r_i(0)\hbox { for all }t\in I. \end{aligned}$$

Then $(r_1^{(n)})'\cdot r_2^{(n)}\equiv 0$, and for $i = 1,2$ we have

$$\begin{aligned} a_{i3}^{(n)} = - r_i^{(n)}\cdot (r_3^{(n)})'\rightarrow a_{i3}\hbox { strongly in }L^p(I), \end{aligned}$$

because $r^{(n)}\rightarrow r$ uniformly and $\beta _n'\rightarrow \beta '$ strongly in $L^p$.

5 Ribbons with Finite Width

The purpose of this section is to prove the following version of Proposition 3.1 for ribbons which are clamped at their left and right edges; compare Freddi et al. (2022). Ribbons are more rigid than generic framed curves. This translates into an additional constraint on the spherical curve $\beta$, which links the parallel transport along $\beta$ to an adapted frame for $\beta$.

Theorem 5.1

For all $\overline{r}\in SO(3)$ and all $\overline{\gamma }\in B_1(0)$ there is a constant $w > 0$ and a $C^{\infty }$ isometric immersion

$$\begin{aligned} u: \overline{I}\times [-w, w]\rightarrow {\mathbb {R}}^3 \end{aligned}$$

satisfying $u(0, 0) = 0$ as well as $u(1, 0) = \overline{\gamma }$ and, for $i = 1, 2$,

$$\begin{aligned} \partial _i u&= e_i\hbox { on }\{0\}\times (-w,w) \\ \partial _i u&= \overline{r}_i\hbox { on }\{1\}\times (-w,w). \end{aligned}$$

The proof of Theorem 5.1 will be obtained by combining facts about isometric immersions with a result involving only the framed curve describing the deformation and the deformation gradient along the center line $I\times \{0\}$. This result about framed curves is Proposition 5.2 below.

5.1 Isometric Immersions

Let $S\subset {\mathbb {R}}^2$ be a bounded domain. The Gauss map of an immersion $u: S\rightarrow {\mathbb {R}}^3$ is the map n from the reference domain $S\subset {\mathbb {R}}^2$ into ${\mathbb {S}}^2$ given by

$$\begin{aligned} n(x) = \frac{\partial _1 u(x)\times \partial _2 u(x)}{|\partial _1 u(x)\times \partial _2 u(x)|}. \end{aligned}$$

If u is an isometric immersion $S\rightarrow {\mathbb {R}}^3$ (where S is endowed with the standard flat metric), i.e., $\partial _i u\cdot \partial _j u = \delta _{ij}$, then the denominator is always 1. Moreover, if $u\in W^{2,1}_{loc}$ satisfies $\partial _i u\cdot \partial _j u = \delta _{ij}$, then we can differentiate this equation. We can apply the product rule because $\nabla u\in L^{\infty }$. Hence we find that $\partial _k u\cdot \partial _i\partial _j u = 0$ almost everywhere, for all i, j, $k = 1, 2$. Hence

$$\begin{aligned} \partial _i\partial _j u(x) = A_{ij}(x) n(x)\hbox { for almost every }x\in S, \end{aligned}$$

(50)

where $A_{ij} = n\cdot \partial _i\partial _j u$ are the components of the second fundamental form of the immersion u.

Now let $b: I\rightarrow S$ be a $W^{2,\infty }$ curve with curvature $\kappa$, and define the frame $r: I\rightarrow SO(3)$ by setting $r_3 = n(b)$ and $r_1 = (D_{b'}u)(b)$. Then (50) shows that

$$\begin{aligned} r_1'(t)\cdot r_2(t) = \kappa , \end{aligned}$$

because u is an isometric immersion. In particular, if b is a straight line, then $r_1$ and $r_2$ are parallel transported along the spherical curve $\beta : I\rightarrow {\mathbb {S}}^2$ defined by $\beta = r_3$.

A spherical curve $\beta : I\rightarrow {\mathbb {S}}^2$ being the Gauss map of some isometric immersion defined in a neighbourhood of $b(I)\subset {\mathbb {R}}^2$ amounts to $\beta$ having bounded geodesic curvature in the sense of Definition 2.8, cf. Hornung (2023). This is an additional constraint that must be taken into account for the proof of Theorem 5.1.

5.2 Framed Curves for Finite Ribbons

We will obtain Theorem 5.1 as a consequence of the following result involving only framed curves.

Proposition 5.2

For all $\overline{r}\in SO(3)$ and all $\overline{\gamma }\in B_1(0)$ there exists a framed curve $(\gamma , r)$ in $C^{\infty }(\overline{I})$ with $r_1'\cdot r_2 \equiv 0$ and such that the following are satisfied:

• (i) $r(0) = I$ and $r(1) = \overline{r}$, as well as $\gamma (0) = 0$ and $\gamma (1) = \overline{\gamma }$.

• (ii) $r_3: I\rightarrow {\mathbb {S}}^2$ has bounded geodesic curvature. More precisely, there exists a function $K_g\in C_0^{\infty }(I)$ with

  $$\begin{aligned} |K_g| \le \frac{\pi }{4}\hbox { on }I \end{aligned}$$

  (51)

  such that

  $$\begin{aligned} {\widetilde{r}}_1 = r_1\cos K_g + r_2\sin K_g \end{aligned}$$

  (52)

  defines an adapted frame ${\widetilde{r}}$ along $r_3: I\rightarrow {\mathbb {S}}^2$.

The proof of Proposition 5.2 will be proven in the following sections. It is similar to that of Proposition 3.1, in that we combine parallel transport with a suitable reparametrization of the frame. However, here we need to satisfy additional conditions related to the finite width of the ribbon and to the rigidity of isometric deformations. Nevertheless, the construction only makes use of piecewise geodesics on the sphere and is elementary geometric.

5.2.1 Piecewise Geodesic Loop

The following construction plays a similar role as the loop in Sect. 3.2. While the curve constructed here is no longer a smooth immersion, it satisfies additional constraints which are essential for Proposition 5.2.

Lemma 5.3

Let $\alpha _0\in (0, \pi /4)$, let x, $y\in {\mathbb {S}}^2$, and let $v^{(0)}\in T_x{\mathbb {S}}^2$, $v^{(1)}\in T_y{\mathbb {S}}^2$ be unit tangent vectors.

Then there exists a continuous curve $\beta : \overline{I}\rightarrow {\mathbb {S}}^2$ with $\beta (0) = x$ and $\beta (1) = y$ such that the following are satisfied:

• (i) There exists a finite set $I'\subset I$ whose cardinality is bounded by a constant depending only on $\alpha _0$, such that for every interval $J\subset I\setminus I'$ the restriction $\beta |_J$ is an immersed geodesic and belongs to $C^{\infty }(\overline{J})$.

• (ii) There is a function $K_g\in C_0^{\infty }(I)$ with $|K_g|\le \alpha _0$ such that the frame ${\widetilde{r}}: \overline{I}\rightarrow SO(3)$ defined by setting ${\widetilde{r}}_3 = \beta$ and

  $$\begin{aligned} {\widetilde{r}}_1(t) = R^{(K_g(t))}_{\beta (t)}\Pi _{\beta |_{(0, t)}}v^{(0)} \hbox { for all }t\in \overline{I} \end{aligned}$$

  (53)

  is an adapted frame for $\beta$.

• (iii) $\Pi _{\beta }v^{(0)} = v^{(1)}$.

Remarks

• (i) The point here is that the angle between the parallel transported vector field $\Pi _{\beta |_{(0,t)}}v^{(0)}$ and the adapted frame ${\widetilde{r}}_1$ never exceeds a prescribed value. This is essential to ensure that $\beta$ can indeed be realized as the Gauss map along the midline of a ribbon with finite width.

• (ii) The conclusion of Lemma 5.3 is invariant under reparametrizations of $\beta$: if $\Psi : I\rightarrow I$ is smooth and strictly monotone, then $\beta \circ \Psi$ and $K_g\circ \Psi$ satisfy the conclusions, too. The function $K_g\circ \Psi$ still belongs to $C_0^{\infty }(I)$.

Fig. 2

A schematic depiction of the proof of Lemma 5.3. The curve $\beta ^{(1)}$ is a geodesic connecting $x\in {\mathbb {S}}^2$ to $y\in {\mathbb {S}}^2$. The loop $\beta ^{(0)}$ consists of a pair of geodesics connecting x to its antipodal point, as constructed in the proof of Lemma 5.4. The initial tangent vector $v^{(0)}$ is parallel transported along $\beta ^{(0)}$ into ${\widetilde{v}}^{(1)}$, which is tangent to $\beta ^{(1)}$ in x. The vector ${\widetilde{v}}^{(1)}$ is then parallel transported along $\beta ^{(1)}$ into ${\widetilde{v}}^{(0)}$. Finally, the loop $\beta ^{(2)}$ consists of a pair of geodesics connecting y to its antipodal point. The tangent vector ${\widetilde{v}}^{(0)}$ is parallel transported along $\beta ^{(0)}$ into $v^{(1)}$

We begin by verifying that Lemma 5.3 can be reduced to the particular case when $x = y$.

Lemma 5.4

Lemma 5.3 is true in the case $x = y$.

Taking this lemma for granted, we can prove Lemma 5.3 in the general case when $x\ne y$. This proof is depicted in Fig. 2.

Proof of Lemma 5.3

Let $\ell > 0$ and $\beta ^{(1)}: [0, \ell ]\rightarrow {\mathbb {S}}^2$ be such that $\beta ^{(1)}$ is the shortest arclength parametrized geodesic connecting x to y. So $\beta ^{(1)}(0) = x$ and $\beta ^{(1)}(\ell ) = y$. (If x and y are antipodal then choose $\beta ^{(1)}$ to be any of the geodesics connecting them.)

We apply Lemma 5.4 first at x, with $v^{(0)}$ as in the hypothesis of Lemma 5.3, but with

$$\begin{aligned} {\widetilde{v}}^{(1)} = (\beta ^{(1)})'(0) \end{aligned}$$

instead of $v^{(1)}$. The loop curve provided by that lemma will be denoted $\beta ^{(0)}$.

We denote by $\beta ^{(2)}$ the loop curve obtained by applying Lemma 5.4 at y with $v^{(1)}$ as in the hypothesis of Lemma 5.3, but with

$$\begin{aligned} {\widetilde{v}}^{(0)} = (\beta ^{(1)})'(\ell ) \end{aligned}$$

instead of $v^{(0)}$. Setting

$$\begin{aligned} \beta = \beta ^{(2)}\bullet \beta ^{(1)} \bullet \beta ^{(0)} \end{aligned}$$

we obtain the desired curve. In fact,

$$\begin{aligned} \Pi _{\beta }v^{(0)}&= \Pi _{\beta ^{(2)}}\Pi _{\beta ^{(1)}}(\beta ^{(1)})'(0) \\&= \Pi _{\beta ^{(2)}}(\beta ^{(1)})'(\ell ) = v^{(1)}, \end{aligned}$$

because $\beta ^{(1)}: [0, \ell ]\rightarrow {\mathbb {S}}^2$ is an arclength parametrized geodesic. On $I^{(0)}$ and $I^{(2)}$ the function $K_g$ is determined by Lemma 5.4, and on $I^{(1)}$ we set $K_g = 0$. $\square$

Proof of Lemma 5.4

Without loss of generality we may assume that x is the north pole, i.e., that $x = e_3$. We may also assume that there exists $\theta \in (-\frac{\pi }{2}, \frac{\pi }{2}]$ such that

$$\begin{aligned} v^{(0)} = R_{e_3}^{-\theta }e_1 = e^{-i\theta } \end{aligned}$$

and

$$\begin{aligned} v^{(1)} = R_{e_3}^{(2\theta )}v^{(0)} = e^{i\theta }. \end{aligned}$$

(54)

Here we identify $e^{i\delta }$ with the tangent vector

$$\begin{aligned} \begin{pmatrix} \cos \delta \\ \sin \delta \\ 0 \end{pmatrix}\in T_{e_3}{\mathbb {S}}^2. \end{aligned}$$

The case $\theta = 0$ is trivial and the case $\theta < 0$ is obtained by reversing the direction of travel. So we may assume without loss of generality that $\theta > 0$. In addition, subdividing the interval $[0, \theta ]$ and iterating the following construction, we may assume without loss of generality that

$$\begin{aligned} \theta \in (0, 2\alpha _0]. \end{aligned}$$

(55)

Geometrically, this iteration means that one adds several loops, where the initial direction of each loop is dictated by a different direction $v^{(0)}$.

After these reductions, we can now simply choose $\beta$ to be a parametrization of two meridians. In order to make this more precise, for $\alpha \in (-\pi , \pi )$ we define $b_{\alpha }: [0, \pi ]\rightarrow {\mathbb {S}}^2$ by

$$\begin{aligned} b_{\alpha }(t) = \begin{pmatrix} \cos \alpha \cdot \sin t \\ \sin \alpha \cdot \sin t \\ \cos t \end{pmatrix}. \end{aligned}$$

This is an arclength parametrization of the meridian with longitude $\alpha$ connecting the north pole $x = e_3$ to the south pole $-e_3$. Now define $\beta ^{(1)}$, $\beta ^{(3)}: [0, \pi ]\rightarrow {\mathbb {S}}^2$ by

$$\begin{aligned} \beta ^{(1)} = b_{-\frac{\theta }{2}} \hbox { and } \beta ^{(3)} = b_{\frac{\theta }{2}}(\pi - \cdot ), \end{aligned}$$

and define $\beta ^{(0)}$, $\beta ^{(2)}$, $\beta ^{(4)}: [0, 1] \rightarrow {\mathbb {S}}^2$ by $\beta ^{(0)} = \beta ^{(4)}\equiv e_3$ and $\beta ^{(2)}\equiv -e_3$. We define $\beta : I\rightarrow {\mathbb {S}}^2$ by

$$\begin{aligned} \beta = \beta ^{(4)} \bullet \beta ^{(3)} \bullet \beta ^{(2)} \bullet \beta ^{(1)} \bullet \beta ^{(0)} \end{aligned}$$

and the intervals $I^{(k)}\subset I$ as in Sect. 3.1.1 (so that, up to reparametrization, $\beta$ equals $\beta ^{(i)}$ on $I^{(i)}$). We define

$$\begin{aligned} v(t) = \Pi _{\beta |_{(0, t)}}v^{(0)} = \Pi _{\beta |_{(0, t)}}e^{-i\theta }. \end{aligned}$$

We claim that

$$\begin{aligned} v(1) = v^{(1)}. \end{aligned}$$

(56)

In order to prove this, first notice that, for all $\alpha$, $\delta \in (-\pi , \pi )$,

$$\begin{aligned} \Pi _{b_{\alpha }|_{(0, t)}}e^{i\delta } = R_{b_{\alpha }(t)}^{(\delta - \alpha )}b_{\alpha }'(t)\hbox { for all }t\in [0, \pi ]. \end{aligned}$$

(57)

In fact, since $b_{\alpha }$ is an arclength parametrized geodesic, it is enough to verify (57) at $t = 0$, where it is obvious because $b_{\alpha }'(0) = e^{i\alpha }$.

Applying (57) at $t = \pi$ we obtain

$$\begin{aligned} \Pi _{b_{\alpha }}e^{i\delta }&= R_{-e_3}^{(\delta -\alpha )}b_{\alpha }'(\pi ) = R_{e_3}^{(\alpha - \delta )} b_{\alpha }'(\pi ) = - e^{i(2\alpha - \delta )}. \end{aligned}$$

Thus

$$\begin{aligned} \Pi _{b_{\theta /2}}v^{(0)} = -e_1 = \Pi _{b_{-\theta /2}} v^{(1)}. \end{aligned}$$

This proves (56), which in turn implies (iii).

In order to define $K_g$ and ${\widetilde{r}}$, let $\varphi$ be a nonnegative $C^{\infty }$ function with ${{\,\textrm{spt}\,}}\varphi \subset \left( \frac{1}{4}, \frac{3}{4}\right)$ and $\int \varphi = 1$. For any interval $J = (t_0, t_1)$ define $\eta _J: J\rightarrow [0, 1]$ by

$$\begin{aligned} \eta _J(t) = \int _0^{\frac{t-t_0}{t_1-t_0}}\varphi . \end{aligned}$$

We define ${\widetilde{K}}_g\in C_0^{\infty }(I)$ by setting

$$\begin{aligned} {\widetilde{K}}_g = {\left\{ \begin{array}{ll} \eta _{I^{(0)}} &{}\hbox { on }I^{(0)} \\ 1 &{}\hbox { on }I^{(1)} \\ 1 - 2\eta _{I^{(2)}} &{}\hbox { on }I^{(2)} \\ -1 &{}\hbox { on }I^{(3)} \\ \eta _{I^{(4)}} - 1 &{}\hbox { on }I^{(4)}, \end{array}\right. } \end{aligned}$$

and we define $K_g = \frac{\theta }{2}\cdot {\widetilde{K}}_g$. Now define ${\widetilde{r}}$ by ${\widetilde{r}}_3 = \beta$ and by (53), that is,

$$\begin{aligned} {\widetilde{r}}_1 = R_{\beta }^{(K_g)}v. \end{aligned}$$

In order to verify that ${\widetilde{r}}$ indeed defines an adapted frame for $\beta : I\rightarrow {\mathbb {S}}^2$, we apply (57) with $\delta = -\theta$ and $\alpha = -\theta /2$ to find

$$\begin{aligned} \Pi _{b_{\frac{\theta }{2}}|_{(0,t)}}v^{(0)} = R_{b_{-\frac{\theta }{2}}(t)}^{(-\theta /2)}b_{-\frac{\theta }{2}}'(t) \hbox { for all }t\in [0, \pi ]. \end{aligned}$$

So

$$\begin{aligned} \beta ' = |\beta '| R_{\beta }^{(\theta /2)}v \hbox { on }I^{(1)} \end{aligned}$$

(58)

because $\beta$ is a constant speed reparametrization of $b_{-\frac{\theta }{2}}$ on $I^{(1)}$. Similarly, (57) shows that

$$\begin{aligned} \beta ' = -|\beta '| R_{\beta }^{(-\theta /2)}v \hbox { on }I^{(3)} \end{aligned}$$

(59)

because $\beta$ is a constant speed reparametrization of $b_{\frac{\theta }{2}}(\pi - \cdot )$ on $I^{(3)}$.

In view of (58), (59) and since

$$\begin{aligned} \beta ' = 0\hbox { on }I^{(0)}\cup I^{(2)}\cup I^{(4)}, \end{aligned}$$

we see that ${\widetilde{r}}_1\times \beta ' = 0$ everywhere. Hence ${\widetilde{r}}$ is an adapted frame for $\beta$. In view of (55) this implies (ii).

Finally, (i) is an immediate consequence of the construction. $\square$

5.2.2 Proof of Proposition 5.2

As in the proof of Proposition 3.1 we choose $N\in {\mathbb {N}}$ and $\overline{r}_1^{(1)},..., \overline{r}_1^{(N)}\in {\mathbb {S}}^2$ such that $\overline{\gamma }$ lies in the interior of the convex hull of $\{\overline{r}_1^{(1)},..., \overline{r}_1^{(N)}\}$; we set $\overline{r}^{(0)}_1 = e_1$ and $\overline{r}_1^{(N+1)} = \overline{r}_1$, and we choose a partition $0 = t_0< t_1< \cdots< t_N < t_{N+1} = 1$ of I. Let $\overline{\beta }$ be the shortest arclength parametrized geodesic with $\overline{\beta }(0) = e_3$ and $\overline{\beta }(1) = \overline{r}_3$.

For $i = 0, \dots , N$ denote by $\beta ^{(i)}: I\rightarrow {\mathbb {S}}^2$ the curve obtained by applying Lemma 5.3 with $x = \overline{\beta }(t_i)$ and $y = \overline{\beta }(t_{i+1})$ as well as

$$\begin{aligned} v^{(0)} = \overline{r}_1^{(i)} \hbox { and } v^{(1)} = \overline{r}_1^{(i+1)}. \end{aligned}$$

Define ${\widehat{\beta }} = \beta ^{(N+1)}\bullet \cdots \bullet \beta ^{(0)}$. Define ${\widehat{r}}: I\rightarrow SO(3)$ by setting ${\widehat{r}}_3 = {\widehat{\beta }}$ and

$$\begin{aligned} {\widehat{r}}_1(t) = \Pi _{{\widehat{\beta }}_{(0, t)}} e_1. \end{aligned}$$

(60)

By Lemma 5.3 there is a function ${\widehat{K}}_g\in C_0^{\infty }(I)$ with (51) and such that

$$\begin{aligned} {\widehat{\beta }}'\times \left( {\widehat{r}}_1\cos {\widehat{K}}_g + {\widehat{r}}_2\sin {\widehat{K}}_g \right) = 0\hbox { on }I. \end{aligned}$$

(61)

By Lemma 5.3 the curve ${\widehat{\beta }}$ is continuous on I, and there exist

$$\begin{aligned} 0 = t_0< t_1< \cdots< t_{N-1} < t_N = 1 \end{aligned}$$

such that, for all $i = 1,..., N$, the restriction ${\widehat{\beta }}|_{(t_{i-1}, t_i)}$ is immersed and belongs to $C^{\infty }\left( [t_{i-1}, t_i]\right)$.

Let $\varphi \in C^{\infty }({\mathbb {R}})$ be a nonnegative function with ${{\,\textrm{spt}\,}}\varphi \subset I{\setminus } \{t_1,..., t_{N-1}\}$ and

$$\begin{aligned} \int _{t_{i-1}}^{t_i}\varphi = t_i - t_{i-1}\hbox { for all }i = 1,..., N. \end{aligned}$$

Define

$$\begin{aligned} \beta (t)&= {\widehat{\beta }}\left( \int _0^t\varphi \right) \\ r(t)&= {\widehat{r}}\left( \int _0^t\varphi \right) \\ K_g(t)&= {\widehat{K}}_g\left( \int _0^t\varphi \right) . \end{aligned}$$

Then $\beta$, r and $K_g$ belong to $C^{\infty }(\overline{I})$. Hence the frame ${\widetilde{r}}$ defined by ${\widetilde{r}}_3 = \beta$ and by (52) is in $C^{\infty }(\overline{I})$, and in view of (61) we have

$$\begin{aligned} \beta '\times {\widetilde{r}}_1 = 0\hbox { on }I. \end{aligned}$$

(62)

Now we reparametrize r as described in Sect. 2.3.2. Then the curve $\gamma : I\rightarrow {\mathbb {R}}^3$ defined by $\gamma (t) = \int _0^t r_1$ satisfies $\gamma (1) = \overline{\gamma }$. The function $K_g$ (hence ${\widetilde{r}}$) is reparametrized by the same change of variables as r. This yields a framed curve $(\gamma , r)$ and a frame ${\widetilde{r}}$ adapted to $r_3: I\rightarrow {\mathbb {S}}^2$ with the desired properties.

5.3 Proof of Theorem 5.1

We can now prove Theorem 5.1 using Proposition 5.2 and the link between isometric immersions and framed curves.

Proof of Theorem 5.1

Define $b: I\rightarrow {\mathbb {R}}^2$ by $b(t) = t e_1$, so that $\kappa = b''\cdot (b')^{\perp } \equiv 0$. Let $\gamma$, r, ${\widetilde{r}}$ and $K_g$ be as in Proposition 5.2 and set $\kappa _g = K_g'$. Then in view of (51) we have in particular

$$\begin{aligned} \left| \int _0^t\kappa _g \right| < \frac{\pi }{2} \hbox { for all }t\in \overline{I}. \end{aligned}$$

(63)

Hence the results in Hornung (2023) imply that there exists an isometric immersion defined on a neighbourhood of b(I) satisfying $n(b) = \beta$.

Let us sketch the construction of this isometric immersion provided in Hornung (2023). Using (63) one can show (Hornung 2023) that there exists $\varepsilon > 0$ such that, defining $M = I\times (-\varepsilon , \varepsilon )$ and ${\widetilde{R}}_1 = (\cos K_g, \sin K_g)^T$ as well as ${\widetilde{R}}_2 = {\widetilde{R}}_1^{\perp }$, the map $\Phi : M\rightarrow {\mathbb {R}}^2$ defined by

$$\begin{aligned} \Phi (t, s) = te_1 + s{\widetilde{R}}_2(t) \end{aligned}$$

is a locally bi-Lipschitz homeomorphism from M onto $U = \Phi (M)$. Hence the map $u: U\rightarrow {\mathbb {R}}^3$ defined by

$$\begin{aligned} u\left( \Phi (t, s)\right) = s{\widetilde{r}}_2(t) + \int _0^t r_1 \hbox { for all }(t, s)\in M \end{aligned}$$

(64)

is well-defined. A computation (Hornung 2023) shows that it belongs to $W^{2,2}(U, {\mathbb {R}}^3)$. Since

$$\begin{aligned} (\nabla u)(\Phi ) = {\widetilde{r}}_1\otimes {\widetilde{R}}_1 + {\widetilde{r}}_2\otimes {\widetilde{R}}_2\hbox { on }M, \end{aligned}$$

(65)

we see that u is indeed an isometric immersion $U\rightarrow {\mathbb {R}}^3$.

Since $K_g$ has compact support, we have ${\widetilde{R}}_2 = e_2$ near $\partial I$ and therefore $\Phi$ is the identity in a neighbourhood of $\partial I\times {\mathbb {R}}$. Moreover, (51) ensures that $e_2\cdot {\widetilde{R}}_2\ge \frac{1}{\sqrt{2}}$ on I. Hence there exists $w > 0$ such that $I\times [-w, w]\subset U$.

Combining (64) and (65) with the boundary conditions satisfied by $\gamma$ and by r, and observing that ${\widetilde{r}} = r$ near $\partial I$, we conclude that u and $\nabla u$ satisfy the asserted boundary conditions. $\square$