# Generalized Bishop frames on curves on $\mathbb{E}^{4}$

Subaru Nomoto  and  Hiraku Nozawa  Subaru Nomoto, Graduate School of Science
and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga,
525-8577, Japan<EMAIL_ADDRESS>Hiraku Nozawa, Department of
Mathematical Sciences, Colleges of Science and Engineering, Ritsumeikan
University, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
<EMAIL_ADDRESS>
###### Abstract.

We introduce and study generalized Bishop frames on regular curves, which are
generalizations of the Frenet and Bishop frames for regular curves on higher
dimensional spaces. There are four types of generalized Bishop frames on
regular curves on $\mathbb{E}^{4}$ up to the change of the order of vectors
fixing the first one which is the tangent vector. One of these four types of
frames is a Bishop frame, and by a result of Bishop, every regular curve
admits such a frame. We show that if a regular curve $\gamma$ on
$\mathbb{E}^{4}$ admits a Frenet frame, then $\gamma$ admits all four types of
generalized Bishop frames. We also show that if the derivative of the tangent
vector of a regular curve is nowhere vanishing, then the curve admits all
three types of generalized Bishop frames except a frame of type F, which is
related to the Frenet frame.

###### Key words and phrases:

Space curve, Frenet frame, Bishop frame, rotation minimizing vector field

The second author is supported by JSPS KAKENHI Grant numbers 17K14195 and
20K03620

##  1\. Introduction

The Frenet frame is fundamental to study space curves, which allows us to
classify the congruence classes of curves in terms of its torsion and
curvature. The Bishop frame (or rotation minimizing frame) is a different type
of frames on curves introduced by Bishop [1]. It is an advantage of the Bishop
frame that every regular curve admits the Bishop frame by a theorem of Bishop
[1], while some regular curves do not admit Frenet frame. Bishop frames are
used in the study of curves, see [9, 11, 3, 7]. For the application of Bishop
frames in differential geometry, see [8, 2, 12, 10]. The open problems on
Bishop frames are listed in [4]. The Bishop frame was used in other domains
including computer graphics and engineering, see [13, 5].

In this article, we study relations between frames on curves on
$\mathbb{E}^{4}$ obtained by generalizing the Frenet and Bishop frames. In
order to motivate us, let us review the idea of Bishop to introduce Bishop
frames. For a regular curve $\gamma:I\to\mathbb{E}^{3}$ defined on an open
interval $I$ with arc-length parametrization, a Bishop frame is a frame of
$\gamma^{*}T\mathbb{E}^{3}$ of the form
$\\{\mathbb{T},\mathbb{B}_{1},\mathbb{B}_{2}\\}$ where $\mathbb{T}$ is the
tangent vector and satisfies the following differential equation:

$\begin{pmatrix}\mathbb{T}^{\prime}\\\ \mathbb{B}_{1}^{\prime}\\\
\mathbb{B}_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}0&b_{1}&b_{2}\\\
-b_{1}&0&0\\\ -b_{2}&0&0\end{pmatrix}\begin{pmatrix}\mathbb{T}\\\
\mathbb{B}_{1}\\\ \mathbb{B}_{2}\end{pmatrix}.$

Throughout this article, regular curves are parametrized by arc-length
parameters unless otherwise stated. We will consider only frames on regular
curves whose first vector is the tangent vector. We regard an orthonormal
frame on a curve $I\to\mathbb{E}^{n}$ as a matrix valued function
$\mathbb{Z}:I\to O(n)$ such that the frame consists of the row vectors of
$\mathbb{Z}$. For a frame on a regular curve, we will call the matrix valued
function $X$ such that $\mathbb{Z}^{\prime}=X\mathbb{Z}$ the coefficient
matrix of the frame. In order to introduce Bishop frames, Bishop considered
three types of frames, whose coefficient matrices have one of the following
forms for some functions $x_{1}$ and $x_{2}$:

$\left(\begin{matrix}0&x_{1}&0\\\ -x_{1}&0&x_{2}\\\
0&-x_{2}&0\end{matrix}\right),$

(1)

$\left(\begin{matrix}0&x_{1}&x_{2}\\\ -x_{1}&0&0\\\
-x_{2}&0&0\end{matrix}\right),$

(2)

$\left(\begin{matrix}0&0&x_{1}\\\ 0&0&x_{2}\\\
-x_{1}&-x_{2}&0\end{matrix}\right).$

(3)

A frame with the coefficient matrix (2) is exactly the Bishop frame. A frame
with coefficient matrix of the form (1) is closely related to the Frenet
frame; it is the Frenet frame if $x_{1}$ is positive.Bishop remarked that if a
frame $\\{\mathbb{T},\mathbb{Z}_{1},\mathbb{Z}_{2}\\}$ has a coefficient
matrix of the form (3), then the coefficient matrix of
$\\{\mathbb{T},\mathbb{Z}_{2},\mathbb{Z}_{1}\\}$ is of the form (1).
Therefore, there are essentially two types among this kinds of frames on
curves on $\mathbb{E}^{3}$, which are the Frenet frame and Bishop frame.
Generalizations of these frames on curves on general Riemannian manifolds are
known [3] : let $\gamma:I\to\mathbb{E}^{4}$ be a regular curve on a Riemannian
manifold. An orthonormal frame
$\mathbb{T},\mathbb{Z}_{1},\mathbb{Z}_{2},\mathbb{Z}_{3}$ on $\gamma$ is the
Frenet frame or Bishop frame if its coefficient matrix $X$ is of the form

$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}b_{1}&\phantom{-}b_{2}&\phantom{-}b_{3}\\\
-b_{1}&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-b_{2}&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-b_{3}&\phantom{-}0&\phantom{-}0&\phantom{-}0\end{pmatrix}$ or
$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}f_{1}&\phantom{-}0&\phantom{-}0\\\
-f_{1}&\phantom{-}0&\phantom{-}f_{2}&\phantom{-}0\\\
\phantom{-}0&-f_{2}&\phantom{-}0&\phantom{-}f_{3}\\\
\phantom{-}0&\phantom{-}0&-f_{3}&\phantom{-}0\end{pmatrix},$

respectively, where $b_{1},b_{2},b_{3},f_{3}$ are functions on $I$ and
$f_{1},f_{2}$ are positive functions on $I$. Generalizing the case of space
curves, we give the following definition:

######  Definition 1.1.

We call an orthonormal frame on a curve on $\mathbb{E}^{4}$ a _generalized
Bishop frame_ if its coefficient matrix has at most three nonzero entries
above the main diagonal.

Except degenerate ones which has a zero column vector, there are 16 kinds of
such frames (see Section 2). We can see that there are the following 4
equivalence classes of these 16 frames on curves up to the change of the order
of vectors fixing the first one which is the tangent vector: If a frame has a
coefficient matrix of the respective form for some functions
$x_{1},x_{2},x_{3}$ up to the change of the order of vectors fixing the first
one, we call it a _generalized Bishop frame_ of type B, C, D and F,
respectively:

$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}x_{1}&\phantom{-}x_{2}&\phantom{-}x_{3}\\\
-x_{1}&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-x_{2}&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-x_{3}&\phantom{-}0&\phantom{-}0&\phantom{-}0\end{pmatrix},$
$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}x_{1}&\phantom{-}x_{2}&\phantom{-}0\\\
-x_{1}&\phantom{-}0&\phantom{-}0&\phantom{-}x_{3}\\\
-x_{2}&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&-x_{3}&\phantom{-}0&\phantom{-}0\end{pmatrix},$ Type B Type C
$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}x_{1}&\phantom{-}0&\phantom{-}0\\\
-x_{1}&\phantom{-}0&\phantom{-}x_{2}&\phantom{-}x_{3}\\\
\phantom{-}0&-x_{2}&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&-x_{3}&\phantom{-}0&\phantom{-}0\end{pmatrix},$
$\displaystyle\begin{pmatrix}\phantom{-}0&\phantom{-}x_{1}&\phantom{-}0&\phantom{-}0\\\
-x_{1}&\phantom{-}0&\phantom{-}x_{2}&\phantom{-}0\\\
\phantom{-}0&-x_{2}&\phantom{-}0&\phantom{-}x_{3}\\\
\phantom{-}0&\phantom{-}0&-x_{3}&\phantom{-}0\end{pmatrix}.$ Type D Type F

A frame of type F is closely related to the Frenet frame; it is Frenet frame
if both $x_{1}$ and $x_{2}$ are positive. A frame of type B is a Bishop frame.
By Bishop’s theorem, every regular curve admits a Bishop frame, but there are
well known examples of regular curves which do not admit the Frenet frame.

In general, the relation between these four types of frames are not clear. But
the following result shows that not all regular curves admit generalized
Bishop frames of type C and D. Therefore it is a geometric property for
regular curves whether they admit these frames or not.

######  Theorem 1.2.

There is a regular curve which admits a generalized Bishop frame of type C but
does not admit a frame of type D. There is a regular curve which does not
admit a frame of type C.

In this article, we focus on the frames on the following types of curves.

######  Definition 1.3.

A curve is said to be _$2$ -regular_ if both the tangent vector and its
derivative are nowhere vanishing.

The main results of this article are the following:

######  Theorem 1.4.

Every $2$-regular curve on $\mathbb{E}^{4}$ admits a frame of type C and D,
respectively.

Note that every regular curve admits a Bishop frame by Bishop’s theorem. The
following result implies that the frame of type F on $2$-regular curves is
distinguished from other types.

######  Theorem 1.5.

There exists a $2$-regular curve on $\mathbb{E}^{4}$ which does not admit a
frame of type F.

A basic tool of the proof is the differential equation of transformations
between two frames (Lemma 3.1). Let $\gamma$ be an arc-length parametrized
regular curve with a Bishop frame $\mathbb{B}$. Assume that $\gamma$ admits a
frame $\mathbb{F}$ of type F. Then we observe that the transformation
$G=\mathbb{F}\mathbb{B}^{-1}$ from $\mathbb{B}$ to $\mathbb{F}$ satisfies the
following differential equation (see Lemma 3.1 below):

(1.1) $G^{\prime}=X_{\mathbb{F}}G-GX_{\mathbb{B}},$

where $X_{\mathbb{B}}$ and $X_{\mathbb{F}}$ are the coefficient matrix of
$\mathbb{B}$ and $\mathbb{F}$, respectively. We can see that, conversely,
given $X_{\mathbb{B}}$, if the equation (1.1) has a solution $G$ for some
$X_{\mathbb{F}}$ of the form of coefficient matrix of a frame of type F, then
$G\mathbb{B}$ is a frame of type F (see Lemma 3.1 below). In this way, more
generally for frames of other types, we can translate the existence problem of
a frame of given type on a curve to the existence of a solution of a
differential equation such that a part of the coefficients, the entries of the
coefficient matrix of the target frame, are undetermined. The main results are
proved by showing the existence of solutions of the differential equations
similar to (1.1) under a suitable choice of the undetermined coefficients.

##  2\. Basic properties of frames

There are 16 kinds of generalized Bishop frames on regular curves on
$\mathbb{E}^{4}$, excluding frames whose coefficient matrix has a zero column
vector. We will study the relation between these frames. It is easy to see
that they are classified into four types up to the action of the symmetric
group $\mathfrak{S}_{3}$ of order $3$ which swaps the second, third and fourth
vector of the frame as follows:

$\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)$

type B

$\displaystyle\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)$
$\displaystyle\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
-\blacksquare&-\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&-\blacksquare&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)$

type C

$\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
-\blacksquare&-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&-\blacksquare&-\blacksquare&\phantom{-}0\\\
\end{smallmatrix}\right)$

type D

$\displaystyle\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&-\blacksquare&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)$
$\displaystyle\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0&\phantom{-}0\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&-\blacksquare&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}\blacksquare\\\
-\blacksquare&-\blacksquare&\phantom{-}0&\phantom{-}0\\\
\phantom{-}0&-\blacksquare&\phantom{-}0&\phantom{-}0\end{smallmatrix}\right)\left(\begin{smallmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare&\phantom{-}0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
-\blacksquare&\phantom{-}0&\phantom{-}0&\phantom{-}\blacksquare\\\
\phantom{-}0&-\blacksquare&-\blacksquare&\phantom{-}0\end{smallmatrix}\right)$

type F

A frame of type B is nothing but a Bishop frame. Clearly a regular curve
admits one of generalized Bishop frames if and only if it admits all
generalized Bishop frames of the same type.

We can characterize the curves which admit a frame of type D as follows.

######  Proposition 2.1.

For a regular curve $\gamma$ on $\mathbb{E}^{4}$, the following are
equivalent:

  1. (1)

$\gamma$ admits a frame of type D.

  2. (2)

There exists a smooth unit normal vector field $\mathbb{D}_{1}$ and a smooth
function $d_{1}$ such that $\mathbb{T}^{\prime}=d_{1}\mathbb{D}_{1}$.

###### Proof.

Clearly (1) implies (2). Let us show (1) assuming (2). Let $\gamma$ be a
regular curve which satisfies the condition (2). Let $\mathbb{D}_{1}$ be a
unit normal vector field on $\gamma$ such that
$\mathbb{T}^{\prime}=d_{1}\mathbb{D}_{1}$. Let us consider a regular curve
$\delta$ whose tangent vector is $\mathbb{D}_{1}$. By a result of Bishop,
$\delta$ admits a Bishop frame of the form
$\mathbb{D}_{1},\mathbb{T},\mathbb{D}_{2},\mathbb{D}_{3}$. Then we have

$\left(\begin{matrix}\mathbb{D}_{1}^{\prime}\\\ \mathbb{T}^{\prime}\\\
\mathbb{D}_{2}^{\prime}\\\
\mathbb{D}_{3}^{\prime}\end{matrix}\right)=\left(\begin{matrix}0&x_{1}&x_{2}&x_{3}\\\
-x_{1}&0&0&0\\\ -x_{2}&0&0&0\\\
-x_{3}&0&0&0\end{matrix}\right)\left(\begin{matrix}\mathbb{D}_{1}\\\
\mathbb{T}\\\ \mathbb{D}_{2}\\\ \mathbb{D}_{3}\end{matrix}\right)$

for some $x_{1},x_{2},x_{3}$, which implies that
$\mathbb{T},\mathbb{D}_{1},\mathbb{D}_{2},\mathbb{D}_{3}$ is a frame of type
D. ∎

With this proposition, it is easy to construct an example of regular curve
which does not admit a frame of type D, which shows the first half of Theorem
1.2:

######  Example 2.2.

Let $\gamma$ be a curve on $\mathbb{E}^{4}$ defined by

(2.1) $\gamma(t)=\begin{cases}(t,e^{-\frac{1}{t}},0,0)&t>0\\\ (0,0,0,0)&t=0\\\
(t,0,e^{\frac{1}{t}},0)&t<0.\end{cases}$

Note that the parameter of $\gamma$ is not an arc-length.Then, it is easy to
see that $\gamma^{\prime\prime}/|\gamma^{\prime\prime}|$ is not continuous at
$t=0$. Therefore, $\gamma$ does not admit a frame of type D. It is easy to see
that this $\gamma$ admits a frame of type C, and hence it shows the first half
of Theorem 1.2.

The later half of Theorem 1.2 is proved in Section 4.

If a regular curve admits a Frenet frame, then the normalization of the
differential of the tangent vector satisfies the condition for
$\mathbb{D}_{1}$ in (2) of the last proposition. Thus we get the following:

######  Corollary 2.3.

If a regular curve admits a frame of type F, then it admits a frame of type D.

######  Corollary 2.4.

Every $2$-regular curve admits a frame of type D.

It is more involved to characterize curves which admits a frame of type C.

######  Proposition 2.5.

A regular curve $\gamma$ admits a frame of type C if and only if there exists
a unit normal vector field $\mathbb{Z}_{0}$ on $\gamma$ such that a curve
whose tangent vector is $\mathbb{Z}_{0}$ admits a frame of type F of the form
$\mathbb{Z}_{0},\mathbb{Z}_{1},\mathbb{T},\mathbb{Z}_{3}$, or of the
form$\mathbb{Z}_{0},\mathbb{T},\mathbb{Z}_{1},\mathbb{Z}_{3}$ whose
coefficient matrix is of the form

(2.2)
$\begin{pmatrix}\phantom{-}0&\phantom{-}f_{1}&\phantom{-}0&\phantom{-}0\\\
-f_{1}&\phantom{-}0&\phantom{-}f_{2}&\phantom{-}0\\\
\phantom{-}0&-f_{2}&\phantom{-}0&\phantom{-}f_{3}\\\
\phantom{-}0&\phantom{-}0&-f_{3}&\phantom{-}0\end{pmatrix}.$

###### Proof.

It is easy to see that the coefficient matrix of the frame
$\mathbb{Z}_{0},\mathbb{Z}_{1},\mathbb{T},\mathbb{Z}_{3}$ is of the form (2.2)
if and only if we have

$\begin{pmatrix}\mathbb{T}^{\prime}\\\ \mathbb{Z}_{1}^{\prime}\\\
\mathbb{Z}_{0}^{\prime}\\\
\mathbb{Z}_{3}^{\prime}\end{pmatrix}=\begin{pmatrix}0&-f_{2}&0&f_{3}\\\
f_{2}&0&-f_{1}&0\\\ 0&f_{1}&0&0\\\
-f_{3}&0&0&0\end{pmatrix}\begin{pmatrix}\mathbb{T}\\\ \mathbb{Z}_{1}\\\
\mathbb{Z}_{0}\\\ \mathbb{Z}_{3}\end{pmatrix}.$

The latter hold if and only if
$\mathbb{T},\mathbb{Z}_{1},\mathbb{Z}_{0},\mathbb{Z}_{3}$ of type C. ∎

By the last proposition, a curve contained in a hyperplane of $\mathbb{E}^{4}$
admits a frame of type C. In particular, the curve $\gamma$ in (2.1) admits a
frame of type C. Therefore it is an example of a regular curve which admits a
frame of type C but does not admit a frame of type D.

Typical examples of curves with generalized Bishop frame are curves with
frames with constant coefficient matrices. For example, let
$X=\begin{pmatrix}0&2&1&0\\\ -2&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{pmatrix}$.
Then $\exp(sX)$ is a generalized Bishop frame of type C on a curve $\gamma$
whose tangent vector is the first column vector of $\exp(sX)$, where

$\gamma(s)=\\\ \left(\frac{\sin\sqrt{2}s\cos s}{\sqrt{2}},\frac{\sin
s\sin\sqrt{2}s}{\sqrt{2}},\frac{-\sin s\sin\sqrt{2}s-\sqrt{2}\cos
s\cos\sqrt{2}s}{\sqrt{2}},\frac{\sin\sqrt{2}s\cos s-\sqrt{2}\sin
s\cos\sqrt{2}s}{\sqrt{2}}\right),$

Let us present an example of a $2$-regular curve which does not admit a frame
of type F, which shows Theorem 1.5:

######  Example 2.6.

Consider a curve $\gamma$ on $\mathbb{E}^{4}$ whose tangent vector field is

(2.3)
$\mathbb{T}(s)=\begin{cases}\displaystyle\left(\frac{2s}{s^{2}+e^{-\frac{2}{s}}+1},\,\frac{2e^{-\frac{1}{s}}}{s^{2}+e^{-\frac{2}{s}}+1},\,0,\,\frac{s^{2}+e^{-\frac{2}{s}}-1}{s^{2}+e^{-\frac{2}{s}}+1}\right)&s>0\\\
(0,0,0,-1)&s=0\\\
\displaystyle\left(\frac{2s}{s^{2}+e^{\frac{2}{s}}+1},\,0,\,\frac{2e^{\frac{1}{s}}}{s^{2}+e^{\frac{2}{s}}+1},\,\frac{s^{2}+e^{\frac{2}{s}}-1}{s^{2}+e^{\frac{2}{s}}+1}\right)&s<0.\end{cases}$

It is easy to see that $\gamma$ is $2$-regular.

Assume that $\gamma$ admits a generalized Bishop frame
$\mathbb{T},\mathbb{F}_{1},\mathbb{F}_{2},\mathbb{F}_{3}$ of type $F$. Since
$\gamma(\mathbb{R}_{\geq 0})$ is contained in a hyperplane
$H_{+}=\\{\,(x,y,z,w)\in\mathbb{E}^{4}\mid z=0\\}$ and the curvature and the
torsion of the Frenet frame of $\gamma$ does not vanish on
$\mathbb{R}_{>0}$,it follows that $\mathbb{T}$, $\mathbb{F}_{1}$,
$\mathbb{F}_{2}$ spans $H_{+}$ at each point on $\gamma$.Similarly, since
$\gamma(\mathbb{R}_{\leq 0})$ is contained in a hyperplane
$H_{-}=\\{\,(x,y,z,w)\in\mathbb{E}^{4}\mid y=0\\}$, by using the Frenet frame
of $\gamma$, we can see that $\mathbb{T}$, $\mathbb{F}_{1}$, $\mathbb{F}_{2}$
spans $H_{-}$ at each point on $\gamma$. Thus, at $t=0$,
$\mathbb{T},\mathbb{F}_{1},\mathbb{F}_{2}$ must span $H_{+}\cap H_{-}$, which
is a contradiction. Therefore, $\gamma$ does not admit a frame of type F.

The tangent vector $\mathbb{T}$ of this example is similar to the curve in
Example 2.1, which can be divided into two curves, each of which is contained
in a hyperplane.

##  3\. Transformations between frames

As in the following lemma, by considering transformations between frames, we
can reduce the problem of the existence of a frame of given coefficient matrix
to the existence of a solution of a differential equation given by the entries
of coeffient matrices.

######  Lemma 3.1.

Let $\gamma:I\to\mathbb{E}^{4}$ be a regular curve. Let $\mathbb{Z}_{0}$ be a
frame on $\gamma$ such that $\mathbb{Z}^{\prime}_{0}=X_{0}\mathbb{Z}_{0}$, and
a function $X_{1}:I\to\mathfrak{o}(4)$. Consider a frame $\mathbb{Z}_{1}$ on
$\gamma$ such that

(3.1) $\mathbb{Z}^{\prime}_{1}=X_{1}\mathbb{Z}_{1}.$

Then the transformation $G:I\to O(4)$ from $\mathbb{Z}_{0}$ to
$\mathbb{Z}_{1}$ given by $G=\mathbb{Z}_{1}\mathbb{Z}_{0}^{-1}$ satisfies the
differential equation

(3.2) ${G}^{\prime}=X_{1}{G}-{G}X_{0}.$

On the contrary, if a function $G:I\to O(4)$ satisfies the differential
equation (3.2), then the frame $\mathbb{Z}_{1}$ given by
$\mathbb{Z}_{1}=G\mathbb{Z}_{0}$ satisfies the differential equation (3.1).

###### Proof.

In order to prove the first part, assume that $\mathbb{Z}_{1}$ satisfies
(3.1). By differentiating both sides of $\mathbb{Z}_{0}\mathbb{Z}_{0}^{-1}=E$,
we get
$(\mathbb{Z}_{0}^{-1})^{\prime}=-\mathbb{Z}_{0}^{-1}\mathbb{Z}^{\prime}_{0}\mathbb{Z}_{0}^{-1}$.
Then we have

${G}^{\prime}=(\mathbb{Z}_{1}\mathbb{Z}_{0}^{-1})^{\prime}=\mathbb{Z}^{\prime}_{1}\mathbb{Z}_{0}^{-1}+\mathbb{Z}_{1}(\mathbb{Z}_{0}^{-1})^{\prime}\\\
=X_{1}\mathbb{Z}_{1}\mathbb{Z}_{0}^{-1}+\mathbb{Z}_{1}(-\mathbb{Z}_{0}^{-1}\mathbb{Z}_{0}^{\prime}\mathbb{Z}_{0}^{-1})=X_{1}{G}-{G}X_{0},$

from which follows the differential equation (3.2). For the proof of the
second part, assume that ${G}$ satisfies (3.2) and consider
$\mathbb{Z}_{1}={G}\mathbb{Z}_{0}$. Then we have

$\mathbb{Z}^{\prime}_{1}=({G}\mathbb{Z}_{0})^{\prime}={G}^{\prime}\mathbb{Z}_{0}+{G}\mathbb{Z}^{\prime}_{0}=(X_{1}{G}-{G}X_{0})\mathbb{Z}_{0}+{G}X_{0}\mathbb{Z}_{0}=X_{1}\mathbb{Z}_{1},$

which is (3.1). ∎

By using the lemma, we can relate curvatures of frames of different types. The
curvature of a frame of type D is related to the curvature of a frame of type
F in a simple way.

######  Proposition 3.2.

Let

(3.3) $X_{\mathbb{F}}=\begin{pmatrix}0&f_{1}&0&0\\\ -f_{1}&0&f_{2}&0\\\
0&-f_{2}&0&f_{3}\\\ 0&0&-f_{3}&0\end{pmatrix},\quad\quad
X_{\mathbb{D}}=\begin{pmatrix}0&d_{1}&0&0\\\ -d_{1}&0&d_{2}&d_{3}\\\
0&-d_{2}&0&0\\\ 0&-d_{3}&0&0\end{pmatrix},$

where $f_{1},f_{2},f_{3},d_{1},d_{2},d_{3}$ are functions on an interval $I$.
Let $\gamma:I\to\mathbb{E}^{4}$ be a $2$-regular curve
$\gamma:I\to\mathbb{E}^{4}$ admits a generalized Bishop frame $\mathbb{F}$ of
type $F$ with coefficient matrix $X_{\mathbb{F}}$, then $\gamma$ admits a
generalized Bishop frame $\mathbb{D}$ of type D with coefficient matrix
$X_{\mathbb{D}}$, where we have

(3.4) $d_{1}=\epsilon f_{1},\quad d_{2}=\epsilon f_{2}\cos\left(\int
f_{3}ds\right),\quad d_{3}=-\kappa f_{2}\sin\left(\int f_{3}ds\right)$

for some $\epsilon,\kappa\in\\{1,-1\\}$.

###### Proof.

By Lemma 3.1, it suffices to show that the solution ${G}:I\to O(4)$ of a
differential equation

(3.5) ${G}^{\prime}=X_{\mathbb{D}}{G}-{G}X_{\mathbb{F}}$

exists if and only if $d_{1},d_{2},d_{3}$ satisfy the relation (3.4).Letting

$G=\begin{pmatrix}1&\bm{0}\\\ {}^{t}\bm{0}&\overline{G}\end{pmatrix},\quad
X_{\mathbb{F}}=\begin{pmatrix}0&{\bm{f}}\\\
-^{t}{\bm{f}}&Y_{\mathbb{F}}\end{pmatrix},\quad
X_{\mathbb{D}}=\begin{pmatrix}0&{\bm{d}}\\\
-^{t}{\bm{d}}&Y_{\mathbb{D}}\end{pmatrix}$

the equation (3.5) is expressed as

$\displaystyle\begin{pmatrix}0&\bm{0}\\\
{}^{t}\bm{0}&\overline{G}^{\prime}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&{\bm{d}}\\\
-^{t}{\bm{d}}&Y_{\mathbb{D}}\end{pmatrix}\begin{pmatrix}1&\bm{0}\\\
{}^{t}\bm{0}&\overline{G}\end{pmatrix}-\begin{pmatrix}1&\bm{0}\\\
{}^{t}\bm{0}&\overline{G}\end{pmatrix}\begin{pmatrix}0&{\bm{f}}\\\
-^{t}{\bm{f}}&Y_{\mathbb{F}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&{\bm{d}}\overline{G}-{\bm{f}}\\\
-^{t}{\bm{d}}+\overline{G}^{t}{\bm{f}}&Y_{\mathbb{D}}\overline{G}-\overline{G}Y_{\mathbb{F}}\end{pmatrix}.$

Therefore (3.5) is equivalent to ${\bm{d}}\overline{G}-{\bm{f}}={\bm{0}}$ and
$\overline{G}^{\prime}=Y_{\mathbb{D}}\overline{G}-\overline{G}Y_{\mathbb{F}}$.
It is easy to see that ${\bm{d}}\overline{G}-{\bm{f}}={\bm{0}}$ is equivalent
to that $\overline{G}$ is of the form $\begin{pmatrix}\epsilon&\bm{0}\\\
{}^{t}\bm{0}&\bar{\bar{G}}\end{pmatrix}$ and $d_{1}=\epsilon f_{1}$ for some
$\epsilon\in\\{1,-1\\}$.Then $\overline{G}$ is $O(3)$-valued if and only if

$\bar{\bar{G}}=\begin{pmatrix}\cos\theta&\kappa\epsilon\sin\theta\\\
\sin\theta&-\kappa\epsilon\cos\theta\end{pmatrix}$

for some function $\theta$ and some $\kappa\in\\{1,-1\\}$. Under this
condition, the equation
$\overline{G}^{\prime}=Y_{\mathbb{D}}\overline{G}-\overline{G}Y_{\mathbb{F}}$
is

$\begin{pmatrix}0&0&0\\\
0&-\theta^{\prime}\sin\theta&\kappa\epsilon\theta^{\prime}\cos\theta\\\
0&\theta^{\prime}\cos\theta&\kappa\epsilon\theta^{\prime}\sin\theta\end{pmatrix}=\\\
\begin{pmatrix}0&-\epsilon
f_{2}+d_{2}\cos\theta+d_{3}\sin\theta&\kappa\epsilon
d_{2}\sin\theta-\kappa\epsilon d_{3}\cos\theta\\\ -\epsilon
d_{2}+f_{2}\cos\theta&\kappa\epsilon f_{3}\sin\theta&-f_{3}\cos\theta\\\
-\epsilon d_{3}+f_{2}\sin\theta&-\kappa\epsilon
f_{3}\cos\theta&-f_{3}\sin\theta\end{pmatrix}$

which is satisfied if and only if $\theta^{\prime}=-\kappa\epsilon
f_{3},\,d_{1}=\epsilon f_{1},\,d_{2}=\epsilon f_{2}\cos\theta,\,d_{3}=\epsilon
f_{2}\sin\theta$, The proof of the proposition is concluded. ∎

We will show that every $2$-regular curve on $\mathbb{E}^{4}$ admits a frame
of type C.

######  Lemma 3.3.

Let

(3.6) $X_{\mathbb{B}}=\begin{pmatrix}0&b_{1}&b_{2}&b_{3}\\\ -b_{1}&0&0&0\\\
-b_{2}&0&0&0\\\ -b_{3}&0&0&0\end{pmatrix},\quad\quad
X_{\mathbb{C}}=\begin{pmatrix}0&c_{1}&c_{2}&0\\\ -c_{1}&0&0&c_{3}\\\
-c_{2}&0&0&0\\\ 0&-c_{3}&0&0\end{pmatrix},$

where $b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ are functions on an interval $I$.
Let $\gamma:I\to\mathbb{E}^{4}$ be a regular curve which has a Bishop frame
whose coefficient matrix is $X_{\mathbb{B}}$. If ${}^{t}\bm{b}$ is nowhere
tangent to ${}^{t}(0,1,0)$, then $\gamma$ admits a generalized Bishop frame
$\mathbb{C}$ of type C whose coefficient matrix $X_{\mathbb{C}}$, where
$c_{1},c_{2},c_{3}$ are given by

(3.7) $c_{1}=\pm\sqrt{b_{1}^{2}+b_{3}^{2}},\quad c_{2}=b_{2},\quad
c_{3}=\pm\frac{b_{3}^{\prime}b_{1}-b_{3}b_{1}^{\prime}}{b_{1}^{2}+b_{3}^{2}},$

and the transformation $G=\mathbb{C}\mathbb{B}^{-1}$ is of the form

(3.8) $\left(\begin{matrix}1&0&0&0\\\ 0&\pm\cos\theta&0&\pm\sin\theta\\\
0&0&1&0\\\ 0&-\sin\theta&0&\cos\theta\\\ \end{matrix}\right)$

for some function $\theta$ such that
$\theta^{\prime}=\pm\frac{b_{3}^{\prime}b_{1}-b_{3}b_{1}^{\prime}}{b_{1}^{2}+b_{3}^{2}}$.

###### Proof.

By Lemma 3.1, it suffices to show that, when (3.7) holds, the equation
${G}^{\prime}=X_{\mathbb{C}}{G}-{G}X_{\mathbb{B}}$ has a solution $G:I\to
O(4)$ of the form in (3.8). Let

${G}=\begin{pmatrix}1&\bm{0}\\\ {}^{t}\bm{0}&\overline{G}\end{pmatrix},\quad
X_{\mathbb{B}}=\begin{pmatrix}0&{\bm{b}}\\\
-{}^{t}{\bm{b}}&O\end{pmatrix},\quad
X_{\mathbb{C}}=\begin{pmatrix}0&{\bm{c}}\\\
-{}^{t}{\bm{c}}&Y_{\mathbb{C}}\end{pmatrix}.$

Then the equation ${G}^{\prime}=X_{\mathbb{C}}{G}-{G}X_{\mathbb{B}}$ is
expressed as

$\begin{pmatrix}0&\bm{0}\\\
{}^{t}\bm{0}&\overline{G}^{\prime}\end{pmatrix}=\begin{pmatrix}0&{\bm{c}}\overline{G}-{\bm{b}}\\\
-^{t}{\bm{c}}+\overline{G}^{t}{\bm{b}}&Y_{\mathbb{C}}\overline{G}\end{pmatrix},$

which is equivalent to a pair of equations
${}^{t}{\bm{c}}=\overline{G}{}^{t}{\bm{b}}$ and
$\overline{G}^{\prime}=Y_{\mathbb{C}}\overline{G}$.

Assume that (3.7) holds. Since $(b_{1},b_{3})$ is nowhere vanishing, we have a
smooth function $\theta$ such that
$(\cos\theta,\sin\theta)=\frac{1}{\sqrt{b_{1}^{2}+b_{3}^{2}}}(b_{1},b_{3})$.
Let $\overline{G}$ be

$\overline{G}=\begin{pmatrix}\pm\cos\theta&0&\pm\sin\theta\\\ 0&1&0\\\
-\sin\theta&0&\cos\theta\\\ \end{pmatrix}.$

By $c_{1}=\pm\sqrt{b_{1}^{2}+b_{3}^{2}}$ and $c_{2}=b_{2}$, we have
${}^{t}{\bm{c}}=\overline{G}{}^{t}{\bm{b}}$. By
$(\cos\theta,\sin\theta)=\frac{1}{\sqrt{b_{1}^{2}+b_{3}^{2}}}(b_{1},b_{3})$,
we have
$\theta^{\prime}=\frac{b_{3}^{\prime}b_{1}-b_{3}b_{1}^{\prime}}{b_{1}^{2}+b_{3}^{2}}=\pm
c_{3}$．Then it follows that

$\overline{G}^{\prime}=\begin{pmatrix}\mp\theta^{\prime}\sin\theta&0&\pm\theta^{\prime}\cos\theta\\\
0&0&0\\\ -\theta^{\prime}\cos\theta&0&-\theta^{\prime}\sin\theta\\\
\end{pmatrix}=\begin{pmatrix}-c_{3}\sin\theta&0&c_{3}\cos\theta\\\ 0&0&0\\\
\mp c_{3}\cos\theta&0&\mp c_{3}\sin\theta\\\
\end{pmatrix}=Y_{\mathbb{C}}\overline{G},$

which concludes the proof of Lemma 3.3.

∎

The following lemma directly follows from Lemma 3.1 in the case where both
$\mathbb{Z}_{0}$ and $\mathbb{Z}_{1}$ are of type B.

######  Lemma 3.4.

If a $2$-regular curve $\gamma$ admits a Bishop frame $\mathbb{B}$ whose
coefficient matrix is $X_{\mathbb{B}}$, then the coefficient matrix of any
other Bishop frame is of the form

(3.9) $\begin{pmatrix}1&\bm{0}\\\
{}^{t}\bm{0}&Q\end{pmatrix}X_{\mathbb{B}}\begin{pmatrix}1&\bm{0}\\\
{}^{t}\bm{0}&Q^{-1}\end{pmatrix}$

for some constant matrix $Q\in O(3)$. On the contrary, for any $Q\in O(3)$,
there exists a Bishop frame whose coefficient matrix is (3.9).

######  Theorem 3.5.

Every $2$-regular curve $\gamma$ admits a frame of type C.

###### Proof.

Take a Bishop frame $\mathbb{B}$ of $\gamma$ and let
$\begin{pmatrix}0&{\bm{b}}\\\ -{}^{t}{\bm{b}}&O\end{pmatrix}$ be the
coefficient matrix. Since $\gamma$ is $2$-regular, $\bm{b}$ is nowhere
vanishing. Consider $f:I\to\mathbb{R}P^{2};s\mapsto[{\bm{b}}(s)]$. By Sard
theorem, we have some $\bm{\xi}\in\mathbb{R}^{3}-\\{\bm{0}\\}$such that
$[\bm{\xi}]\in\mathbb{R}P^{2}\setminus\operatorname{Image}f$. Take $Q\in O(3)$
so that $\bm{\xi}Q=(0,1,0)$. Then, by Lemma 3.4, $\gamma$ admits a Bishop
frame whose coefficient matrix is $\begin{pmatrix}0&{\bm{b}}Q\\\
-{}^{t}Q{}^{t}{\bm{b}}&O\end{pmatrix}$. Since $(0,1,0)$ is never tangent to
$\bm{b}Q$, by Lemma 3.3, $\gamma$ admits a frame of type C. ∎

By Bishop’s theorem, every regular curve admits a Bishop frame. Therefore, by
Proposition 3.2 and Lemma 3.3, we get the following consequence, which implies
Theorem 1.4:

######  Corollary 3.6.

If a $2$-regular curve $\gamma$ on $\mathbb{E}^{4}$ admits a frame of type F,
then it admits generalized Bishop frames of all other types B, C and D. In
particular, if $\gamma$ admits the Frenet frame, then it admits frames of all
other types.

##  4\. Some examples of curves

We will present an example of a regular curve which does not admit a
generalized Bishop frame of type C. Together with Example 2.2, it shows
Theorem 1.2.

######  Proposition 4.1.

Let $\gamma$ be a curve on $\mathbb{E}^{4}$ with a Bishop frame $\mathbb{B}$
whose coefficient matrix is
$X_{\mathbb{B}}=\begin{pmatrix}0&b_{1}&b_{2}&b_{3}\\\ -b_{1}&0&0&0\\\
-b_{2}&0&0&0\\\ -b_{3}&0&0&0\end{pmatrix}$, where $b_{1},b_{2},b_{3}$ are
given by

$\displaystyle b_{1}(s)$ $\displaystyle=\begin{cases}e^{\frac{1}{s}}&s<0\\\
e^{-\frac{1}{s-2}}&2<s\\\ 0&\text{else},\end{cases}$ $\displaystyle b_{2}(s)$
$\displaystyle=\begin{cases}e^{-\frac{1}{s(-s+1)}}&0<s<1\\\
0&\text{else},\end{cases}$ $\displaystyle b_{3}(s)$
$\displaystyle=\begin{cases}e^{-\frac{1}{(s-1)(-s+2)}}&1<s<2\\\
0&\text{else}.\end{cases}$

Then $\gamma$ is a regular curve which does not admit a frame of type C.

###### Proof.

Assume that $\gamma$ admits a generalized Bishop frame $\mathbb{C}$ of type C
whose coefficient matrix is $\begin{pmatrix}0&c_{1}&c_{2}&0\\\
-c_{1}&0&0&c_{3}\\\ -c_{2}&0&0&0\\\ 0&-c_{3}&0&0\end{pmatrix}$. By Lemma 3.1,
the transformation $G=\mathbb{C}\mathbb{B}^{-1}$ satisfies the differential
equation $G^{\prime}=X_{\mathbb{C}}G-GX_{\mathbb{B}}$. Let

${G}=\begin{pmatrix}1&\bm{0}\\\ {}^{t}\bm{0}&\overline{G}\end{pmatrix},\quad
X_{\mathbb{B}}=\begin{pmatrix}0&{\bm{b}}\\\
-{}^{t}{\bm{b}}&O\end{pmatrix},\quad
X_{\mathbb{C}}=\begin{pmatrix}0&{\bm{c}}\\\
-{}^{t}{\bm{c}}&Y_{\mathbb{C}}\end{pmatrix}.$

Then $G^{\prime}=X_{\mathbb{C}}G-GX_{\mathbb{B}}$ is equivalent to
${}^{t}\bm{c}=\overline{G}{}^{t}\bm{b}$ and
$\overline{G}^{\prime}=Y_{\mathbb{C}}\overline{G}$. By
$\overline{G}^{\prime}=Y_{\mathbb{C}}\overline{G}$, we have

$\overline{G}=\begin{pmatrix}\cos\theta&0&\sin\theta\\\ 0&1&0\\\
-\sin\theta&0&\cos\theta\\\ \end{pmatrix}\overline{G}_{0}$

for a function $\theta$ and a constant matrix $\overline{G}_{0}\in O(3)$. Take
$\bm{\tilde{b}}$ so that ${}^{t}\bm{\tilde{b}}=\overline{G}_{0}{}^{t}\bm{b}$
and $\tilde{b}_{i}$ be the $i$-th component of $\bm{\tilde{b}}$ $(i=1,2,3)$.
Then, by ${}^{t}\bm{c}=\overline{G}{}^{t}\bm{b}$, we have

(4.1) $\left(\begin{matrix}c_{1}\\\ c_{2}\\\
0\end{matrix}\right)=\left(\begin{matrix}\cos\theta&0&\sin\theta\\\ 0&1&0\\\
-\sin\theta&0&\cos\theta\\\
\end{matrix}\right)\left(\begin{matrix}\tilde{b}_{1}\\\ \tilde{b}_{2}\\\
\tilde{b}_{3}\end{matrix}\right).$

Let $e_{1}={}^{t}(1,0,0)$, $e_{2}={}^{t}(0,1,0)$, $e_{3}={}^{t}(0,0,1)$. Then,
${}^{t}\bm{b}$ is tangent to $e_{1}$ for $s<0$, tangent to $e_{2}$ for $0<s<1$
and tangent to $e_{3}$ for $s>1$.

First let us show a contradiction assuming that ${}^{t}\bm{\tilde{b}}$ is
nowhere tangent to $e_{2}$. In this case, since
$(\tilde{b}_{1},\tilde{b}_{3})$ is nowhere vanishing, we have
$(\cos\theta,\sin\theta)=\frac{1}{\sqrt{\widetilde{b}_{1}^{2}+\widetilde{b}_{3}^{2}}}(\widetilde{b}_{1},\widetilde{b}_{3})$.
Since $\bm{b}$ is tangent to $e_{1}$ for $s<0$, it is clear that $\theta$ is
constant for $s<0$. Similarly we have that $\theta$ is constant also for
$0<s<1$ and for $1<s$. Therefore $G$ is constant everywhere. By (4.1), it
follows that $\overline{G}{}^{t}\bm{b}$ is contained in the subspace
$\mathbb{R}^{2}\oplus 0\subset\mathbb{R}^{3}$ for a constant orthogonal matrix
$G$. But this is not possible, since the image of ${}^{t}\bm{b}$ is clearly
not contained in any $2$-dimensional subspace of $\mathbb{R}^{3}$. Thus we
have a contradiction.

Now let us consider the case where $\overline{G}_{0}e_{1}$ is tangent to
$e_{2}$. In this case, ${}^{t}\bm{\tilde{b}}$ is tangent to $e_{2}$ for $s<0$
and $s>2$, and not tangent to $e_{2}$ for $0<s<1$ and $1<s<2$. It follows that
$(\tilde{b}_{1},\tilde{b}_{3})$ is nowhere vanishing for $0<s<1$ and $1<s<2$.
As in the last paragraph, it follows that $\theta$ is constant for $0<s<2$.
Since $\overline{G}_{0}e_{1}$ is tangent to $e_{2}$, $\overline{G}_{0}$ is of
the form $\begin{pmatrix}0&g_{23}&g_{24}\\\ \pm 1&0&0\\\
0&g_{43}&g_{44}\end{pmatrix}$. Then
${}^{t}\bm{\tilde{b}}=e^{-\frac{1}{s(1-s)}}\begin{pmatrix}g_{23}\\\ 0\\\
g_{43}\end{pmatrix}$ for $0<s<1$ and
${}^{t}\bm{\tilde{b}}=e^{-\frac{1}{(s-1)(-s+2)}}\begin{pmatrix}g_{24}\\\ 0\\\
g_{44}\end{pmatrix}$ for $1<s<2$. By (4.1), it follows that $c_{2}$ is
identically zero for $0<s<2$. Then the image of $\overline{G}{}^{t}\bm{b}$ for
$0<s<2$ is contained in the subspace $\mathbb{R}\oplus 0\oplus
0\subset\mathbb{R}^{3}$. On the other hand, ${}^{t}\bm{b}$ is tangent to
$e_{2}$ for $0<s<1$ and ${}^{t}\bm{b}$ is tangent to $e_{3}$ for $1<s<2$.
Therefore it is a contradiction.

We omit the detail of the proof of the remaining two cases; the case where
$\overline{G}_{0}e_{2}$ is tangent to $e_{2}$, and the second case where
$\overline{G}_{0}e_{3}$ is tangent to $e_{2}$. In both cases, we can show a
contradiction by an argument similar to the last two cases. Therefore, we
conclude that $\gamma$ does not admit a generalized Bishop frame of type C.

∎

Note that a curve with a Bishop frame with the following curvature, which is
simpler than the last example, admits a frame of type C:

$\displaystyle b_{1}(s)$ $\displaystyle=\begin{cases}e^{\frac{1}{s}}&s<0\\\
0&\text{else},\end{cases}$ $\displaystyle b_{2}(s)$
$\displaystyle=\begin{cases}e^{-\frac{1}{s(-s+1)}}&0<s<1\\\
0&\text{else},\end{cases}$ $\displaystyle b_{3}(s)$
$\displaystyle=\begin{cases}e^{-\frac{1}{(s-1)(-s+2)}}&1<s<2\\\
0&\text{else}.\end{cases}$

Indeed, we can take a transformation $G$ from the Bishop frame to a frame of
type C so that $\overline{G}_{0}e_{2}$ is tangent to $e_{2}$.

## References

  * [1] R.L. Bishop, There is more than one way to frame a curve. Amer. Math. Monthly 82 (1975), 246–251.
  * [2] D. Brander, J. Gravesen, Monge surfaces and planar geodesic foliations. J. Geom. 109 (2018), no. 1, Paper No. 4, 14 pp.
  * [3] F. Etayo, Rotation Minimizing vector fields and frames in Riemannian manifold. Springer, Proc. Math. Stat. 161, (2016), pp. 91-100.
  * [4] R.T. Farouki, Rational rotation-minimizing frames—recent advances and open problems. Appl. Math. Comput. 272 (2016), part 1, 80–91.
  * [5] R.T. Farouki, S. Li, Optimal tool orientation control for 5-axis CNC milling with ball-end cutters. Comput. Aided Geom. Design 30 (2013), no. 2, 226–239.
  * [6] F. Gökçelik, Z. Bozkurt, İ. Gök, F.N. Ekmekci, Y. Yaylı, Parallel transport frame in $4$-dimensional Euclidean space $\mathbb{E}^{4}$. Casp. J. Math. Sci. 3 (2014), no. 1, 91–103.
  * [7] M. Grbović, E. Nešović, On the Bishop frames of pseudo null and null Cartan curves in Minkowski $3$-space. J. Math. Anal. Appl. 461 (2018), no. 1, 219–233.
  * [8] T. Körpınar, E. Turhan, On characterization of $B$-canal surfaces in terms of biharmonic $B$-slant helices according to Bishop frame in Heisenberg group $\operatorname{Heis}^{3}$. J. Math. Anal. Appl. 382 (2011), no. 1, 57–65.
  * [9] by same author, A new approach on timelike biharmonic slant helices according to Bishop frame in Lorentzian Heisenberg group $\operatorname{Heis}^{3}$. Kochi J. Math. 6 (2011), 139–148.
  * [10] E.L. Mansfield, A. Rojo-Echeburúa, On the use of the rotation minimizing frame for variational systems with Euclidean symmetry. Stud. Appl. Math. 143 (2019), no. 3, 244–271.
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  * [13] W. Wang, B. Jüttler, D. Zheng, Y. Liu, Computation of rotation minimizing frames. ACM Trans. Graph. 27, Article 2, 2008.


# Doing it right: It’s time for addressing issues in citation context analysis

Anne Lauscher 1 Brandon Ko2 Bailey Kuehl3 Sophie Johnson3
David Jurgens4 Arman Cohan3 Kyle Lo3
1Data and Web Science Research Group, University of Mannheim, Germany
2University of Washington, Seattle WA
3Allen Institute for AI, Seattle WA
4School of Information, University of Michigan, Ann Arbor MI
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Part of the work was conducted during an internship at the Allen Institute for
AI.

#  MultiCite:
It’s Time for Addressing Issues in Citation Context Analysis

Anne Lauscher 1 Brandon Ko2 Bailey Kuehl3 Sophie Johnson3
David Jurgens4 Arman Cohan3 Kyle Lo3
1Data and Web Science Research Group, University of Mannheim, Germany
2University of Washington, Seattle WA
3Allen Institute for AI, Seattle WA
4School of Information, University of Michigan, Ann Arbor MI
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Part of the work was conducted during an internship at the Allen Institute for
AI.

#  MultiCite:
Revisiting Citation Context Analysis in Natural Language Processing

Anne Lauscher 1 Brandon Ko2 Bailey Kuehl3 Sophie Johnson3
David Jurgens4 Arman Cohan3 Kyle Lo3
1Data and Web Science Research Group, University of Mannheim, Germany
2University of Washington, Seattle WA
3Allen Institute for AI, Seattle WA
4School of Information, University of Michigan, Ann Arbor MI
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Part of the work was conducted during an internship at the Allen Institute for
AI.

#  MultiCite: Modeling realistic citations requires
moving beyond the single-sentence single-label setting

Anne Lauscher 1 Brandon Ko2 Bailey Kuehl3 Sophie Johnson3
David Jurgens4 Arman Cohan3 Kyle Lo3
1Data and Web Science Research Group, University of Mannheim, Germany
2University of Washington, Seattle WA
3Allen Institute for AI, Seattle WA
4School of Information, University of Michigan, Ann Arbor MI
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Part of the work was conducted during an internship at the Allen Institute for
AI.

###### Abstract

Citation context analysis (CCA) is an important task in natural language
processing that studies _how_ and _why_ scholars discuss each others’ work.
Despite decades of study, traditional frameworks for CCA have largely relied
on overly-simplistic assumptions of how authors cite, which ignore several
important phenomena. For instance, scholarly papers often contain rich
discussions of cited work that span multiple sentences and express multiple
intents concurrently. Yet, CCA is typically approached as a single-sentence,
single-label classification task, and thus existing datasets fail to capture
this interesting discourse. In our work, we address this research gap by
proposing a novel framework for CCA as a _document-level context extraction
and labeling_ task. We release MultiCite, a new dataset of 12,653 citation
contexts from over 1,200 computational linguistics papers. Not only is it the
largest collection of expert-annotated citation contexts to-date, MultiCite
contains multi-sentence, multi-label citation contexts within full paper
texts. Finally, we demonstrate how our dataset, while still usable for
training classic CCA models, also supports the development of new types of
models for CCA beyond fixed-width text classification. We release our code and
dataset at https://github.com/allenai/multicite.

##  1 Introduction

In scientific writing, citations are answers to questions. Citing authors
preemptively respond to questions such as— _why is this work needed_ , _where
has a fact been previously shown_ , or _what technique is used_ —in order to
construct and justify an argument about the correctness of the claims in their
work (Gilbert, 1977; Teufel, 2014). Together, these citations connect the
current paper to the broader discourse of science (e.g., Garfield, 1955;
Siddharthan and Teufel, 2007), help signal future impact and uses (e.g.,
McKeown et al., 2016), and, in downstream applications, can aid in summarizing
a work’s contributions (e.g., Qazvinian and Radev, 2008; Lauscher et al.,
2017). However, the function a particular citation serves—what questions it
answers—is often implicit and relies on the reader to understand that
citation’s purpose. Here, we introduce a new dataset and formalism of citation
function, showing that a citation can answer multiple questions
simultaneously, that the citation context answering these questions often
extends beyond a single sentence, and that recognizing a citation’s intent can
be reframed as a question-answering task, unifying its place within larger
reading comprehension tasks.

The importance and role of citations in understanding scholarly work has been
recognized across multiple disciplines, from sociology (e.g., Garfield et al.,
1964, 1970) to computer science (e.g., McKeown et al., 2016; Yasunaga et al.,
2019). Prior computational work has attempted analyze citations through
classifying aspects of what questions they answer, e.g., their function (e.g.,
Teufel et al., 2006; Jurgens et al., 2018), sentiment (e.g., Athar, 2011; Jha
et al., 2016), or centrality to a paper (Valenzuela et al., 2015). However,
these works have varied significantly in how they define a citation’s
contexts—i.e., the text relating to a citation necessary to understand its
intent—with most only examining a single sentence (e.g., Athar, 2011; Dong and
Schäfer, 2011; Ravi et al., 2018; Cohan et al., 2019) and some leaving the
context notion undefined (Jurgens et al., 2018; Vyas et al., 2020). Further,
few have recognized that often a citation’s purpose is not singular, that the
citation may be simultaneously answering multiple questions. We revisit these
assumptions to show the complexity of citations’ discourse functions.

#### Contributions.

Our contributions are three-fold: (i) We first demonstrate the existence of
the described phenomena and discuss their importance in a qualitative analysis
of NLP publications. Building upon the identified issues, we then propose a
new framework for citation context analysis (CCA). (ii) In order to allow for
training models under our proposed framework, we next present our new Multi-
Sentence Multi-Intent Citation (MultiCite) corpus, a carefully-annotated
collection of full-text publications with citation context and citation intent
information. MultiCite substantially advances our ability to perform CCA by
being the only dataset which captures multi-sentence and multi-intent citation
contexts. (iii) Finally, we establish a range of computational baselines on
our new data set under two different task formulations: We start with intent
classification as the most traditional setup and demonstrate the importance of
using gold citation contexts. Then, we reframe understanding citation function
as a form of question answering: Given a pair of papers, what is the reasoning
behind the citation? We hope that our work fuels and inspires more research on
accurate CCA models, and enables computational linguists to study more complex
and overlooked phenomena of citation usage.

##  2 Related Work

Author/ Year | Concept | Size | Context notion? | Multi-label?
---|---|---|---|---
Pride and Knoth (2020) | Purpose & Influence | 11,233 | Single sentence | ✗
Vyas et al. (2020) | Sentiment | Reanno. | No context annotated | ✗
Tuarob et al. (2019) | Function (Algorithm) | 8,796 | 3 Sentences | ✗
Cohan et al. (2019) | Intent | 9,159+1,861 | Single sentence | ✗
Ravi et al. (2018) | Sentiment | 8,925 | Single sentence | ✗
Jurgens et al. (2018) | Function | 1,969 | No context annotated | ✗
Jha et al. (2016) | Purpose & Polarity | 3,271 | Flexible within 4 sentences | ✗
Valenzuela et al. (2015) | Meaningfulness | 465 | ? | ✗
Abu-Jbara et al. (2013) | Purpose & Polarity | 3,271 | Flexible within 4 sentences | ✗
Li et al. (2013) | Function | ? | ? | ?
Jochim and Schütze (2012) | Citation Facets | 2,008 | No context annotated | ✗
Athar and Teufel (2012) | Sentiment | 1,741 | Flexible within 4 sentences | 1 label/sent.
Dong and Schäfer (2011) | Function & Polarity | ? | Single sentence | ✗
Athar (2011) | Sentiment | 8,736 | Single sentence | ✗
Teufel et al. (2006) | Function | 548 | ? | ✗
MultiCite (this work) | Function | 12,653 | Flexible | ✓
Table 1: Existing citation context analysis data sets with their properties in
comparison to this work.

We describe traditional works on CCA with respect to task forms and published
resources. An overview of these works is provided in Table 1.111For detailed
reviews of CCA, we refer to (Iqbal et al., 2020) and (Hernández-Alvarez and
Gomez, 2016).

The problem of citation analysis dates back to the seminal work of Teufel et
al. (2006), who addressed the task as a multi-class classification problem
without a clear notion of the citation context. Building upon this,
researchers initially operated on a single-sentence citation context (e.g.,
Athar, 2011; Dong and Schäfer, 2011), but later acknowledged the importance of
precise and flexible context windows (e.g., Athar and Teufel, 2012; Abu-Jbara
et al., 2013). Most recently, in-line with the advent of deep neural model
architectures, research efforts focused on increasing the size of the
published resources (Cohan et al., 2019; Tuarob et al., 2019; Pride and Knoth,
2020). However, larger data sets came at the expense of annotating precise
context notions, which naturally leads to increased complexity of the
annotation task. In §3, we demonstrate the importance of flexible contexts and
further show that contexts expressing multiple intents exist. None of the
preceding works provides a multi-label data set: though Athar and Teufel
(2012) allow for multiple labels around a citation marker, they assign exactly
one label to each sentence within the context. In contrast, we present a new
framework for CCA together with the largest, flexible-context, and only multi-
label annotated resource for CCA to-date.

##  3 Multi-Sentence Multi-Intent Framework for Citation Analysis

We recap the framework under which preceding works conducting citation context
analysis (CCA) have been operating. Based on this, we analyze its shortcomings
and propose to move to our multi-sentence multi-intent framework.

#### Traditional Citation Analysis Framework.

As outlined before, preceding works have mostly been focusing on single-
sentence single-intent citation context classification (e.g., Athar, 2011;
Jurgens et al., 2018; Cohan et al., 2019; Pride and Knoth, 2020, _inter
alia_). Accordingly, these works can be subsumed under a single notion, which
we refer to as the _traditional framework_ of CCA:

Let $B$ be a target paper cited by a citing paper $A=[s_{0},...,s_{N}]$, which
corresponds to a sequence of $N$ sentences $s_{i}$, and let $c$ be the
corresponding citation marker (a single token, e.g., _[1]_), indicating the
citation of $B$ in $A$. The citation context corresponds to a single sentence
$s_{c}$ with $c\in s_{c}$, often referred to as _the citance_ (Nakov et al.,
2004). Then, given a set of labels $L=\\{l^{(j)}\\}^{M}_{j=1}$ of size $M$,
the output is a single label $l^{(j)}$ assigned to $s_{c}$.

#### Overseen Phenomena.

We claim that the traditional framework does not account for two important
phenomena: multi-sentence contexts, and multi-label intents. We will support
our claim by providing examples, which we discuss under a citation labeling
scheme inspired by Jurgens et al. (2018), depicted in Table 2.222We later
adopt this labeling scheme for the purpose of creating our data set.

_Multi-Sentence._ First, consider the following citance $s_{c}$ with the
underlined citation marker $c$:

_“(Gliozzo et al., 2005) succeeded eliminating this requirement by using the
category name alone as the initial keyword, yet obtaining superior performance
within the keywordbased approach.”_

This citance alone provides background information about a previous approach.
Consequently, one would label it as _Background_. However, in the subsequent
sentence, we can read the following:

_“The goal of our research is to further improve the scheme of text
categorization from category name, which was hardly explored in prior work.”_

Only in this sentence, the underlying intent of the authors of $A$ is exposed:
the cited publication $B$ is used as a _Motivation_ for the presented
research. Under the traditional framework, we fail to correctly classify this
example.333We acknowledge that positional features could help. In this work,
however, we focus on textual semantics only.

_Multi-Intent._ Next, consider the following citance $s_{c}$ with the
underlined citation marker $c$:

_“In our experiments we use the same definition of structural locality as was
proposed for the ISBN dependency parser in(Titov and Henderson, 2007b).”_

Under the citation labeling scheme inspired by Jurgens et al. (2018), depicted
in Table 2, this sentence can be labeled as _Similarities_. However, another
possibility is to label this citance as _Uses_ , as the authors are adopting a
definition of the cited work. In some of the preceding works, ambiguous
citances were reportedly removed (e.g., Cohan et al., 2019), leading to an
artificial simplification of the task.

_Multi-Sentence Multi-Intent._ While we noticed that the two phenomena
outlined above exist in isolation, we also observe instances which combine
both:

_“Results Table 1 compares the published BERT BASE results from Devlin et al.
(2019) to our reimplementation with either static or dynamic masking. We find
that our reimplementation with static masking performs similar to the original
BERT model, and dynamic masking is comparable or slightly better than static
masking.”_

Here, the published results from the well-known BERT paper, a research
artefact, are _Used_ as a baseline ($s_{c}$). Then, the authors compare their
reimplementation as well as their extension to these scores (sentence 2),
resulting in expressed _Similarities_ as well as _Differences_.

#### Multi-Sentence Multi-Intent CCA.

As discussed, all the examples above exhibit phenomena, which can not be fully
expressed under the traditional framework. While we acknowledge that instances
falling under this category are less common, we argue that the community
should not longer ignore this “long tail” and address the phenomena
highlighted above. To this end, we propose a new multi-sentence multi-intent
framework for CCA:

Let $B$ be a target paper cited by a citing span of text
$T=[s_{0},...,s_{N}]$, which corresponds to a sequence of $N$ sentences
$s_{i}$ in a citing paper $A$. Given a set of labels
$L=\\{l^{(j)}\\}^{M}_{j=1}$ of size $M$, the output consists of _all_
(dis)continuous sequences $E=[s_{e0},...,s_{eO}]$ of size $O$, with $1<=O<=N$
(the set of elements on $E$ build a subset of the set of elements in $T$),
which contain _all_ evidentiary sentences $s_{e}$ for at most $k<=M$ citation
intents $l^{(j)}$ expressed in this context.

Under this notion, we can still instantiate the traditional framework and
approach CCA as a single-sentence multi-class classification task by
restricting $T$ to only contain $s_{c}$, i.e., the citance, for any citation
marker $c$ in $A$ and restricting $k$ to $k=1$, i.e., assigning a single label
$l^{(j)}$ only. However, aiming towards a more holistic picture, we are
longing to set $T=A$, thereby feeding the whole full text of a paper $A$.
Similarly, we want to set $k=M$, thereby allowing for full multi-intent
classification. By setting $T=A$ and $k=M$ we cast the task as full-blown
multi-sentence multi-intent CCA, returning all underlying intents of $B$ being
cited by $A$ with all evidentiary contexts.

##  4 MultiCite: A New Corpus for Citation Analysis

Intent |  Description
---|---
_Background_ |  The target paper provides relevant information for this domain.
_Motivation_ |  The target paper provides motivation for the source paper. For instance, it illustrates the need for data, goals, methods etc.
_Uses_ |  The source paper uses an idea, method, tool, etc. of the target paper.
_Extends_ |  The source paper extends an idea, method, tool, etc. of the target paper.
_Similarities_ |  The source paper expresses similarities towards the target paper. Either similarities between the source and the target paper or similarities between another publication and the target paper.
_Differences_ |  The source paper expresses differences towards the target paper. Either differences between the source and the target paper or differences between another publication and the target paper.
_Future Work_ |  The target paper is a potential avenue for future research. Often corresponds to hedging or speculative language about work not yet performed.
Table 2: Our citation intent labeling scheme based on Jurgens et al. (2018).

As it can be seen from the discussion of the related work, to-date there
exists no single data set which allows for training models under our proposed
framework. We close this research gap and present MultiCite, the first multi-
sentence multi-intent corpus for citation analysis consisting of $1,200$
computational linguistics publications.

###  4.1 Annotation Study

We describe the creation of MultiCite.

#### Sampling Procedure.

We procure an initial corpus of candidate papers from S2ORC Lo et al. (2020),
a large collection of open-access full-text papers annotated with inline
citation mentions resolved to their cited papers, and filtering to 50K papers
from the ACL Anthology or from arXiv with a cs.CL category.

To capture interesting cases, which, ideally, exhibit the phenomena we are
targeting with our research, we initially experimented with several paper
sampling strategies and found the following strategy to yield a variety of
interesting publication pairs: we compute the number of paragraphs in which
the target marker appears divided by the total number of paragraphs and
retrieve the top $k$ papers. This way, we capture publication pairs, where the
target paper is cited many times and therefore, in many different ways. We
hypothesize that these papers play a central role in the citing paper.

#### Mention Annotation.

To guide our annotators in finding passages in the citing paper A which are
talking about the cited publication B, we highlight mentions of B in A. We
automate this by highlighting all markers given in the respective S2ORC file.
To compute the reliability of this method, we let annotators manually identify
all references to B including citation markers, scientific entity names, and
other co-references such as _“The authors …”_ in a small sample of 262
publication pairs. We then compute the agreement with the gamma tool Mathet et
al. (2015) and obtain a mean score of 0.60 gamma macro averaged over the
publications. We therefore explicitly instruct our annotators to use the
highlighting as a rough guidance but to manually check for other mentions and
co-references.

Figure 1: The interface of our dedicated annotation platform: on the left hand
side, the annotator can browse through their assigned publications; in the
center, each sentence (choosable via checkboxes) of the citing publication is
displayes (mentions of the cited publication are highlighted in yellow); on
the right hand side, applying intents can be selected.

#### Annotation Task, Scheme, and Platform.

Our annotation task consists of two main steps: (1) Given a paper $A$ and a
paper $B$ (identified via some marker), identify all citing contexts. (2)
Given each individual citation context, assign a label to the context
answering the question why $A$ is mentioning $B$. Aiming to improve upon the
issues outlined before, we make sure that our guidelines include _explicit_ as
well as _implicit_ citations (proxied via a coreference, e.g., via pronoun or
name) into our task. Regarding the label scheme, we focus on intent
classification and start from the annotation scheme of Jurgens et al. (2018),
which we chose due to its relative simplicity compared to the one of Teufel et
al. (2006). We then iteratively adapted the guidelines according to the
discussions with the annotators. In the end, we only split the _Comparison or
Contrast_ class from the original scheme into two classes, _Similarities_ and
_Differences_. The complete scheme is depicted in Table 2.

To facilitate the process as far as possible, we developed a dedicated
annotation platform. A screenshot of the interface is provided in Figure 1.

#### Annotation Process.

For our annotation study, we hired nine graduate students in NLP recruited via
Upwork. With each of them, we conduct an hour of one-on-one training. The
annotators independently completed an hour of annotations, which were manually
reviewed and used for a second hour of one-on-one training focused on feedback
and correcting common mistakes. Annotators were then allowed to work
independently on batches of 20 papers at a time with manual annotation review
after each batch for quality control. Annotators were encouraged to indicate
“Unsure” for citation contexts with ambiguous labels, and leave comments
describing their thoughts. For these cases, two of the nine students were
recruited to do a subsequent adjudication round resolving to one or more
existing labels, if possible, else leaving as “Unsure” if unable to come to
consensus. Annotators were paid between $25-35 USD per hour, based on their
indicated offer on Upwork.

#### Inter-Annotator Agreement

Producing a single measure of inter-annotator agreement (IAA) is difficult for
data collected in this manner. Annotators might agree on the same intents but
disagree on which sentences belong in the citation context, vice versa, or
disagree on both fronts. While some prior work has developed IAA measures that
capture both context selection and labeling, e.g., $\gamma$ by Mathet et al.
(2015), such methods aren’t widely adopted in NLP and thus resulting IAA
values can be difficult to interpret. We opt instead to report two measures:
(a) mean accuracy of humans predicting the intent labels given gold contexts,
and (b) mean F1 score of humans predicting the context sentences given gold
intents.

For (a), we sample a set of 54 gold intent-context pairs across 5 papers. For
each example, two annotators who haven’t seen these papers previously were
shown the gold context and asked to select all possible intents from 8
categories (including an “Unsure” option). Mean accuracy is 0.76 when counting
_any_ gold label match as correct, and 0.70 when only counting cases when
_all_ predicted labels match the gold annotations as correct.

For (b), we sample 120 single gold intent-context pairs, each from a different
annotated paper. For each example, two annotators who haven’t seen these
papers previously were shown the gold intent and asked to select the context
sentences from among a randomly chosen window of 20 sentences encapsulating
the gold context. Mean sentence-level F1 scores are 0.64, 0.63 and 0.65,
respectively for gold contexts of length 1, 2 or 3+ sentences.

###  4.2 Corpus Analysis

(a) Context length distribution.

(b) Intent distribution.

(c) Context length per intent.

(d) PMI between intents.

Figure 2: Results of the quantitative corpus analysis: (a) distribution of
context lengths (log scale), (b) distribution of intents, (c) distribution of
context lengths per intent class, (d) pointwise mutual information (PMI)
between intent classes. We use the following intent abbreviations:
_Background_ (bg), _Motivation_ (mot), _Uses_ (use), _Similarities_ (sim),
_Extends_ (ext), _Differences_ (diff), _Future Work_ (fw).

Our new MultiCite consists of 1,193 publications, which are annotated with in
total 12,653 citation contexts representing 4,555 intents per publication. To
obtain a deeper insight into the nature of our annotations, we show the
results of a more extensive quantative analysis in Figure 2.

Unsurprisingly, the context length distribution (Figure 2(a)) indicates that
most annotations consist of single-sentence contexts only. However, a
substantial number of contexts went beyond the single sentence. Indeed, we
were able to capture contexts consisting of up to 14 sentences, which
highlights that artificially restricting the context length will lead to loss
of potentially relevant information.

The intent distribution (Figure 2(b)) is highly skewed. This is in-line with
results reported in previous work (Jurgens et al., 2018): the most dominant
class is _Background (bg)_ , followed by _Uses (use)_ , while _Future work
(fw)_ occurs least often.

The distribution of context lengths per intent class (Figure 2(c)) reveals
that each of the intents can be expressed with a one to three sentences only.
However, for instance for _Motivation (mot)_ , contexts clearly exhibit a
variety in length. We hypothesize that this intent is of higher complexity and
therefore often requires more context.

Finally, we compare the label co-occurrences. To this end, we compute
pointwise mutual information (PMI) between the intents (Figure 2(d)). The
highest PMI is observed between _Extends (ext)_ and _Differences (dif)_ ,
which points to the strong interrelation between those two classes: when
authors introduce an extension they made to a research artefact from preceding
work, they often express at the same time, that exactly this aspect makes the
work different from what was previously presented.

##  5 Importance of Multi-Sentence Contexts

In order to assess the importance of the correct multi-sentence citation
context for an intent classification, we conduct a series of experiments
aligned with previous works (e.g., Jha et al., 2016), in which we feed various
amounts of context sentences to a multi-label classification model.

###  5.1 Experimental Setup

We describe the experimental setup for our multi-label classification
experiments.

#### Standard Split.

For all our experiment, we use the same standard split on the publication-
level of MultiCite, to not leak any information from the articles. Concretely,
$30\%$ of the publications are reserved for testing, $70\%$ for training, from
which, in turn, $30\%$ are reserved for model validation. This results in
5,491 training instances, 2,447 development instances, and 3,313 test
instances.

#### Models and Baselines.

As our aim is not to beat previous performances on other benchmarks, but to
assess the difficulty of our data set, understand the importance of our gold
citation context annotations, and to provide strong baselines for future
research, we resort to the most natural model and baseline choices: (1) we
report a majority vote baseline, in which we simply predict the majority
label. (2) As domain-specific transformer (Vaswani et al., 2017)-based model,
we employ a SciBERT (Beltagy et al., 2019) encoder, on top of which we place a
multi-label classification head consisting of a set of sigmoid classifiers
(one for each of the classes) to which we feed the sequence start token. (3)
Last, we replace the encoder with a RoBERTa Model (Liu et al., 2019), a more
rebustly trained version of BERT (Devlin et al., 2019).

#### Input Preparation.

For the two transformer-based models, we prepare the input as follows: we
first highlight the target citation marker by inserting an opening and closing
cite tag around the citation, e.g., <cite>Author (Year)</cite>. We next sample
$s$ consecutive sentences around the target citation444The position of the
citing sentence within the context varies, to not bias the models towards the
mid of the citation context. as citation context. For instance, for $s=1$,
only the citing sentence is sampled and for $s=2$, the citing sentence and the
preceding or following sentence is extracted. Alternatively, we employ the
annotated gold context. We then follow the standard input procedure for each
model, i.e., we apply WordPiece (Johnson et al., 2017) tokenization for
SciBERT and byte-level BPE (Radford et al., 2019) for RoBERTa, and add the
corresponding special tokens for each model.

#### Training and Optimization.

SciBERT is only available in base configuration ($12$ layers, $12$ attention
heads, $768$ as hidden size, cased vocabulary with size $31,116$). For
RoBERTa, we employ the large version ($24$ layers, $16$ attention heads,
hidden size $1024$, cased vocabulary with size $50,265$). We conduct standard
fine-tuning of the models with a mean over $N$ binary cross-entropy losses:

$\scriptsize{L=\frac{1}{N}\sum_{n=1}^{N}-[y_{n}\cdot\log\hat{y}_{n}+(1-y_{n})\cdot\log(1-\hat{y}_{n})]\,,}$
(1)

with $N$ as the number of classes, $\hat{y}_{n}\in\mathbb{R}$ as the sigmoid
activated output for class $n$, and $y_{n}$ as the true label. We optimize
using Adam (Kingma and Ba, 2015). To select the best hyperparameters we grid
search for the best learning rate $\lambda\in\\{1\cdot 10^{-5},2\cdot
10^{-5}\\}$ and number of epochs $e\in[1,9]$ based on the development set
performance. The effective batch size is fixed to $32$ and the sigmoid
prediction threshold to $0.5$.

#### Evaluation Measures.

We compute two types of accuracies: a _strict_ version, in which a prediction
is correct iff all predicted labels match exactly the gold annotation; and a
_weak_ version, in which a prediction is correct if at least one of the
predicted labels matches the gold classes. The weak measure reflects an upper
bound on performance (i.e., whether the model can detect any of the correct
intents) and allows us to compare our multilabel models with existing single-
label models. Additionally, we break down the performance in different
categories according to the gold context size.

###  5.2 Results

|  | size = 1 | size = 2 | size = 3 | size = 4 | all
---|---|---|---|---|---|---
|  | support = 2795 | support = 335 | support = 112 | support = 39 | support = 3313
| input size |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict
SC Majority | – |  0.39 |  0.37 |  0.31 |  0.16 |  0.36 |  0.19 |  0.41 |  0.13 |  0.39 |  0.34
SC Oracle | – |  1.00 |  0.89 |  1.00 |  0.77 |  1.00 |  0.74 |  1.00 |  0.56 |  1.00 |  0.87
SC ACL-ARC | – |  0.68 |  0.60 |  0.54 |  0.41 |  0.52 |  0.34 |  0.51 |  0.15 |  0.66 |  0.56
SciBERT | 1 |  0.78 |  0.69 |  0.45 |  0.28 |  0.47 |  0.24 |  0.51 |  0.18 |  0.74 |  0.62
3 |  0.74 |  0.64 |  0.59 |  0.39 |  0.54 |  0.29 |  0.62 |  0.23 |  0.72 |  0.60
5 |  0.71 |  0.61 |  0.50 |  0.33 |  0.46 |  0.27 |  0.54 |  0.18 |  0.68 |  0.57
7 |  0.62 |  0.54 |  0.43 |  0.28 |  0.48 |  0.27 |  0.51 |  0.15 |  0.60 |  0.50
9 |  0.56 |  0.50 |  0.37 |  0.25 |  0.37 |  0.21 |  0.56 |  0.18 |  0.53 |  0.46
gold |  0.80 |  0.70 |  0.68 |  0.46 |  0.66 |  0.39 |  0.64 |  0.26 |  0.78 |  0.66
RoBERTa | 1 |  0.80 |  0.69 |  0.46 |  0.29 |  0.46 |  0.25 |  0.56 |  0.18 |  0.75 |  0.63
3 |  0.78 |  0.66 |  0.59 |  0.41 |  0.50 |  0.27 |  0.62 |  0.18 |  0.75 |  0.61
5 |  0.75 |  0.63 |  0.54 |  0.39 |  0.54 |  0.32 |  0.59 |  0.21 |  0.72 |  0.59
7 |  0.73 |  0.62 |  0.53 |  0.37 |  0.44 |  0.24 |  0.56 |  0.21 |  0.70 |  0.58
9 |  0.71 |  0.59 |  0.54 |  0.36 |  0.46 |  0.26 |  0.54 |  0.15 |  0.68 |  0.55
gold |  0.81 |  0.69 |  0.70 |  0.50 |  0.67 |  0.45 |  0.59 |  0.28 |  0.79 |  0.66
Table 3: Results of the multi-label citation context classification with
SciBERT and RoBERTa for different input context sizes (input) across all test
instances (all) and spread out for different gold context sizes with their
respective support in the test set. We report _strict_ and _weak_ accuracies
for gold context sizes with number of supporting instances > $30$. Scores in
bold highlight the best performances per model per column. Note that the
results for SC ACL-ARC are computed with the reduced label set.

The results are shown in Table 3. We report the results for context sizes with
more than $30$ instances in the test set only.

Across all test instances (all) the best weak and strict accuracies are
achieved when feeding the gold context. When employing the citing sentence
only, the results drop $4$ percentage points for SciBERT and $3$ (strict) to
$4$ (weak) percentage points for RoBERTA. The scores decrease even more, the
more sentences are serving as input to the model training with drops up to
$25$ percentage points (strict accuracy for SciBERT trained on 9-sentence
contexts). When categorizing the prediction instances according to their gold
context size, more interesting patterns emerge: for contexts consisting of the
citing sentence only, the delta between the model trained on gold context and
the model trained on the citing sentence is naturally smaller, but
surprisingly, they seem to be still existent ($2$ percentage points for weak,
$1$ percentage point for strict accuracy with SciBERT; $1$ percentage point
for weak accurcy with RoBERTa). We hypothesize that this indicates that the
models trained on the gold contexts are able to learn more about the intent
classes than the single-sentence models. This, again, highlights the
importance of considering precisely-sized contexts in training, even if the
majority of the prediction instances are single-sentence instances only. For
gold context sizes $3$ and $4$, the gap between the models trained on gold
contexts and the ones trained on the citing sentence increases even more, but
we can also see that models trained on $3$ and $4$ instances are better able
to capture true intents than the $1$-sentence models.

##  6 Citation Analysis as Question Answering

Consider the following application scenario: a user might want to know why a
paper cites another paper or for which reasons a paper is referenced. In both
cases, they expect a paper-level evaluation (i.e., the model should retrieve a
set of intents), and, at the same time, they would want to see evidence
supporting the results (the model should retrieve a gold context for each
intent). We propose to fulfill these two desiderata by resorting to a question
answering (QA) formulation as another instance of our Multi-sentence Multi-
intent framework. The advantages of this approach are (a) its flexibility, as,
theoretically, users can any questions, and (b) its compatibility with general
attempts on scientific QA (e.g., Dasigi et al., 2021). Further, this QA
reframing allows CCA to be solved using QA models, treating CCA as a
challenging case of natural language understanding in the scientific domain.

###  6.1 Task Formulation and Methodology

We adapt the Qasper (e.g., Dasigi et al., 2021) task formulation and model for
our purposes. Qasper is a document-grounded QA model that has been pre-trained
to answer a variety of questions on scientific texts, e.g., extractive QA or
yes/no questions, making it ideal as an initial model to adapt for CCA.

#### Task.

The task is formulated as follows: given a paper pair ($A$ and $B$, as before)
and a question related to the intent of $A$ citing $B$, output the answer
based on the full text of $A$ as well as an evidence for the given answer,
i.e., a single gold context $E=[s_{e0},...,s_{eO}]$ as a (dis)continuous
sequence of sentences $s_{i}$ from $A$. Concretely, we ask binary _yes/no_
-questions for each of the intents, e.g., for the _background_ intent, we ask
_“Does the paper cite … for background information?”_ (dots are replaced with
the respective citation marker).

#### Model.

We use the Qasper evidence scaffold model: this model is a multi-task
sequence-to-sequence Longformer-Encoder-Decoder (LED; Beltagy et al., 2020)
model. LED is a variant of the Transformer (Vaswani et al., 2017) encoder-
decoder model which supports processing long inputs, e.g., scientific papers,
due to a sparse attention mechanism which scales linearly with the input
sequence length. The original LED parameters are initialized from BART (Lewis
et al., 2020). As model input, the question and paper context are concatenated
to a single string. We prepend each sentence in the context with a </s> token
and globally attend over all question tokens and </s> tokens. The model
objective consists of two parts: a generative answering component, and an
evidence extraction component. (1) The generative answering component is a
classifier over the model’s vocabulary trained with a cross-entropy loss to
generate _“Yes”_ or _“No”_.555Using the generative component as in the
original work allows us to (a) keep the flexibility of generating other
answers and (b) reuse Qasper’s head weights if desired. (2) The evidence
extraction component is a classifier over the </s> tokens, trained with a
binary cross-entropy loss for evidence/non-evidence sentences. As the class of
positive, i.e., evidentiary, sentences is underrepresented in the data set, we
follow the original work and scale the loss proportional to the ratio of
positive/negative gold sentences per batch. The total loss corresponds to the
sum of the two task losses.

###  6.2 Experimental Setup

We describe the experimental setup of our QA experiments.

#### Data.

We use the same standard split as in our classification experiments. However,
we preprocess the data according the model’s required input format, i.e., we
create for each paper-pair seven questions (one for each of our seven
intents). For all positive intents, we create a _“Yes”_ -answer and provide
the first gold context as evidence. For the negative intents, we create a
_“No”_ -answer without evidence. This way, we end up with 4,074 training,
1,764 development, and 2,499 test questions.

#### Model Configuration and Optimization.

We employ two LED models: first, we start from the original LED base model (12
attention heads and 6 layers in encoder and decoder, respectively; 768 as
hidden size; 50,265 as vocabulary size, maximum input length 16,384 tokens).
Secondly, to estimate the complementarity of the knowledge needed for CCA QA
with more general scientific QA as in Qasper, we start from the LED base model
trained on the Qasper data set shared by the authors of the paper (encoder
only). For all models, we use the code from Dasigi et al. (2021). We train all
models for maximum $5$ epochs with early stopping based on the validation set
performance (span-level Answer-F1 as described below, patience of 2 epochs)
and grid search over the following hyperparameters: batch size
$b\in\\{2,4,8,16\\}$ and initial learning rate $\lambda\in\\{3\cdot
10^{-5},5\cdot 10^{-5}\\}$. We optimize all models with Adam (Kingma and Ba,
2015).

#### Evaluation Measures and Baselines.

For evaluating answer performance, we follow Dasigi et al. (2021) and report a
binary F1 measure micro averaged across the papers. Similarly, to evaluate the
evidence extraction performance, we compute an evidence F1 where we compare
the gold context for an answer with the per-sentence predictions of the model.
Additionally, as for all _“No”_ -answers, i.e., for all citation intents which
do not apply, also no evidence is given, we compute an F1 only considering the
positive intents, dubbed _Evidence F1 w/o No_. Similarly, we compute an
additional evidence score, which we condition on the correct model answer
predictions only (_Evidence F1 correct_). To estimate the difficulty of the
task, we report a two simple majority vote baselines. In the first variant,
dubbed _Majority SL_ , we only predict _“yes”_ for the majority class,
_Background_. In the second variant, to which we refer to as _Majority ML_ ,
we compute for each question type each of which relates to a single intent,
the majority answer (_“yes”_ or _“no”_).

###  6.3 Results

|  A F1 |  E F1 |  E F1 w/o No |  E F1 correct
---|---|---|---|---
Majority SL |  0.61 |  0.48 |  0.00 |  0.47
Majority ML |  0.72 |  0.48 |  0.00 |  0.38
LED Qasper |  0.75 |  0.48 |  0.00 |  0.35
LED Base |  0.78 |  0.08 |  0.04 |  0.07
Table 4: Citation Context Analysis QA results. We report Answer-F1 (A F1),
Evidence-F1 (E F1), as well as its variants Evidence-F1 w/o No (E F1 w/o No)
and Evidence-F1 for correct answers only (E F1 correct) .

The results are shown in Table 4. Both LED models surpass the baselines in
terms of Answer F1. Surprisingly, the knowledge from the Qasper LED encoder
does not seem to lead to strong performance gains. Relating to the Evidence F1
scores, interesting patterns emerge: overall the scores from LED Qasper are
comparable to the ones reported in the original work for general scientific
QA. However, when negative answers are not considered in the LED Qasper
model’s performance (excluding contexts which no answer should be given), the
resulting score shows that the model does not properly extract context
sentences. The LED Base model, in contrast, learns to extract context, though
the scores are very low. We attribute this observation to the amount of
context given. Overall, the results indicate the difficulty of the task under
this natural formulation and we propose that our initial experiments open up
an interesting and challenging research avenue for QA in the context of CCA.

##  7 Conclusion

In this work, we have presented a new Multi-Sentence Multi-Intent CCA
framework. In a qualitative analysis of citation contexts, we demonstrated the
importance of considering multiple sentences and multiple labels for citation
intent classification. In lack of a data set which allows to operate under our
new framework, we presented MultiCite, an annotated corpus consisting of
12,653 citations across 1,193 fully-annotated NLP publications. Next, we
employed MultiCite in traditional classification experiments, which showed the
importance of using our annotated gold contexts as model inputs. Finally, we
proposed to cast citation context analysis as question answering task, which
allows for more flexibility and end-to-end modeling of the problem. This way,
the task integrates with other reading comprehension tasks on scientific
publications. We hope that our work draws more attention to research on
previously overlooked phenomena in citation context analysis to support more
accurate research evaluation studies.

## Acknowledgements

We would like to thank Sam Skjonsberg and Mark Neumann for their help on the
annotation interface. We would also like to thank Noah Smith for helping us
connect with collaborators on this project.

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|  | size = 1 | size = 2 | size = 3 | size = 4 | size = 5 | size = 6 | size = 7 | size = 8 | size = 10 | size = 14 | all
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | supp. = 2795 | supp. = 335 | supp. = 112 | supp. = 39 | supp. = 17 | supp. = 7 | supp. = 3 | supp. = 2 | supp. = 2 | supp. = 1 | supp. = 3313
| Input |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict |  weak |  strict
S | 1 |  0.78 |  0.69 |  0.45 |  0.28 |  0.47 |  0.24 |  0.51 |  0.18 |  0.59 |  0.29 |  0.86 |  0.57 |  1.00 |  0.67 |  1.00 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.74 |  0.62
| 3 |  0.74 |  0.64 |  0.59 |  0.39 |  0.54 |  0.29 |  0.62 |  0.23 |  0.65 |  0.35 |  0.86 |  0.57 |  1.00 |  0.67 |  1.00 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.72 |  0.60
| 5 |  0.71 |  0.61 |  0.50 |  0.33 |  0.46 |  0.27 |  0.54 |  0.18 |  0.53 |  0.24 |  0.86 |  0.57 |  1.00 |  0.67 |  1.00 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.68 |  0.57
| 7 |  0.62 |  0.54 |  0.43 |  0.28 |  0.48 |  0.27 |  0.51 |  0.15 |  0.53 |  0.24 |  0.57 |  0.43 |  1.00 |  0.67 |  1.00 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.60 |  0.50
| 9 |  0.56 |  0.50 |  0.37 |  0.25 |  0.37 |  0.21 |  0.56 |  0.18 |  0.53 |  0.29 |  0.71 |  0.57 |  1.00 |  0.67 |  0.50 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.53 |  0.46
| gold |  0.80 |  0.70 |  0.68 |  0.46 |  0.66 |  0.39 |  0.64 |  0.26 |  0.65 |  0.24 |  0.86 |  0.86 |  1.00 |  0.67 |  1.00 |  1.00 |  0.50 |  0.00 |  1.00 |  0.00 |  0.78 |  0.66
R | 1 |  0.80 |  0.69 |  0.46 |  0.29 |  0.46 |  0.25 |  0.56 |  0.18 |  0.65 |  0.35 |  0.86 |  0.57 |  1.00 |  0.67 |  0.50 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.75 |  0.63
| 3 |  0.78 |  0.66 |  0.59 |  0.41 |  0.50 |  0.27 |  0.62 |  0.18 |  0.76 |  0.41 |  0.86 |  0.71 |  1.00 |  0.67 |  1.00 |  1.00 |  0.50 |  0.00 |  0.00 |  0.00 |  0.75 |  0.61
| 5 |  0.75 |  0.63 |  0.54 |  0.39 |  0.54 |  0.32 |  0.59 |  0.21 |  0.65 |  0.41 |  0.71 |  0.57 |  1.00 |  0.67 |  1.00 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.72 |  0.59
| 7 |  0.73 |  0.62 |  0.53 |  0.37 |  0.44 |  0.24 |  0.56 |  0.21 |  0.53 |  0.35 |  0.71 |  0.57 |  1.00 |  0.67 |  0.50 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.70 |  0.58
| 9 |  0.71 |  0.59 |  0.54 |  0.36 |  0.46 |  0.26 |  0.54 |  0.15 |  0.65 |  0.35 |  0.86 |  0.86 |  1.00 |  0.67 |  0.50 |  0.50 |  0.50 |  0.00 |  0.00 |  0.00 |  0.68 |  0.55
| gold |  0.81 |  0.69 |  0.70 |  0.50 |  0.67 |  0.45 |  0.59 |  0.28 |  0.71 |  0.35 |  0.86 |  0.86 |  0.67 |  0.33 |  1.00 |  1.00 |  1.00 |  0.50 |  0.00 |  0.00 |  0.79 |  0.66
Table 5: Full results of the multi-label citation context classification with
SciBERT (S) and RoBERTa (R) for different input context sizes (input) across
all test instances (all) and spread out for different gold context sizes with
their respective support in the test set. We report _strict_ and _weak_
accuracies. Scores in bold highlight the best performances per column.

##  Appendix A Full Results for Section 5

We provide the full experimental results for the classification experiments
described in Section 5 in Table 5.