Clifford structures, bilegendrian surfaces, and extrinsic curvature

11th August 2023

Graham SmithDepartamento de Matemática, Pontifícia Universidade Católica do
Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil

Dedicated to François Labourie on the occasion of his 60th birthday

Abstract: We use Clifford algebras to construct a unified formalism for
studying constant extrinsic curvature immersed surfaces in riemannian and
semi-riemannian $3$-manifolds in terms of immersed bilegendrian surfaces in
their unitary bundles. As an application, we provide full classifications of
both complete and compact immersed bilegendrian surfaces in the unit tangent
bundle ${\text{\rm U}}\mathbb{S}^{3}$ of the $3$-sphere.

AMS Classification: 53A05, 12E15, 35J96

Key words : Clifford algebras, extrinsic curvature, holomorphic curves,
bilegendrian curves

1 - Introduction. 1.1 - Introduction.  The symplectic nature of riemannian
geometry is well-known but rarely emphasized. One theory, however, in which
the symplectic perspective has proven itself to be particularly fruitful is
that of immersed surfaces of constant extrinsic curvature in riemannian
$3$-manifolds. In this paper, we describe a framework for studying constant
extrinsic curvature surfaces in terms of bilegendrian surfaces in the unit
tangent bundle. We study the interplay between these two classes of surfaces
and a third, which we will call $J$-invariant surfaces. This interplay,
mediated by Clifford structures over the contact distribution of the total
space of the unit tangent bundle, provides a unified perspective on many known
results together with new insights into others. This paper is intended as a
survey, although some new results will also be proven.

1.2 - Discussion.  We define an immersed surface in a $3$-manifold $X$ to be a
pair $S:=(S,e)$, where $S$ is a surface and $e:S\rightarrow X$ is a smooth
immersion. When $X$ is riemannian, we define the extrinsic curvature of $S$ to
be equal to the determinant of its shape operator. For
$k\in\mathbb{R}\setminus\\{0\\}$, we define a CEC-$k$ surface in $X$ to be an
immersed surface in $X$ of constant extrinsic curvature equal to $k$.CEC-$0$
surfaces, which also exhibit fascinating geometric phenomena, will not be
addressed in this paper. When the curvature is not specified, we will simply
speak of CEC surfaces.

CEC surfaces can be viewed as solutions of Monge-Ampère type partial
differential equations, which may be elliptic or hyperbolic, according to
whether the curvature is positive or negative. The use of symplectic
structures over the total space of the tangent bundle $TX$ to study such
partial differential equations was pioneered by Lychagin–Rubtsov in $[32]$
(see also $[33]$ and $[37]$). However, it was in a series of papers - $[26]$,
$[27]$, $[28]$, $[29]$ \- that Labourie used contact structures over the total
space of the unit tangent bundle $UX$ of $X$ to apply these ideas to the study
of CEC surfaces. More precisely, given an immersed surface $(S,e)$, we denote
by $\hat{e}:=\nu_{e}$ its unit normal vector field, and we call the surface
$\hat{S}:=(S,\hat{e})$ its Gauss lift. Labourie showed that, for all positive
$k$, there exists an almost complex structure $J_{k}$ over the contact bundle
of $UX$ such that $(S,e)$ is a CEC-$k$ surface if and only if its Gauss lift
is $J_{k}$-pseudoholomorphic. He then used Gromov’s theory of
pseudoholomorphic curves to prove a simple, yet powerful, compactness result
for families of such surfaces. Labourie’s work has since revolutionized our
understanding of positively-curved CEC surfaces, and has found applications
across a variety of fields.For applications to hyperbolic geometry and
dynamical systems theory, we refer the reader to Labourie’s own work $[26]$
and $[29]$. For applications to general relativity and Teichmüller theory, we
refer the reader to $[1]$, $[3]$, $[4]$, $[49]$, and our survey $[13]$.
Recently, in $[44]$, $[46]$ and $[47]$, we showed how Labourie’s ideas yield a
complete classification of positively-curved CEC surfaces in $3$-dimensional
space-forms, subject to a natural completeness condition.

In $[46]$, we showed how Labourie’s almost complex structures can be seen as
components of quaternionic structures over the contact bundle of $UX$. We had
already adopted this point of view implicitely in $[42]$, where we used almost
Calabi-Yau structures to generalize Labourie’s ideas to hypersurfaces in
higher-dimensional manifolds (the reader may also consult the work $[19]$ of
Harvey–Lawson for an alternative perspective on these ideas). In the present
paper, by replacing quaternionic structures with Clifford structures, we
extend Labourie’s framework to include, not only negatively-curved CEC
surfaces, but also mixed-signature CEC surfaces in semi-riemannian
$3$-manifolds.

We will study CEC surfaces in terms of what we will call bilegendrian
surfaces, which build on the concept of bilagrangian surfaces introduced by
Hitchin in $[20]$ and $[21]$. We will also divide bilegendrian surfaces into
two classes, namely those of elliptic type, which generate elliptic partial
differential equations, and those of hyperbolic type, which generate
hyperbolic ones. Hitchin’s bilagrangian surfaces, for example, are of elliptic
type, as are Labourie’s $k$-surfaces. We will encounter noteworthy
bilegendrian surfaces of hyperbolic type presently.

It is interesting to see how our framework of bilegendrian surfaces unifies
known results for CEC surfaces. For example, when $X$ is a space-form, we
construct in Theorem 3.4.4 a canonical flat metric, which may be riemannian or
semi-riemannian, over the complement of a certain exceptional set of every
bilegendrian surface in $UX$. When the curvature is positive, this identifies
with the canonical flat structure obtained upon integrating the square root of
the Hopf differential, whilst, when the curvature is negative, it identifies
with the asymptotic Chebyshev net, used in Hilbert’s proof of the non-
existence of complete negatively-curved CEC surfaces in Euclidean $3$-space
$\mathbb{R}^{3}$ (see Sections 3.5, 3.6, and 3.7).

The theory of flat fronts is also encompassed within our framework. This
theory, developed by Kokubu–Umehara–Yamada in $[24]$, building on ideas
elegantly presented by Gálvez–Martínez–Milán in $[14]$, serves to analyse the
structure of singularities of CEC-$1$ surfaces in hyperbolic $3$-space
$\mathbb{H}^{3}$, and has since been studied by various authors in a variety
of settings (see, for example, $[15]$, $[25]$, and $[38]$). More recently, in
$[16]$, Gálvez–Mira introduced a second class of flat fronts in order to study
singularities of CEC-$(-1)$ surfaces in the $3$-sphere $\mathbb{S}^{3}$. In
fact, flat fronts of the first type form a special case of the surfaces
studied by Labourie (more precisely, they are $J_{1}$-holomorphic curves).
Flat fronts of the second type are not covered by Labourie’s theory, but
instead constitute an interesting example of bilegendrian surfaces of
hyperbolic type.

We conclude this paper by classifying complete and compact bilegendrian
surfaces in ${\text{\rm U}}\mathbb{S}^{3}$, which correspond to the flat
fronts studied in $[16]$. This generalizes the classifications of compact flat
surfaces in $\mathbb{S}^{3}$ given by Kitigawa in $[22]$ and by Weiner in
$[51]$. We underline, however, that this generalization is not automatic, as
the techniques of Kitigawa and Weiner, which rely on the geometry of the
projection onto the base space, break down at points where this projection
degenerates. Instead, we will show how the bilegendrian condition naturally
interacts with the Lie group structure of $\mathbb{S}^{3}$ to yield canonical
factorizations of the immersions and of their period lattices in the compact
case. Furthermore, we will see that these factorization results readily follow
from a general factorization criterion for Lie group-valued functions over
$\mathbb{R}^{2}$, making them, in a sense, more algebraic in nature than
geometric.

Finally, our bilegendrian framework also has interesting applications to the
semi-riemannian case. For example, CEC-$1$ surfaces of mixed signature in the
anti de-Sitter space ${\text{AdS}}^{2,1}$, studied by Dajczer–Nomizu in $[9]$
and León-Guzmán–Mira–Pastor in $[30]$, correspond to bilegendrian surfaces of
hyperbolic type. Likewise, negatively-curved CEC surfaces of mixed signature
in spacetimes yield bilegendrian surfaces of elliptic type, and should
therefore be amenable to study via Labourie’s techniques, with potentially
interesting applications. In fact, the elliptic nature of such surfaces was
already observed by T. K. Milnor in $[35]$, but we are not aware of this
property having been further examined by later authors.

1.3 - Happy 60th Birthday François!  It is now more than 20 years since I
first attended François’ course on Teichmüller theory in the Bâtiment 425 of
what was then known as l’Université Paris-Sud (Paris-XI, now Paris-Saclay).
His intuitive approach to geometry, which I later learned had been influenced
by Gromov, was for me both novel and inspiring. Over the three years I spent
as his PhD student, I learned many important things, including, amongst
others, how to correctly serve a glass of wineNever more than half-full..
François knew how to provide the right encouragement and the right advice at
those key moments during my doctoral research which to me seemed the darkest.
Above all, however, from that time on, and throughout the rest of my career,
the distance of his mathematical vision, and the profundity of his insights,
have been an inspiration, impelling me to strive, within the limits of my
ability, to achieve similar excellence in my own work. The subject of this
paper, as I hope François will recognize, is a tribute to that experience: a
new perspective on the original problem that he proposed to me when we first
met. With this paper, I wish François a very happy birthday, and many more
fruitful years of mathematical creation to come!

1.4 - Acknowledgements.  I am grateful to Pablo Mira, Carlos Tomei and Pierre
Pansu for helpful comments made to earlier drafts of this paper. I am also
grateful to Richard Wentworth and Jean-Marc Schlenker for having invited me to
contribute to the volume in which this paper should appear, and for assuring
me that the proposed topic would genuinely be of interest. Finally, of course,
I am grateful to François Labourie for having introduced me, all those years
ago, to CEC surfaces and the marvels and wonders they conceal.

2 - Clifford algebras. 2.1 - Overview.  Clifford algebras were introduced by
William Clifford in $[6]$ and $[7]$ with the aim of unifying Hamilton’s theory
of quaternions with that of Grassman algebras. In the present paper, they will
be of interest to us as a source of endomorphisms $J$ such that
$J^{2}=-\epsilon{\text{Id}}$, for some $\epsilon\in\\{\pm 1\\}$. We will call
any such $J$ a pseudo-involution, we will call $\epsilon$ its sign, and we
will say, furthermore, that $J$ is balanced whenever its trace vanishes.
Pseudo-involutions of positive sign are just complex structures. Balanced
pseudo-involutions of negative sign are paracomplex structures.

Given a pseudo-involution $J$ of a $4$-dimensional real vector space $E$, we
will be interested in the planes in $E$ that $J$ preserves. By mild abuse of
terminology, we will say that a plane $P$ is $J$-invariant whenever it is
preserved by $J$, but is not a real $J$-eigenspace. The interest in working
with Clifford structures is now precisely that they yield pseudo-involutions
whose invariant planes are bilegendrian and vice-versa. More precisely, given
two symplectic forms $\omega_{i}$ and $\omega_{k}$ over $E$, we say that a
plane $P$ in $E$ is $(\omega_{i},\omega_{k})$-bilagrangian whenever it is
lagrangian with respect to both $\omega_{i}$ and $\omega_{k}$. In Theorem
2.3.2, we will show that, for all the pseudo-involutions $J$ of interest to
us, there exists a pair $(\omega_{i},\omega_{k})$ of symplectic forms such
that any plane $P$ in $E$ is $J$-invariant if and only if it is
$(\omega_{i},\omega_{k})$-bilagrangian.

We conclude this section by studying the geometries of $J$-invariant planes.
In particular, we identify special vectors which exist for all but a certain
set of exceptional planes. These vectors correspond to eigenvectors of
symmetric matrices, and will be of use in studying bilegendrian surfaces in
the next section.

2.2 - Pseudo-isometries and pseudo-involutions.  Before introducing Clifford
structures, it is useful to study the properties of the objects of which they
are comprised, namely pseudo-involutions. Let $E$ be a $2d$-dimensional real
vector space. We will say that an endomorphism $M$ of $E$ is a pseudo-
involution whenever

$M^{2}=-\epsilon{\text{Id}}\ ,$

for some $\epsilon\in\\{\pm 1\\}$, which we call its sign. We will say that a
pseudo-involution $M$ is balanced whenever

${\text{Tr}}(M)=0\ .$

Pseudo-involutions of positive sign, which are always balanced, are just
complex structures. They are associated with elliptic partial differential
equations, and thus have widespread and well-known geometric applications
(see, for example, $[34]$). Balanced pseudo-involutions of negative sign are
also known as para-complex structures. They are associated with hyperbolic
partial differential equations, and have also attracted the interest of many
authors (see, for example, $[37]$, and the survey $[8]$).

The following construction of pseudo-involutions will be particularly useful.
Let ${{b}}$ be a non-degenerate, symmetric, bilinear form over $E$. We will
say that ${{b}}$ has positive sign whenever it has an even number of positive
directions, and we will say that it has negative sign otherwise. We say that
an endomorphism $M$ of $E$ is a pseudo-isometry of ${{b}}$ whenever

${{b}}(M\cdot,M\cdot)=\pm{{b}}(\cdot,\cdot)\ .$

We denote the Lie group of pseudo-isometries of ${{b}}$ by
$\hat{{\text{O}}}({{b}})$, and we denote its Lie algebra by
${\mathfrak{so}}({{b}})$. Note that every element $M$ of
${\mathfrak{so}}({{b}})\mathop{\cap}\hat{{\text{O}}}({{b}})$ satisfies

$M^{2}=-M^{*}M=\pm{\text{Id}}\ ,$

and is thus a pseudo-involution.

Lemma & Definition 2.2.1 If $M$ is a pseudo-involution of negative sign, then
$M$ is diagonalizable with eigenvalues $\\{\pm 1\\}$. In particular, $M$ is
balanced if and only if its two eigenspaces have the same dimension.

Proof: Indeed, if $S+N:=M$ denotes the Jordan decomposition of $M$, then

${\text{Id}}=\pm M^{2}=\pm S^{2}\pm(2SN+N^{2})\
.\eqnum{\text{{\rm(2.1)}}\hbox{}}$

Trivially $S^{2}$ is diagonal, $(2SN+N^{2})$ is nilpotent, and the two commute
with one-another, so that (2.1) is the Jordan decomposition of the identity.
Since the Jordan decomposition is unique, it follows that

$S^{2}={\text{Id}}\ \text{and}\ (2SN+N^{2})=0\ .$

It follows from the first relation that the eigenvalues of $S$ are $\pm 1$.
Since $SN$ is nilpotent, the second relation yields

$N=\frac{1}{2}S\cdot\bigg{(}2SN+N^{2}\bigg{)}\cdot\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2^{k}}(SN)^{k}=0\
,$

so that $M$ is diagonalizable, as desired. $\square$For any pseudo-involution
$M$ of $E$, let ${\text{Inv}}(M)$ denote the space of $M$-invariant
$d$-dimensional subspaces of $E$, furnished with the topology that it inherits
as a subspace of the grassmannian. When $M$ has negative sign and is balanced,
this set also contains its two real eigenspaces, which we will find preferable
to exclude in the sequel. For this reason, we denote by ${\text{Inv}}^{*}(M)$
the set of $M$-invariant $d$-dimensional subspaces of $E$ which are not
$M$-eigenspaces. Note that, when $M$ has positive sign, it has no real
eigenspaces, and ${\text{Inv}}^{*}(M)$ coincides with ${\text{Inv}}(M)$. By
abuse of terminology, in later sections, we will say that a plane is
$M$-invariant whenever it is an element of ${\text{Inv}}^{*}(M)$.

In this paper, we will be concerned with the case where $E$ is
$4$-dimensional.

Lemma 2.2.2 Let $M$ be a pseudo-involution of some $4$-dimensional real vector
space $E$.  (1) If $M$ has positive sign, then ${\text{Inv}}^{*}(M)$ is
homeomorphic to $\mathbb{S}^{2}$, and  (2) if $M$ has negative sign and is
balanced, then ${\text{Inv}}^{*}(M)$ is homeomorphic to
$\mathbb{S}^{1}\times\mathbb{S}^{1}$.

Remark 2.2.1. In particular, by compactness of the torus, whenever $M$ has
negative sign and is balanced, its two eigenspaces are isolated in
${\text{Inv}}(M)$. Proof: Suppose first that $M$ has positive sign. Then $M$
is a complex structure, and ${\text{Inv}}^{*}(M)$ identifies with
$1$-dimensional complex projective space, which is homeomorphic to
$\mathbb{S}^{2}$, as desired. Suppose now that $M$ is negative and balanced,
and let $E_{-}$ and $E_{+}$ denote respectively its negative and positive
eigenspaces. With respect to the decomposition $E=E_{-}\oplus E_{+}$, $M$ has
the form

$M=\begin{pmatrix}-{\text{Id}}\hfill&0\hfill\cr
0\hfill&{\text{Id}}\hfill\cr\end{pmatrix}\ .$

Now let $P$ be an $M$-invariant subspace of $E$ which is not an eigenspace. In
particular, $P$ contains a vector $x:=(u,v)$ both of whose components are non-
zero. Since $P$ is $M$-invariant, the vector $y:=(-u,v)$ is also an element of
$P$, so that $P$ is generated by the pair $(u,0)$ and $(0,v)$. Conversely, any
pair of non-vanishing vectors of this form trivially generates an
$M$-invariant subspace which is not an eigenspace. It follows that
${\text{Inv}}^{*}(M)$ is homeomorphic to the cartesian product of the
projective space of $E_{-}$ with that of $E_{+}$. Since each of these
projective spaces is homeomorphic to $\mathbb{S}^{1}$, the result follows.
$\square$

2.3 - Clifford algebras.  We now present the main properties of Clifford
algebras that will be required in the sequel. For a comprehensive introduction
to this fascinating theory, we refer the reader to the excellent works $[17]$
of Garling and $[31]$ of Lounesto. Throughout what follows, we will only be
concerned with the relatively trivial case of Clifford algebras defined over
$2$-dimensional real vector spaces.

Let ${{b}}$ be a symmetric, bilinear form defined over a $2$-dimensional real
vector space $E$. We define the Clifford algebra ${\text{Cl}}({{b}})$ to be
the quotient of the tensor algebra $T^{*}E$ of $E$ by the two-sided ideal

$\Cal{I}:=\langle u\otimes v+v\otimes u+2{{b}}(u,v)\ |\ u,v\in E\rangle\ .$

When ${{b}}$ is trivial, ${\text{Cl}}({{b}})$ is none other than the exterior
algebra of $E$, also known as its Grassmann algebra, and when ${{b}}$ is
positive-definite, it is straightforward to show that ${\text{Cl}}({{b}})$ is
isomorphic to the quaternion algebra.

We will henceforth only be concerned with the case where ${{b}}$ is non-
degenerate. Let $(e_{1},e_{2})$ be a ${{b}}$-orthonormal basis of $E$. The
algebra ${\text{Cl}}({{b}})$ is generated as a vector space by the identity
together with the elements

${{i}}:=e_{1}\ ,\ {{j}}:=e_{2}\ ,\ \text{and}\ {{k}}:=e_{1}\cdot e_{2}\ .$

Note that each of these elements has unit (positive or negative) square, and
that any two of them anticommute with one-another. The Clifford algebra
carries a natural $\mathbb{Z}_{2}$-grading

${\text{Cl}}({{b}})={\text{Cl}}^{+}({{b}})\oplus{\text{Cl}}^{-}({{b}})\ ,$

where ${\text{Cl}}^{+}({{b}})$ and ${\text{Cl}}^{-}({{b}})$ denote the vector
subspaces generated respectively by products of even and odd numbers of
elements of $E$. In the present case,

${\text{Cl}}^{+}({{b}})=\langle 1,{{k}}\rangle\ \text{and}\
{\text{Cl}}^{-}({{b}})=\langle{{i}},{{j}}\rangle\ .$

We denote by $\pi^{+}:{\text{Cl}}({{b}})\rightarrow{\text{Cl}}^{+}({{b}})$ and
$\pi^{-}:{\text{Cl}}({{b}})\rightarrow{\text{Cl}}^{-}({{b}})$ the canonical
projections onto each of the factors. This decomposition will play a
significant role in the sequel.

The Clifford algebra carries one natural involution and two natural anti-
involutions which we now describe. The involution, called the grade
involution, is defined by

$\hat{x}:=\pi^{+}(x)-\pi^{-}(x)\ .$

That is, when

$x:=a+b{{i}}+c{{j}}+d{{k}}\ ,\eqnum{\text{{\rm(2.2)}}\hbox{}}$

its image under the grade involution is

$\hat{x}=a-b{{i}}-c{{j}}+d{{k}}\ .$

The first anti-involution, called reversion, is first defined on products of
elements of $E$ by reversing the order of their factors, and is then extended
to ${\text{Cl}}({{b}})$ by linearity. With $x$ as in (2.2), its image under
reversion is thus

$x^{\dagger}=a+b{{i}}+c{{j}}-d{{k}}\ .$

Finally, the second anti-involution, called Clifford conjugation, is simply
the composition of grade involution with reversion, so that, with $x$ again as
in (2.2), its Clifford conjugate is

$\overline{x}=a-b{{i}}-c{{j}}-d{{k}}\ .$

These three operations trivially commute with one another.

In the present, $2$-dimensional case, it makes sense to deepen the analogy
with quaternions. We thus denote the $(+1)$-eigenspace of Clifford conjugation
by $\Cal{R}$, and we call it the real subspace. We likewise denote the
$(-1)$-eigenspace of Clifford conjugation by $\Cal{I}$, and we call it the
imaginary subspace. Trivially,

$\Cal{R}=\langle 1\rangle\ \text{and}\
\Cal{I}=\langle{{i}},{{j}},{{k}}\rangle\ .$

In particular, $\Cal{R}$ is the centre of ${\text{Cl}}({{b}})$, and $\Cal{I}$
is a Lie sub-algebra. We denote also by $\Cal{R}$ and $\Cal{I}$ the respective
projections onto the real and imaginary subspaces. Note that $\Cal{R}$ can be
viewed as a trace in the sense that, for all $x,y\in{\text{Cl}}({{b}})$,

$\Cal{R}(x\cdot y)=\Cal{R}(y\cdot x)\ .$

Since Clifford conjugation is an anti-involution, for all
$x\in{\text{Cl}}({{b}})$,

$\overline{x\cdot\overline{x}}=\overline{\overline{x}}\cdot\overline{x}=x\cdot\overline{x}\
,$

so that $x\cdot\overline{x}$ is always real. We thus define the inner-product
$g$ over ${\text{Cl}}({{b}})$ by

$g(x,y):=\Cal{R}(x\cdot\overline{y})\ .$

Note that the anticommutator of any two imaginary elements $x,y$ satisfies

$\\{x,y\\}=2g(x,y)\ ,$

so that two imaginary elements anticommute if and only if they are orthogonal
to one another.

We denote the norm-squared of $g$ by $\|\cdot\|_{g}^{2}$. With $x$ as in
(2.2),

$\|x\|_{g}^{2}=a^{2}-b^{2}{{i}}^{2}-c^{2}{{j}}^{2}-d^{2}{{k}}^{2}\ ,$

so that the inner-product $g$ is non-degenerate with signature determined by
the signs of the squares of ${{i}}$, ${{j}}$ and ${{k}}$. We will say that a
unit-length imaginary element of ${\text{Cl}}({{b}})$ is positive or negative
according to the sign of its norm-squared. Note that this terminology is
consistent with that introduced for pseudo-involutions in the preceding
section.

Given $x\in{\text{Cl}}({{b}})$, we denote by $m_{x}$ the operation of
multiplication on the left by this element. For all
$x,y,z\in{\text{Cl}}({{b}})$,

$g(m_{x}(y),m_{x}(z))=\|x\|_{g}^{2}g(y,z)\ ,$

so that $m_{x}$ is a pseudo-isometry whenever $x$ has unit length.
Furthermore, when, in addition, $x$ is imaginary, for all
$y,z\in{\text{Cl}}({{b}})$,

$g(y,m_{x}(z))=-g(m_{x}(y),z)\ ,$

so that $m_{x}$ is $g$-antisymmetric. In particular, for all unit-length,
imaginary $x$, we define the non-degenerate alternating form $\omega_{x}$ by

$\omega_{x}(y,z):=g(y,m_{x}(z))\ .$

Finally, we define the complementary inner-product $\hat{g}$ over
${\text{Cl}}({{b}})$ by

$\hat{g}(y,z):=g(\hat{y},z)=\Cal{R}(\hat{y}\cdot\overline{z})\ .$

Note that ${\text{Cl}}^{+}({{b}})$ and ${\text{Cl}}^{-}({{b}})$ are both $g$\-
and $\hat{g}$-orthogonal to one another. Furthermore

$\hat{g}|_{{\text{Cl}}^{+}({{b}})}=g|_{{\text{Cl}}^{+}({{b}})}\ \text{and}\
\hat{g}|_{{\text{Cl}}^{-}({{b}})}=-g|_{{\text{Cl}}^{-}({{b}})}\ .$

As before, for all $x\in{\text{Cl}}^{-}({{b}})$, and for all
$y,z\in{\text{Cl}}({{b}})$,

$\hat{g}(m_{x}(y),m_{x}(z))=-\|x\|_{g}^{2}\hat{g}(y,z)\ ,$

so that $m_{x}$ is a $\hat{g}$-pseudo-isometry whenever it has unit length.
Likewise for all such $x$, $y$ and $z$,

$\hat{g}(y,m_{x}(z))=g(m_{x}(y),z)\ ,$

so that $m_{x}$ is also $\hat{g}$-symmetric.

We now characterize planes preserved by the action of multiplication on the
left by unit-length imaginary elements of ${\text{Cl}}({{b}})$.

Lemma 2.3.1 If $x$ is a unit-length element of $\Cal{I}$ of negative sign,
then $m_{x}$ is balanced. Furthermore, the eigenspaces of $m_{x}$ are given by

$E_{+}=\langle 1+x,y+x\cdot y\rangle\ \text{and}\ E_{-}=\langle 1-x,y-x\cdot
y\rangle\ ,\eqnum{\text{{\rm(2.3)}}\hbox{}}$

where $y$ is any other unit-length element of $\Cal{I}$ orthogonal to $x$.

Proof: Indeed, let $y$ be another unit-length element of $\Cal{I}$ orthogonal
to $x$. Since $y$ anticommutes with $x$, $m_{y}$ interchanges the eigenspaces
of $m_{x}$, so that $m_{x}$ is balanced, as desired. We verify (2.3) by
inspection, and this completes the proof. $\square$We now relate $J$-invariant
planes to bilagrangian planes. The following simple observation will play a
fundamental role throughout the rest of this paper.

Theorem 2.3.2, $J$-invariance of bilagrangian planes Let $x$ be a unit-length
element of $\Cal{I}$. A real plane $P\subseteq{\text{Cl}}({{b}})$ is
$m_{x}$-invariant if and only if it is $\omega_{y}$-lagrangian for all
imaginary $y$ orthogonal to $x$.

Proof: Note first that if $y$ is a unit-length element of $\Cal{I}$ orthogonal
to $x$ then, since $y$ anticommutes with $x$, $(y\cdot x)$ is also a unit-
length element of $\Cal{I}$ orthogonal to $x$. Suppose now that $P$ is an
element of ${\text{Inv}}^{*}(m_{x})$. Let $z\in P$ be a non-trivial element
such that $w:=m_{x}(z)\in P$ is not colinear with $z$. With $y$ as above,

$\omega_{y}(z,w)=g(z,m_{y}(m_{x}(z)))=g(z,m_{(y\cdot x)}(z))=\omega_{(y\cdot
x)}(z,z)=0\ ,$

so that $\omega_{y}$ vanishes over $P$, as desired.

Conversely, suppose that $P$ is $\omega_{y}$-lagrangian for all imaginary $y$
orthogonal to $x$. With $y$ again as above, for all $z,w\in P$,

$\omega_{y}(z,m_{x}(w))=g(z,y\cdot x\cdot w)=\omega_{(y\cdot x)}(z,w)=0.$

Since $P$ is $\omega_{y}$-lagrangian, in particular, it is its own
$\omega_{y}$-annihilator. It follows that $m_{x}(w)\in P$ for all $w\in P$, so
that $P$ is $m_{x}$-invariant, as desired.

Finally, we verify by inspection of (2.3) that when $x$ has negative sign, and
when $y$ is a unit-length element of $\Cal{I}$ orthogonal to $x$, no
eigenspace of $m_{x}$ is $\omega_{y}$-lagrangian, and this completes the
proof. $\square$We conclude this section by comparing the present framework
with that developed by Lychagin–Rubtsov in $[32]$ (see also $[33]$ and
$[37]$). Since the following constructions will not be used in the sequel, we
will only sketch the main ideas, leaving the reader to complete the details.
Viewing $({\text{Cl}}({{b}}),g)$ as an inner-product space in its own right,
we define $\alpha:\Cal{I}\times\Cal{I}\rightarrow{\mathfrak{so}}(g)$ by

$\alpha(x,y)\cdot z:=x\cdot z-z\cdot y\ .$

The reader may verify that, for all
$(x,y),(x^{\prime},y^{\prime})\in\Cal{I}\times\Cal{I}$,

$g(x,x^{\prime})+g(y,y^{\prime})=-\frac{1}{4}{\text{Tr}}(\alpha(x,y)\cdot\alpha(x^{\prime},y^{\prime}))\
,$

so that, in particular, $\alpha$ is, up to a scalar factor, a linear
isomorphism. Since $g$ is non-degenerate, we may also define the linear
isomorphism $\beta:{\mathfrak{so}}(g)\rightarrow\Lambda^{2}{\text{Cl}}({{b}})$
by

$\beta(M):=g(\cdot,M\cdot)\ .$

The metric $g$ also induces a non-degenerate inner-product over
$\Lambda^{2}{\text{Cl}}({{b}})$, and the reader may verify that, for all
$M,M^{\prime}\in{\mathfrak{so}}({\text{Cl}}({{b}}))$,

$g(\beta(M),\beta(M^{\prime}))=-\frac{1}{2}{\text{Tr}}(M\cdot M^{\prime})\ .$

Note now that the Hodge $*$ operator define a negative pseudo-involution of
$\Lambda^{2}{\text{Cl}}({{b}})$, and the reader may also verify that its
$(+1)$ and $(-1)$ eigenspaces are respectively the images under
$(\beta\circ\alpha)$ of $\Cal{I}\times\\{0\\}$ and $\\{0\\}\times\Cal{I}$.
Combining these facts, it follows that, for all $x,x^{\prime}\in\Cal{I}$,

$\omega_{x}\wedge\omega_{x^{\prime}}=g(\omega_{x},*\omega_{x^{\prime}}){\text{dVol}}=g((\beta\circ\alpha)(x,0),*(\beta\circ\alpha)(x^{\prime},0)){\text{dVol}}=2g(x,x^{\prime}){\text{dVol}}\
,$

so that, in the terminology of Lychagin–Rubtsov, the pair
$(\omega_{x},\omega_{x^{\prime}})$ of symplectic forms is effective if and
only if the elements $x$ and $x^{\prime}$ are $g$-orthogonal to one another.
Note now that, in Theorem 2.3.2, the space of imaginary elements of
${\text{Cl}}({{b}})$ orthogonal to $x$ is $2$-dimensional, and therefore has
an orthonormal basis $(y_{1},y_{2})$ of cardinality $2$. It follows by the
preceding discussion that the pair $(\omega_{y_{1}},\omega_{y_{2}})$ is
effective, whilst, by Theorem 2.3.2, a real plane is $m_{x}$-invariant if and
only it is $(\omega_{y_{1}},\omega_{y_{2}})$-bilegendrian.

2.4 - Invariant planes.  We now study the restrictions of $g$ and $\hat{g}$ to
invariant planes. Let $x$ be a unit-length element of
${\text{Cl}}^{-}({{b}})$. We first study the eigenspaces of $m_{x}$ in the
case where $x$ has negative sign.

Lemma 2.4.1 If $x$ is a unit-length element of ${\text{Cl}}^{-}({{b}})$ of
negative sign, then  (1) the eigenspaces of $m_{x}$ are $g$-null,  (2) the
eigenspaces of $m_{x}$ are $\hat{g}$-orthogonal to one-another, and  (3)
$\pi_{+}$ restricts to isometries from $(E_{\pm},\hat{g})$ to
$({\text{Cl}}^{+}({{b}}),2g)$.

Proof: The first and second assertions follow respectively from the
$g$-antisymmetry and $\hat{g}$-symmetry of $m_{x}$. The third assertion
follows by inspection of (2.3) with $y={{k}}$, and this completes the proof.
$\square$We now study elements of ${\text{Inv}}^{*}(m_{x})$. We will say that
an element $P$ of ${\text{Inv}}^{*}(m_{x})$ is exceptional whenever the
restriction of $\hat{g}$ to this plane degenerates. We first classify
exceptional invariant planes.

Lemma 2.4.2 Let $x$ be a unit-length element of ${\text{Cl}}^{-}({{b}})$ of
positive sign, and let $P$ be an element of ${\text{Inv}}^{*}(m_{x})$. The
restriction of $\hat{g}$ to $P$ degenerates if and only if it vanishes
identically. Furthermore, when this holds, either  (1) $P$ is the direct sum
of a $g$-null line of ${\text{Cl}}^{+}({{b}})$ with a $g$-null line of
${\text{Cl}}^{-}({{b}})$, or  (2) $P$ is the graph of $\pm m_{(x\cdot{{k}})}$
over ${\text{Cl}}^{+}({{b}})$.

Remark 2.4.1. Note that ${\text{Cl}}^{+}({{b}})$ and ${\text{Cl}}^{-}({{b}})$
only have $g$-null lines when ${{b}}$ has negative sign. Furthermore, when
this holds, $(1)$ accounts for precisely $2$ planes in ${\text{Cl}}({{b}})$
since, although $4$ different planes may be obtained by the different direct
sums of these null lines, only two of them will be elements of
${\text{Inv}}^{*}(m_{x})$. Proof: Suppose that $\hat{g}$ degenerates over $P$.
There exists a non-trivial element $z\in P$ such that, for all other $w\in P$,

$\hat{g}(z,w)=0\ .$

Since $m_{x}$ is $\hat{g}$-symmetric, for all such $w$,

$\hat{g}(m_{x}(z),w)=\hat{g}(z,m_{x}(w))=0\ .$

Since $x$ has positive sign, $m_{x}$ defines a complex structure over $P$. In
particular, $z$ and $m_{x}(z)$ are linearly independent, so that $\hat{g}$
vanishes identically, as desired.

Suppose now that $\hat{g}$ vanishes identically over $P$. Let
$\alpha:=z+x\cdot w$ be a non-trivial element of $P$, where
$z,w\in{\text{Cl}}^{+}({{b}})$. Straightforward calculations yield

$\|z+x\cdot w\|_{\hat{g}}^{2}=\|z\|_{g}^{2}-\|w\|_{g}^{2}\ ,$

and

$\hat{g}(x\cdot(z+x\cdot w),(z+x\cdot w))=-2g(z,w)\ .$

Since $\hat{g}$ vanishes identically over $P$, it follows that

$\|z\|_{g}^{2}-\|w\|_{g}^{2}=g(z,w)=0\ .$

There are now two possibilities. Either $z$ and $w$ are both $g$-null, in
which case $P$ is the direct sum of a $g$-null line of
${\text{Cl}}^{+}({{b}})$ with a $g$-null line of ${\text{Cl}}^{-}({{b}})$.
Otherwise, $z$ and $w$ have the same, non-zero $g$-norm-squared and are
$g$-orthogonal to one another, so that $w=\pm{{k}}\cdot z$, and $P$ is the
graph of $\pm m_{(x\cdot{{k}})}$ over ${\text{Cl}}^{+}({{b}})$, as desired.
$\square$

Lemma 2.4.3 Let $x$ be a unit-length element of ${\text{Cl}}^{-}({{b}})$ of
negative sign, and let $P$ be an element of ${\text{Inv}}^{*}(m_{x})$. The
restriction of $\hat{g}$ to $P$ degenerates if and only if $P$ contains a
$\hat{g}$-null real eigenvector of $m_{x}$.

Remark 2.4.2. Note that it is only possible for $m_{x}$ to have $\hat{g}$-null
real eigenvectors when ${{b}}$ has negative sign. In particular, when ${{b}}$
has positive sign, $\hat{g}$ never degenerates over $P$. Proof: Indeed, by
Lemma 2.2.2, $P=\langle z,w\rangle$, where $z$ and $w$ are eigenvectors of
$m_{x}$ with respective eigenvalues $(-1)$ and $(+1)$. By Item $(2)$ of Lemma
2.4.1, $z$ and $w$ are $\hat{g}$-orthogonal to one another, so that $\hat{g}$
degenerates over $P$ if and only if one of $z$ or $w$ is $\hat{g}$-null, as
desired. $\square$Having classified exceptional invariant planes, we now study
the properties of non-exceptional invariant planes.

Lemma & Definition 2.4.4 Let $x$ be a unit-length element of
${\text{Cl}}^{-}({{b}})$ of positive sign, and let $P$ be a non-exceptional
element of ${\text{Inv}}^{*}(m_{x})$. There exist precisely $4$ vectors $v\in
P$ such that $(v,m_{x}(v))$ is $g$-orthogonal and $\hat{g}$-orthonormal. We
call any such vector a principal vector of $E$.

Proof: Note first that, since $m_{x}$ is $g$-antisymmetric,
$g(\cdot,m_{x}\cdot)$ vanishes identically over $P$. Consider now the bilinear
form $h:=\hat{g}(\cdot,m_{x}\cdot)$. Since $m_{x}$ is $\hat{g}$-symmetric, $h$
is also symmetric. Since $m_{x}$ is a $\hat{g}$-anti-isometry, it is also an
$h$-anti-isometry. Since $P$ is non-exceptional, by Lemma 2.4.2, $\hat{g}$,
and therefore also $h$, is non-degenerate. Since $h$ admits a non-trivial
anti-isometry, it has mixed signature and thus has $2$ null directions.
Furthermore, neither of these null directions is $\hat{g}$-null, for otherwise
$\hat{g}$ would vanish identically over $P$. We readily verify that any
$h$-null vector $v$ with unit $\hat{g}$-norm-squared has the desired
properties, and the result follows. $\square$

Lemma & Definition 2.4.5 Let $x$ be a unit-length element of
${\text{Cl}}^{-}({{b}})$ of negative sign, and let $P$ be a non-exceptional
element of ${\text{Inv}}^{*}(m_{x})$. There exist precisely $8$ vectors $v\in
P\otimes\mathbb{C}$ such that $(v,m_{x}(v))$ is $g$-orthogonal and
$\hat{g}$-orthonormal. Furthermore, $4$ of these vectors are real unless
${{b}}$ has negative sign and $P$ contains eigenvectors of $m_{x}$ of opposing
sign. We call any such vector a principal direction of $E$.

Remark 2.4.3. We will not be explicitly concerned with the case of imaginary
principal directions in the sequel. Proof: Let $z,w\in{\text{Cl}}^{+}({{b}})$
be such that $\hat{z}:=(z,m_{x}(z))$ and $\hat{w}:=(w,m_{x}(w))$ are the
$m_{x}$-eigenvectors which generate $P$. Since $m_{x}$ is $\hat{g}$-symmetric,
these two eigenvectors are $\hat{g}$-orthogonal to one another. Furthermore,
since $P$ is non-exceptional, neither vector is $\hat{g}$-null, and we may
therefore suppose that they each have unit norm-squared. If their norms-
squared have the same sign, then the vectors $(\hat{z}\pm\hat{w})/\sqrt{2}$
have the desired properties. Otherwise, their norms-squared have opposing
signs, and the vectors $(\hat{z}\pm i\hat{w})/\sqrt{2}$ have the desired
properties. The remaining vectors are obtained by multiplying these vectors by
$(\pm 1)$ and $(\pm i)$, and the result follows. $\square$

Lemma & Definition 2.4.6 Let $x$ be a unit-length element of
${\text{Cl}}^{-}({{b}})$ of negative sign, let $P$ be a non-exceptional
element of ${\text{Inv}}^{*}(m_{x})$, and let $v$ be a real principal
direction of $P$. The plane $E_{v}:=\langle v,{{k}}\cdot x\cdot v\rangle$ has
$1$-dimensional intersections with each of ${\text{Cl}}^{+}({{b}})$ and
${\text{Cl}}^{-}({{b}})$. We call the intersection
$E_{v}\cap{\text{Cl}}^{+}({{b}})$ the principal line of $P$ associated to $v$.

Remark 2.4.4. An analogous result also holds for imaginary principal
directions, although we will have no use for this in the present paper. Proof:
Indeed, let $z,w\in{\text{Cl}}^{+}({{b}})$ be such that $v=z+x\cdot w$. A
straightforward calculation yields

$\hat{g}(v,m_{x}(v))=(1-\|x\|_{g}^{2})g(z,w)=2g(z,w)\ .$

Since $v$ and $m_{x}(v)$ are $\hat{g}$-orthogonal to one another, it follows
that $z=\pi_{+}(v)$ and $w=x\cdot\pi_{-}(v)$ are $g$-orthogonal to one-
another. In particular, $z$ and $({{k}}\cdot w)$ are linearly dependent, and
the result follows. $\square$

3 - Legendrian Clifford structures. 3.1 - Overview.  We now apply the
algebraic theory developed in Section 2 to the setting of contact geometry.
Recall that the unit sphere bundle $UX$ of any riemannian $3$-manifold $X$
carries a natural contact structure $\alpha$, with contact bundle $W$, and
symplectic form $\omega_{i}:=d\alpha$. We will construct natural families of
Clifford structures over $W$ such that, for all
$k\in\mathbb{R}\setminus\\{0\\}$, there exists a pseudo-involution $J_{k}$,
associated with a fibrewise symplectic form $\omega_{k}$, having the property
that $(S,e)$ is a CEC-$k$ surface if and only if its Gauss lift $(S,\hat{e})$
is $(\omega_{i},\omega_{k})$-bilegendrian, and thus also $J_{k}$-invariant.

When $k>0$, the pseudo-involution $J_{k}$ is nothing other than the Monge-
Ampère structure introduced by Labourie in $[28]$, showing how his framework
embeds within our own. When the ambient space is either $\mathbb{H}^{3}$ or
$\mathbb{S}^{3}$, and when $k$ is respectively equal to $(+1)$ or $(-1)$,
$(\omega_{i},\omega_{k})$-bilegendrian surfaces are respectively flat fronts
of the first and second types discussed in the introduction.

The main result of this section is Theorem 3.4.4, which proves that, whenever
$X$ is a space form, every immersed bilegendrian surface in $UX$ carries a
canonical flat metric over the complement of a certain set of exceptional
points. We underline, above all, how this follows in an elementary manner from
the many symmetries of the Clifford structure. Finally, in Sections 3.5 and
3.6 we show how this metric is related to well-known structures over CEC
surfaces, and in Section 3.7, we use this metric to prove Hilbert’s theorem
concerning the non-existence of complete, negatively-curved CEC surfaces in
euclidean $3$-space.

3.2 - The Clifford structure.  We now describe the explicit Clifford structure
over $\mathbb{R}^{2}\times\mathbb{R}^{2}$ which we will use to study surfaces
of constant extrinsic curvature. Let ${{b}}$ be a non-degenerate, symmetric
bilinear form over $\mathbb{R}^{2}$, and let $A$ be an element of
${\mathfrak{so}}({{b}})\cap\hat{{\text{O}}}({{b}})$. Note that

${\text{Vol}}:=-{{b}}(\cdot,A\cdot)\eqnum{\text{{\rm(3.1)}}\hbox{}}$

is a unit-length alternating form, so that, in particular, the choice of $A$
also defines a preferred orientation of $\mathbb{R}^{2}$.

Given $\eta\in\\{\pm 1\\}$, we define the symplectic form $\omega_{i}$, the
inner-product $g^{\eta}$, and the grade involution $\alpha$ over
$\mathbb{R}^{2}\times\mathbb{R}^{2}$ by

$\eqalign{\omega_{i}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))&:={{b}}(\xi,\mu^{\prime})-{{b}}(\mu,\xi^{\prime})\
,\cr
g^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))&:={{b}}(\xi,\xi^{\prime})+\eta{{b}}(\mu,\mu^{\prime})\
,\ \text{and}\cr\alpha(\xi,\mu)&:=(\xi,-\mu)\ .\cr}$

We also define the complementary inner product $\hat{g}^{\eta}$ by
$g^{\eta}(\cdot,\alpha\cdot)$, so that

$\hat{g}^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime})):={{b}}(\xi,\xi^{\prime})-\eta{{b}}(\mu,\mu^{\prime})\
.$

Recall now that geometric structures can also be defined in terms of their
stabilizer subgroup. Let ${\text{Stab}}(\omega_{i},g^{\eta},\alpha)$ denote
the stabilizer subgroup of the subgroup of the triple
$(\omega_{i},g^{\eta},\alpha)$ in
${\text{End}}(\mathbb{R}^{2}\times\mathbb{R}^{2})$.

Lemma 3.2.1 ${\text{Stab}}(\omega_{i},g^{\eta},\alpha)$ consists of all
matrices $M$ of the form

$M=\begin{pmatrix}N\hfill&0\hfill\cr 0\hfill&N\hfill\cr\end{pmatrix}\
,\eqnum{\text{{\rm(3.2)}}\hbox{}}$

where $N\in{\text{O}}({{b}})$.

Proof: Indeed, choose $M\in{\text{Stab}}(\omega_{i},\hat{g}^{\eta},\alpha)$.
Since $M$ preserves $\alpha$,

$M=\begin{pmatrix}N\hfill&0\hfill\cr 0\hfill&N^{\prime}\hfill\cr\end{pmatrix}\
,$

for some $N,N^{\prime}\in{\text{End}}(\mathbb{R}^{2})$. Since $M$ preserves
$g^{\eta}$,

$N,N^{\prime}\in{\text{O}}({{b}})\ .$

Finally, since $M$ preserves $\omega_{i}$, denoting by $N^{*}$ the
${{b}}$-dual of $N$,

$N^{\prime}=(N^{*})^{-1}=N\ ,$

so that $M$ has the desired form. Conversely, we verify that every linear map
of this form preserves $(\omega_{i},g^{\eta},\alpha)$, and this completes the
proof. $\square$We now show how the triple $(\omega_{i},g^{\eta},\alpha)$
defines a Clifford structure over $\mathbb{R}^{2}\times\mathbb{R}^{2}$.

Lemma 3.2.2 Suppose that ${{b}}$ has a positive direction $v$, and let ${{i}}$
denote the corresponding element of ${\text{Cl}}(\eta{{b}})$. There exists a
linear isomorphism
$\phi:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow{\text{Cl}}(\eta{{b}})$
sending $\omega_{i}$, $g^{\eta}$ and $\alpha$ respectively to $\omega_{i}$,
$g$ and the grade involution. Furthermore, $\phi$ is unique up to
precomposition by an element of ${\text{Stab}}(\omega_{i},g^{\eta},\alpha)$.

Proof: Since ${{k}}$ has the same sign as ${{b}}$, there exists an isometry
$\phi_{0}:(\mathbb{R}^{2},{{b}})\rightarrow({\text{Cl}}^{+}(\eta{{b}}),g)$. We
define
$\phi:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow{\text{Cl}}(\eta{{b}})$ by

$\phi(\xi,\mu):=\phi_{0}(\xi)+{{i}}^{-1}\cdot\phi_{0}(\mu)\ .$

This isomorphism trivially sends $\alpha$ to the grade involution.
Furthermore, for all
$(\xi,\mu),(\xi^{\prime},\mu^{\prime})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$,

$\omega_{i}(\phi(\xi,\mu),\phi(\xi^{\prime},\mu^{\prime}))=g(\phi_{0}(\xi)+{{i}}^{-1}\cdot\phi_{0}(\mu),{{i}}\cdot\phi_{0}(\xi^{\prime})+\phi_{0}(\mu^{\prime}))\
.$

Since ${\text{Cl}}^{\pm}(\eta{{b}})$ are $g$-orthogonal to one another, and
since $m_{{{i}}}$ is $g$-antisymmetric, it follows that

$\omega_{i}(\phi(\xi,\mu),\phi(\xi^{\prime},\mu^{\prime}))=g(\phi_{0}(\xi),\phi_{0}(\mu^{\prime}))-g(\phi_{0}(\mu),\phi_{0}(\xi^{\prime}))={{b}}(\xi,\mu^{\prime})-{{b}}(\mu,\xi^{\prime})=\omega_{i}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))\
,$

as desired. Similarly, for all
$(\xi,\mu),(\xi^{\prime},\mu^{\prime})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$,

$g(\phi(\xi,\mu),\phi(\xi^{\prime},\mu^{\prime}))=g(\phi_{0}(\xi)+{{i}}^{-1}\cdot\phi_{0}(\mu),\phi_{0}(\xi^{\prime})+{{i}}^{-1}\cdot\phi_{0}(\mu))=g(\phi_{0}(\xi),\phi_{0}(\xi^{\prime}))+\|{{i}}\|_{g}^{2}g(\phi_{0}(\mu),\phi_{0}(\mu^{\prime}))\
.$

Since

$\|{{i}}\|_{g}^{2}=-{{i}}^{2}=\eta{{b}}(e_{1},e_{1})=\eta\ ,$

it follows that

$g(\phi(\xi,\mu),\phi(\xi^{\prime},\mu^{\prime}))=g^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))\
,$

as desired. This proves existence, uniqueness follows by definition of
${\text{Stab}}(\omega_{i},g^{\eta},\alpha)$, and this completes the proof.
$\square$We now pull back through $\phi$ the objects constructed in Section 2
to obtain what we call a Clifford structure over
$\mathbb{R}^{2}\times\mathbb{R}^{2}$. Since ${\text{O}}({{b}})$ has more than
one connected component${\text{O}}({{b}})$ has $2$ connected components when
${{b}}$ has positive sign, and $4$ when ${{b}}$ has negative sign., there is,
in fact, a sign ambiguity, which we resolve upon setting

$\omega_{k}^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime})):={\text{Vol}}(\xi,\mu)-\eta{\text{Vol}}(\xi^{\prime},\mu^{\prime})\
.$

Explicitly, the Clifford structure is given by the matrices $I^{\eta}$,
$J^{\eta}$, and $K$, defined by

$\omega_{i}=:g^{\eta}(\cdot,I^{\eta}\cdot)\ ,\
\omega_{k}^{\eta}=:\omega_{i}(\cdot,J^{\eta}\cdot)\ ,\ \text{and}\
\omega_{k}^{\eta}=:g^{\eta}(\cdot,K\cdot)\ ,$

and we also denote

$\hat{I}^{\eta}:=\alpha\circ I^{\eta}\ ,\ \hat{J}^{\eta}:=\alpha\circ
J^{\eta}\ ,\ \text{and}\ \hat{K}:=\alpha\circ K\ .$

We readily verify that

$I^{\eta}=\begin{pmatrix}0\hfill&{\text{Id}}\hfill\cr-\eta{\text{Id}}\hfill&0\hfill\cr\end{pmatrix}\
,\ J^{\eta}=\begin{pmatrix}0\hfill&\eta A\hfill\cr
A\hfill&0\hfill\cr\end{pmatrix}\ ,\ \text{and}\
K=\begin{pmatrix}A\hfill&0\hfill\cr 0\hfill&-A\hfill\cr\end{pmatrix}\
,\eqnum{\text{{\rm(3.3)}}\hbox{}}$

and we verify by inspection that these matrices satisfy the algebraic
relations of a Clifford structure, namely

$I^{\eta}J^{\eta}=K\ ,$

their squares satisfy

$(I^{\eta})^{2}=-\eta{\text{Id}}\ ,\ (J^{\eta})^{2}=-\eta\epsilon{\text{Id}}\
,\ \text{and}\ K^{2}=-\epsilon{\text{Id}}\ ,$

and their anti-commutators satisfy

$\\{I^{\eta},J^{\eta}\\}=\\{I^{\eta},K\\}=\\{J^{\eta},K\\}=0\ .$

3.3 - The Clifford bundle.  We now extend the construction of the preceding
section to define Clifford structures over bundles. This formalism can, in
fact, be fruitfully applied to bundles over general riemannian manifolds (see,
for example, $[28]$ and $[46]$). However, we will limit ourselves here to a
simple case where all objects in question will be covariant constant. This
will nonetheless be sufficient to derive interesting properties of CEC
surfaces in space-forms of riemannian and lorentzian signature.

Let ${{b}}$ be a non-degenerate, symmetric bilinear form over
$\mathbb{R}^{4}$. For $\eta\in\\{\pm 1\\}$, we define the symplectic form
$\omega_{i}$, the inner product $g^{\eta}$, and the grade involution $\alpha$
over $\mathbb{R}^{4}\times\mathbb{R}^{4}$ by

$\eqalign{\omega_{i,(x,y)}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))&={{b}}(\xi,\mu^{\prime})-{{b}}(\mu,\xi^{\prime})\
,\cr
g^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime}))&:={{b}}(\xi,\xi^{\prime})+\eta{{b}}(\mu,\mu^{\prime})\
,\ \text{and}\cr\alpha(\xi,\mu)&:=(\xi,-\mu)\ .\cr}$

We likewise define the complementary inner product $\hat{g}^{\eta}$ by
$\hat{g}^{\eta}:=g(\cdot,\alpha\cdot)$, so that

$\hat{g}^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime})):={{b}}(\xi,\xi^{\prime})-\eta{{b}}(\mu,\mu^{\prime})\
.$

We define the open, codimension $1$ submanifold $M$ of
$\mathbb{R}^{4}\times\mathbb{R}^{4}$ by

$M:=\\{(x,y)\in\mathbb{R}^{4}\times\mathbb{R}^{4}\ |\
{{b}}(x,x),{{b}}(y,y)\neq 0\ ,\ {{b}}(x,y)=0\\}\ .$

We define the $4$-dimensional subbundle $W$ of $TM$ such that, at every point
$(x,y)$ of $M$,

$W_{(x,y)}:=\\{(\xi,\mu)\in\mathbb{R}^{4}\times\mathbb{R}^{4}\ |\
{{b}}(\xi,x)={{b}}(\xi,y)={{b}}(\mu,x)={{b}}(\mu,y)=0\\}\ .$

At each point $(x,y)$ of $M$, the triple $(\omega_{i},g^{\eta},\alpha)$ yields
a Clifford structure over $W_{(x,y)}$ as in Section 3.2, and we resolve the
sign ambiguity by setting,

$\omega_{k}^{\eta}((\xi,\mu),(\xi^{\prime},\mu^{\prime})):={\text{Vol}}(x,\xi,\mu,y)-\eta{\text{Vol}}(x,\xi,\mu,y)\
,$

where Vol here denotes the canonical volume form of $\mathbb{R}^{4}$. Let
$I^{\eta}$, $J^{\eta}$ and $K$ denote the pseudo-involutions of $W$ obtained
as in (3.3), and let $\hat{I}^{\eta}$, $\hat{J}^{\eta}$ and $\hat{K}^{\eta}$
denote their respective compositions with the grade-involution $\alpha$.

We now show that this Clifford structure is covariant constant. To this end,
we define the $4$-dimensional bundle $N$ over $M$ such that, at every point
$(x,y)$ of $M$,

$N_{(x,y)}:=\langle(x,0),(y,0),(0,x),(0,y)\rangle\ .$

Since $N$ is transverse to $W$ in
${\text{T}}(\mathbb{R}^{4}\times\mathbb{R}^{4})$, composing the canonical
derivative of $\mathbb{R}^{4}\times\mathbb{R}^{4}$ with projection along $N$
yields a covariant derivative over $W$ which we denote by $\nabla$.Although it
is usual in riemannian geometry to restrict covariant derivatives to
subbundles by composing with orthogonal projection, it is in fact sufficient
to project along any transverse subbundle, as is standard practice in affine
geometry (see, for example, $[40]$) Observe that, at every point $(x,y)$ of
$M$, the two bundles $W$ and $N$ are $\omega_{i}$-, $g^{\eta}$\- and
$\hat{g}^{\eta}$-orthogonal to one another, so that, for all sections $\sigma$
and $\tau$ of $W$ over $M$, and for every tangent vector field $\xi$ over $M$,

$(\nabla_{\xi}\omega_{i})(\sigma,\tau)=(\nabla_{\xi}g^{\eta})(\sigma,\tau)=(\nabla_{\xi}\hat{g}^{\eta})(\sigma,\tau)=0\
.$

The analogous result also holds for $\omega_{k}^{\eta}$, although it is less
immediate.

Lemma 3.3.1 For all sections $\sigma$ and $\tau$ of $W$ over $M$, for every
tangent vector field $\xi$ over $M$ taking values in $W$, and for each
$\eta\in\\{\pm 1\\}$,

$(\nabla_{\xi}\omega_{k}^{\eta})(\sigma,\tau)=0\
.\eqnum{\text{{\rm(3.4)}}\hbox{}}$

Corollary 3.3.2 For every section $\sigma$ of $W$ over $M$, and for every
tangent vector field $\xi$ over $M$ taking values in $W$,

$(\nabla_{\xi}I^{\eta})\sigma=(\nabla_{\xi}J^{\eta})\sigma=(\nabla_{\xi}K)\sigma=(\nabla_{\xi}\alpha)\sigma=0\
.\eqnum{\text{{\rm(3.5)}}\hbox{}}$

Proof of Lemma 3.3.1: It suffices to prove the result for
$\omega_{k}^{\prime}:=\omega_{k}^{+1}+\omega_{k}^{-1}$, as the proof for
$\omega_{k}^{\prime\prime}:=\omega_{k}^{+1}-\omega_{k}^{-1}$ is identical.
Note first that, since $\xi$ takes values in $W$,

$D_{\xi}x,D_{\xi}y\in\langle x,y\rangle^{\perp}\ .$

Since $\pi_{1}(\sigma)$ and $\pi_{1}(\tau)$ also take values in $\langle
x,y\rangle^{\perp}$, and since this space is $2$-dimensional, it follows that

${\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(\tau),D_{\xi}x,y)={\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(\tau),x,D_{\xi}y)=0\
.$

Thus,

$D_{\xi}\big{(}{\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(\tau),x,y)\big{)}={\text{Vol}}(\pi_{1}(D_{\xi}\sigma),\pi_{1}(\tau),x,y)+{\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(D_{\xi}\tau),x,y)\
.$

Let $p$ denote the projection onto $N$ along $W$, and note that

$\pi_{1}(D_{\xi}\sigma)-\pi_{1}(\nabla_{\xi}\sigma)=\pi_{1}(p(D_{\xi}\sigma))\in\langle
x,y\rangle\ ,$

with a similar identity holding for $\tau$. It follows that

$\eqalign{D_{\xi}\omega_{k}^{\prime}(\sigma,\tau)&=2D_{\xi}\big{(}{\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(\tau),x,y)\big{)}\cr&=2{\text{Vol}}(\pi_{1}(\nabla_{\xi}\sigma),\pi_{1}(\tau),x,y)+2{\text{Vol}}(\pi_{1}(\sigma),\pi_{1}(\nabla_{\xi}\tau),x,y)\cr&=\omega_{k}^{\prime}(\nabla_{\xi}\sigma,\tau)+\omega_{k}^{\prime}(\sigma,\nabla_{\xi}\tau)\
,\cr}$

so that

$\big{(}\nabla_{\xi}\omega_{k}^{\prime})(\sigma,\tau)=0\ ,$

as desired. $\square$

3.4 - Bilegendrian surfaces.  We now introduce the class of surfaces of
interest to us. We define an immersed surface in $M$ to be a pair $(S,\phi)$,
where $S$ is a smooth surface, and $\phi:S\rightarrow M$ is a smooth
immersion. We define an integral surface of $W$ to be an immersed surface
$(S,\phi)$ whose every tangent plane is contained in $W$, that is, such that
${\text{Im}}(D\phi(x))\subseteq W_{\phi(x)}$ for all $x\in S$. We define a
$(\omega_{i},\omega^{\eta}_{k})$-bilegendrian surface to be an integral
surface of $W$ such that

$\phi^{*}\omega_{i}=\phi^{*}\omega^{\eta}_{k}=0\ .$

In what follows, we will refer to such surfaces simply as bilegendrian
surfaces when no ambiguity arises. By Lemma 2.3.2, an integral surface of $W$
is a bilegendrian surface if and only if every one of its tangent planes is
$J^{\eta}$-invariant. We will say that a bilegendrian surface is of elliptic
type (resp hyperbolic type) whenever $J^{\eta}$ is positive (resp. negative).
Note that bilegendrian surfaces of elliptic type are immersed
$J^{\eta}$-holomorphic curves, justifying our terminology.

Given a bilegendrian surface $S$, we define its cubic form by

$C(\xi,\mu,\nu):=\omega_{i}(\nabla_{\xi}D\phi\cdot\mu,D\phi\cdot\nu)\ .$

Since $\omega_{i}$ vanishes over every tangent plane of $S$, its cubic form is
a tensor. It will also be useful to define its complementary cubic form by

$\hat{C}(\xi,\mu,\nu):=\omega_{i}(\nabla_{\xi}D\phi\cdot\mu,\alpha(D\phi\cdot\nu))\
,$

even though this object is not a tensor. The cubic form plays the same role in
lagrangian and legendrian geometries as the second fundamental form does in
riemannian hypersurface geometry. Indeed,

$C(\xi,\mu,\nu)=\hat{g}^{\eta}(\nabla_{\xi}D\phi\cdot\mu,\hat{I}^{\eta}D\phi\cdot\nu)\
,$

and since $\hat{I}^{\eta}$ sends every $\hat{J}^{\eta}$-invariant plane to its
$\hat{g}^{\eta}$-orthogonal complement, $C$ indeed encodes the normal
component of $\nabla_{\xi}D\phi\cdot\mu$ in $W$ at every $\hat{g}$-non-
degenerate point of $S$.

Lemma 3.4.1 The cubic form $C$ is symmetric.

Proof: Indeed, since $S$ is $\omega_{i}$-lagrangian,

$\eqalign{C(\xi,\mu,\nu)&=\omega_{i}(\nabla_{\xi}D\phi\cdot\mu,D\phi\cdot\nu)\cr&=\omega_{i}(\nabla_{\mu}D\phi\cdot\xi+D\phi\cdot[\xi,\mu],D\phi\cdot\nu)\cr&=\omega_{i}(\nabla_{\mu}D\phi\cdot\xi,D\phi\cdot\nu)\cr&=C(\mu,\xi,\nu)\
.\cr}$

Likewise, since $\omega_{i}$ is covariant constant and vanishes over $TS$,

$\eqalign{C(\xi,\mu,\nu)&=\omega_{i}(\nabla_{\xi}D\phi\cdot\mu,D\phi\cdot\nu)\cr&=D_{\xi}\omega_{i}(D\phi\cdot\mu,D\phi\cdot\nu)-\omega_{i}(D\phi\cdot\mu,\nabla_{\xi}D\phi\cdot\nu)\cr&=\omega_{i}(\nabla_{\xi}D\phi\cdot\nu,D\phi\cdot\mu)\cr&=C(\xi,\nu,\mu)\
.\cr}$

Since the transpositions $(12)$ and $(23)$ generate the group of permutations
of $\\{1,2,3\\}$, it follows that $C$ is symmetric, as desired. $\square$

Lemma 3.4.2 $J^{\eta}$ is $C$-symmetric. That is, for all $\xi$, $\mu$ and
$\nu$,

$C(J^{\eta}\xi,\mu,\nu)=C(\xi,J^{\eta}\mu,\nu)=C(\xi,\mu,J^{\eta}\nu)\
.\eqnum{\text{{\rm(3.6)}}\hbox{}}$

Proof: Indeed, since $\nabla J^{\eta}=0$,

$C(\xi,J^{\eta}\mu,\nu)=g^{\eta}(\nabla_{\xi}J^{\eta}\mu,I^{\eta}\nu)=g^{\eta}(J^{\eta}\nabla_{\xi}\mu,I^{\eta}\nu)=-g^{\eta}(\nabla_{\xi}\mu,J^{\eta}I^{\eta}\nu)=g^{\eta}(\nabla_{\xi}\mu,I^{\eta}J^{\eta}\nu)=C(\xi,\mu,J^{\eta}\nu)\
,$

and the result follows by symmetry of $C$. $\square$

Lemma 3.4.3 Let $\xi$ and $\mu$ be tangent vector fields over $S$. If $\mu$ is
a field of $J^{\eta}$-eigenvectors, then

$\hat{C}(\xi,\mu,\mu)=0\ .\eqnum{\text{{\rm(3.7)}}\hbox{}}$

Proof: Let $\epsilon\in\\{\pm 1\\}$ denote the $J^{\eta}$-eigenvalue of $\mu$.
Since $\hat{g}$ and $I$ are covariant constant, and since $I$ is
$\hat{g}$-symmetric,

$\hat{g}(\nabla_{\xi}D\phi\cdot\mu,I\cdot
D\phi\cdot\mu)=\frac{1}{2}D_{\xi}\hat{g}(D\phi\cdot\mu,I\cdot
D\phi\cdot\mu)=\frac{\epsilon}{2}D_{\xi}\hat{g}(D\phi\cdot\mu,I\cdot J\cdot
D\phi\cdot\mu)=\frac{\epsilon}{2}D_{\xi}\hat{g}(D\phi\cdot\mu,K\cdot
D\phi\cdot\mu)\ .$

Since $K$ is $\hat{g}$-antisymmetric, this vanishes, and the result follows.
$\square$Let $S$ be a bilegendrian surface. We say that a point $x\in S$ is
exceptional whenever $\hat{g}^{\eta}$ degenerates over the tangent space of
$S$ at this point. We denote the set of exceptional points of $S$ by
${\text{Exc}}(S)$. By Lemmas 2.4.4 and 2.4.5, in a neighbourhood of every non-
exceptional point, we may define a pair $(E_{1},E_{2})$ of (possibly complex)
fields of principal vectors, which is unique up to reordering and change of
sign. Furthermore, by Lemma 2.4.6, the principal vectors likewise define a
pair $(L_{1},L_{2})$ of line-valued functions such that, at each point,
$L_{1}(x)$ and $L_{2}(x)$ are orthogonal to one-another and also to the plane
$\langle(\pi_{1}\circ\phi)(x),(\pi_{2}\circ\phi)(x)\rangle$. We will call
these functions principal lines of $S$. These will be of use in the sequel.

Finally, when the principal directions are real, they integrate to yield two
transverse foliations of $S\setminus{\text{Exc}}(S)$ whose leaves we will call
lines of curvature of $S$.

Theorem 3.4.4, Existence of a flat metric The metric $\phi^{*}\hat{g}^{\eta}$
is flat over $S\setminus{\text{Exc}}(S)$, and the lines of curvature, when
they exist, are geodesics of this metric.

Proof: Let $\hat{\nabla}^{\eta}$ denote the Levi-Civita covariant derivative
of $\phi^{*}\hat{g}^{\eta}$ over $S\setminus{\text{Exc}}(S)$. By definition of
the principal vectors, we may suppose that $E_{2}=J^{\eta}E_{1}$. By
$\hat{g}^{\eta}$-symmetry of $J^{\eta}$, for all $\xi$ tangent to $S$,

$\hat{g}^{\eta}(\nabla_{\xi}E_{1},E_{2})=\hat{g}^{\eta}(\nabla_{\xi}E_{1},J^{\eta}\cdot
E_{1})=\frac{1}{2}D_{\xi}\hat{g}^{\eta}(E_{1},J^{\eta}\cdot
E_{1})-\hat{g}^{\eta}(E_{1},(\nabla_{\xi}J^{\eta})\cdot E_{1})=0\ .$

On the other hand,

$\hat{g}^{\eta}(\nabla_{\xi}E_{1},E_{1})=\frac{1}{2}D_{\xi}\hat{g}^{\eta}(E_{1},E_{1})=0\
.$

At every point where $\hat{g}^{\eta}$ is non-degenerate, this yields

$\hat{\nabla}^{\eta}_{\xi}E_{1}=0\ .$

In a similar manner, we show that

$\hat{\nabla}^{\eta}_{\xi}E_{2}=0\ .$

It follows that $(E_{1},E_{2})$ is a covariant constant frame of
$\phi^{*}\hat{g}^{\eta}$, and this completes the proof. $\square$

3.5 - Example I - Surfaces of constant positive curvature in $\mathbb{H}^{3}$.
Bilegendrian surfaces may be used to study CEC surfaces of non-zero curvature
in space forms. We will not provide an exhaustive treatment of this subject,
and will instead limit ourselves to two illustrative cases, namely positively
curved CEC surfaces in hyperbolic $3$-space $\mathbb{H}^{3}$, and negatively
curved CEC surfaces in euclidian $3$-space $\mathbb{R}^{3}$.

Let ${{b}}$ be a non-degenerate, symmetric bilinear form of signature $(3,1)$.
Recall that $\mathbb{H}^{3}$ identifies with a connected component of the
submanifold

$X:=\\{x\in\mathbb{R}^{4}\ |\ {{b}}(x,x)=-1\\}\ .$

Let $(S,e)$ be an immersed surface in $X$, let $\nu_{e}$ denote its unit
normal vector field, let ${\text{I}}_{e}$, ${\text{II}}_{e}$ and
${\text{III}}_{e}$ denote its three fundamental forms, and let $A_{e}$ denote
its shape operator. Since the tangent bundle of $X$ naturally embeds into the
trivial bundle $X\times\mathbb{R}^{4}$, we view $\nu_{e}$ as a function taking
values in $\mathbb{R}^{4}$. For $k>0$, we define the $k$-Gauss lift of $e$ by

$\hat{e}_{k}(x):=\bigg{(}e(x),\frac{1}{\sqrt{k}}\nu_{e}(x)\bigg{)}\ ,$

and we call $(S,\hat{e}_{k})$ the $k$-Gauss lift of $(S,e)$.

Lemma 3.5.1 For all $k>0$, $\hat{e}_{k}$ takes values in $M$. Furthermore, for
all $x\in S$, and for every tangent vector $\xi$ to $S$ at $x$,

$D\hat{e}_{k}(x)\cdot\xi=\bigg{(}De(x)\cdot\xi,\frac{1}{\sqrt{k}}De(x)\cdot
A_{e}(x)\cdot\xi\bigg{)}\ .\eqnum{\text{{\rm(3.8)}}\hbox{}}$

In particular, $(S,\hat{e}_{k})$ is $W$-horizontal and
$\omega_{i}$-lagrangian.

Proof: Indeed, by hypothesis

${{b}}(e(x),e(x))=0\ ,\ {{b}}(\nu_{e}(x),e(x))=0\ ,\ \text{and}\
{{b}}(\nu_{e}(x),\nu_{e}(x))=1\ ,$

from which it follows that the $k$-Gauss lift takes values in $M$.
Differentiating the second two relations yields

${{b}}(D\nu_{e}(x)\cdot\xi,e(x))=-{{b}}(De(x)\cdot\xi,\nu_{e}(x))\ \text{and}\
{{b}}(D\nu_{e}(x)\cdot\xi,\nu_{e}(x))=0\ .$

However, by definition

${{b}}(De(x)\cdot\xi,\nu_{e}(x))=0\ ,$

so that

${{b}}(D\nu_{e}(x)\cdot\xi,e(x))=0\ ,$

and $D\nu_{e}(x)$ therefore takes values in $\langle
e(x),\nu_{e}(x)\rangle^{\perp}$. Since $e$ is an immersion in
$\mathbb{H}^{3}$, this subspace is equal to the image of $De(x)$. However, for
all other tangent vectors $\mu$ of $S$ at $x$,

${{b}}(D\nu_{e}(x)\cdot\xi,De(x)\cdot\mu)={\text{II}}_{e}(\xi,\mu)={{b}}(De(x)\cdot
A_{e}(x)\cdot\xi,De(x)\cdot\mu)\ ,$

from which it follows that

$D\nu_{e}(x)\cdot\xi=De(x)\cdot A_{e}(x)\cdot\xi\ ,$

and this completes the proof. $\square$

Lemma 3.5.2 For $k>0$, $(S,e)$ has constant extrinsic curvature equal to $k$
if and only if its $k$-Gauss lift is $J^{+1}$-invariant.

Proof: Indeed, $(S,e)$ has constant extrinsic curvature equal to $k$ if and
only if ${\text{Det}}(A_{e})=k$. By (3.8), this holds if and only if the
$k$-Gauss lift is $\omega^{+1}_{k}$-lagrangian, and the result now follows by
Theorem 2.3.2. $\square$Theorem 3.4.4 now manifests itself in the following
form.

Theorem 3.5.3, Existence of a translation structure If $(S,e)$ has constant
extrinsic curvature equal to $k$, then the metric

$h_{e}:={\text{I}}_{e}-\frac{1}{k}{\text{III}}_{e}\eqnum{\text{{\rm(3.9)}}\hbox{}}$

is flat away from the set of umbilic points of $(S,e)$. Furthermore, the lines
of curvature are geodesics of this metric.

Remark 3.5.1. This metric has mixed signature away from the set of umbilic
points of $e$. Proof: Indeed, by (3.8),

$\hat{e}_{k}^{*}g^{-\eta}={\text{I}}_{e}-\frac{1}{k}{\text{III}}_{e}\ ,$

and the result now follows by Theorem 3.4.4. $\square$The metric constructed
in Theorem 3.5.3 is interpreted geometrically as follows. Let $x$ be a non-
umbilic point of $S$. Upon suitably reparametrizing a neighbourhood of $x$ by
an open subset $\Omega$ of $\mathbb{R}^{2}$, we may suppose that

$h_{e}=dx^{2}-dy^{2}\ ,$

and that the horizontal and vertical lines are lines of curvature of this
immersion. It is then straightforward to show that the fundamental forms of
$e$ are

${\text{I}}_{e}=\begin{pmatrix}{\text{cosh}}^{2}(\theta)\hfill&0\hfill\cr
0\hfill&{\text{sinh}}^{2}(\theta)\hfill\cr\end{pmatrix}\ ,\
{\text{II}}_{e}=\frac{\sqrt{k}}{2}\begin{pmatrix}{\text{sinh}}(2\theta)\hfill&0\hfill\cr
0\hfill&{\text{sinh}}(2\theta)\hfill\cr\end{pmatrix}\ ,\ \text{and}\
{\text{III}}_{e}=k\begin{pmatrix}{\text{sinh}}^{2}(\theta)\hfill&0\hfill\cr
0\hfill&{\text{cosh}}^{2}(\theta)\hfill\cr\end{pmatrix}\ ,$

for some smooth function $\theta:\Omega\rightarrow]0,\infty[$.

Recall now that the Hopf differential $\psi_{e}$ of $e$ is the trace-free
component of the first fundamental form with respect to the second fundamental
form. An elementary calculation then yields

$\psi_{e}=dzdz=(dx^{2}-dy^{2})+i(dxdy+dydx)\ .$

In particular, Theorem 3.5.3 shows that the Hopf differential is holomorphic.
This observation is in itself not very interesting, since holomorphicity of
$\psi_{e}$ is readily proven using more direct techniques. It is more
intriguing, however, to observe that $h_{e}$ is none other than the real part
$\psi_{e}$. In other words, $h_{e}$ is the metric of the well-known flat
$(\mathbb{Z}_{2}\ltimes\mathbb{R}^{2})$-structure defined over the complement
of the umbilic set of $S$ upon integrating the square root of its Hopf
differential.

3.6 - Example II - Surfaces of constant negative curvature in
$\mathbb{R}^{3}$.  Since $\mathbb{R}^{3}$ cannot be expressed as the level set
of a non-degenerate bilinear form over $\mathbb{R}^{4}$, it is necessary to
modify slightly our construction of $M$ and $W$. Although we could treat
$\mathbb{R}^{3}$ as the limit of hyperbolic space as the length of the time
direction tends to infinity, it is simpler to proceed more directly as
follows. Let ${{b}}$ be a symmetric, non-degenerate bilinear form over
$\mathbb{R}^{3}$. We define

$M:=\\{(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\ |\ b(y,y)\neq 0\\}\ ,$

and, for all $(x,y)\in M$, we define

$W_{(x,y)}:=\\{(\xi,\mu)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\ |\
b(\xi,x)=b(\mu,x)=0\\}\ .$

The Clifford structure is then constructed as before, with the pseudo-
involutions $I^{\eta}$, $J^{\eta}$ and $K$ defined as in (3.3). The reader may
verify that the results of Sections 3.3 and 3.4 continue to apply in this
case.

As before, let $(S,e)$ be an immersed surface in $\mathbb{R}^{3}$, let
$\nu_{e}$ denote its unit normal vector field, let ${\text{I}}_{e}$,
${\text{II}}_{e}$ and ${\text{III}}_{e}$ denote its three fundamental forms,
and let $A_{e}$ denote its shape operator. We now view $\nu_{e}$ as a function
taking values in $\mathbb{S}^{2}$ and, for $k>0$, we define the $k$-Gauss lift
of $e$ by

$\hat{e}_{k}(x):=\bigg{(}e(x),\frac{1}{\sqrt{k}}\nu_{e}(x)\bigg{)}\ ,$

and we call $(S,\hat{e}_{k})$ the $k$-Gauss lift of $(S,e)$. We readily verify
that (3.8) continues to hold in this context, and we obtain the following
result.

Lemma 3.6.1 For $k>0$, $(S,e)$ has constant extrinsic curvature equal to
$(-k)$ if and only if its $k$-Gauss lift is $J^{-1}$-invariant.

Remark 3.6.1. As in Lemma 3.5.2, the sign of $J$ is the same as that of the
curvature. When the surfaces in question are of mixed signature, the situation
is reversed, and the sign of $J$ is the opposite of that of the curvature. We
will not address this case in the present paper. Proof: Indeed, $(S,e)$ has
constant extrinsic curvature equal to $(-k)$ if and only if
${\text{Det}}(A_{e})=(-k)$. By (3.8), this holds if and only if the $k$-Gauss
lift is $\omega^{-1}_{k}$-lagrangian, and the result now follows by Theorem
2.3.2. $\square$The manifestation of Theorem 3.4.4 in the present context is
as follows.

Theorem 3.6.2, Existence of an asymptotic Chebyshev net If $(S,e)$ has
constant extrinsic curvature equal to $(-k)$, then the metric

$h_{e}:={\text{I}}_{e}+\frac{1}{k}{\text{III}}_{e}\eqnum{\text{{\rm(3.10)}}\hbox{}}$

is flat. Furthermore, the lines of curvature are geodesics of this metric.

Remark 3.6.2. In this case, there are no umbilic points and $h_{e}$ is
everywhere positive-definite. Proof: Indeed, by (3.8),

$\hat{e}_{k}^{*}g^{-\eta}={\text{I}}_{e}+\frac{1}{k}{\text{III}}_{e}\ ,$

and the result now follows by Theorem 3.4.4. $\square$It remains only to
discuss the geometric interpretation of the metric constructed in Theorem
3.6.2. We first suppose that $(S,e)$ is simply-connected. We will say that
$(S,e)$ is quasicomplete whenever it is complete with respect to the metric
${\text{I}}_{e}+{\text{III}}_{e}$. Note that this property follows trivially
from completeness. When this holds, $S$ may be globally parametrized by
$\mathbb{R}^{2}$ such that

$h_{e}=dx^{2}+dy^{2}\ .$

It is now straightforward to show that the fundamental forms of $e$ are

${\text{I}}_{e}=\begin{pmatrix}{\text{cos}}^{2}(\theta)\hfill&0\hfill\cr
0\hfill&{\text{sin}}^{2}(\theta)\hfill\cr\end{pmatrix}\ ,\
{\text{II}}_{e}=\frac{\sqrt{k}}{2}\begin{pmatrix}{\text{sin}}(2\theta)\hfill&0\hfill\cr
0\hfill&-{\text{sin}}(2\theta)\hfill\cr\end{pmatrix}\ ,\ \text{and}\
{\text{III}}_{e}=k\begin{pmatrix}{\text{sin}}^{2}(\theta)\hfill&0\hfill\cr
0\hfill&{\text{cos}}^{2}(\theta)\hfill\cr\end{pmatrix}\
.\eqnum{\text{{\rm(3.11)}}\hbox{}}$

for some smooth function $\theta:\mathbb{R}^{2}\rightarrow]0,\pi/2[$. In
particular, the diagonals in $\mathbb{R}^{2}$ are the asymptotic lines of $e$,
so that this parametrization is none other than the asymptotic Chebyshev net
of $S$, used in the proof of Hilbert’s theorem concerning the non-existence of
complete, negatively curved CEC surfaces in $\mathbb{R}^{3}$ (see $[48]$).

3.7 - Example III - Hilbert’s theorem.  Our initial aim in writing this paper
was to review the proof of Hilbert’s theorem as part of a survey of recent
developments in the theory of CEC surfaces in $3$-dimensional space forms.
Since the proof of this result is both interesting and elementary once the
asymptotic Chebyshev net has been constructed, for the sake of completeness,
we discuss it here.

We will address a more general case than that treated in Section 3.6. For
$c\in\mathbb{R}$, we denote by $X^{3}_{c}$ the $3$-dimensional riemannian
space-form of constant sectional curvature equal to $c$. For $k>0$, let
$(S,e)$ be a complete CEC surface in $X^{3}_{c}$ of curvature equal to $(-k)$.
Reasoning as in Section 3.6, we show that $S$ is globally parametrized by
$\mathbb{R}^{2}$ in such a manner that the first, second and third fundamental
forms are given by (3.11), for some smooth function
$\theta:\mathbb{R}^{2}\rightarrow]0,\pi/2[$.

Lemma 3.7.1 The function $\theta$ satisfies

$(\theta_{xx}-\theta_{yy})dxdy=(k-c){\text{dArea}}_{e}\
,\eqnum{\text{{\rm(3.12)}}\hbox{}}$

where ${\text{dArea}}_{e}$ here denotes the area form of the first fundamental
form of $e$.

Proof: Indeed, using the above notation, we define the oriented, orthonormal
frame $(u_{x},u_{y})$ of ${\text{I}}_{e}$ by

$u_{x}:=\frac{1}{{\text{cos}}(\theta)}\partial_{x}\ \text{and}\
u_{y}:=\frac{1}{{\text{sin}}(\theta)}\partial_{y}\ .$

The commutator of this frame is given by

$[u_{x},u_{y}]=\frac{1}{{\text{cos}}(\theta){\text{sin}}(\theta)}\big{(}-{\text{cot}}(\theta)(\partial_{x}\theta)\partial_{y}-{\text{tan}}(\theta)(\partial_{y}\theta)\partial_{x}\big{)}\
,$

and its connection form is therefore

$\alpha=[u_{x},u_{y}]^{\flat}=-(\partial_{y}\theta)dx-(\partial_{x}\theta)dy\
,$

where $\flat$ here denotes Berger’s musical isomorphism. It follows by Gauss’
equation that

$(\theta_{xx}-\theta_{yy})dxdy=-d\alpha=(k-c){\text{dA}}_{e}\ ,$

as desired. $\square$

Corollary 3.7.2 The function $\theta$ satisfies the hyperbolic sine-Gordon
equation

$\theta_{xx}-\theta_{yy}=\frac{(k-c)}{2}{\text{sin}}(2\theta)\
.\eqnum{\text{{\rm(3.13)}}\hbox{}}$

Proof: Indeed, with the above notation, the area form of ${\text{I}}_{e}$ is

${\text{dArea}}_{e}=\frac{1}{2}{\text{sin}}(2\theta)dxdy\ ,$

and the result follows by (3.12). $\square$Hilbert’s theorem now follows as a
straightforward consequence of Hazzidaki’s lemma, which we now recall. Let
$[\theta]_{0}$ denote the total oscillation of $\theta$, that is

$[\theta]_{0}:=\mathop{{\text{Sup}}}_{x\neq
y\in\mathbb{R}^{2}}\left|\theta(x)-\theta(y)\right|\ .$

Lemma 3.7.3, Hazzidaki Let $\theta:\mathbb{R}^{2}\rightarrow]0,\pi/2[$ be a
smooth function, consider the metric

$g_{\theta}:={\text{cos}}^{2}(\theta)dx^{2}+{\text{sin}}^{2}(\theta)dy^{2}\
,\eqnum{\text{{\rm(3.14)}}\hbox{}}$

and let $\kappa_{\theta}$ denote its curvature. If $\kappa_{\theta}$ does not
change sign, then

$\int_{\mathbb{R}^{2}}\left|\kappa_{\theta}\right|{\text{dA}}_{\theta}\leq
4[\theta]_{0}\ .\eqnum{\text{{\rm(3.15)}}\hbox{}}$

Proof: Indeed, by (3.12),

$\kappa_{\theta}{\text{dArea}}_{\theta}=(\theta_{xx}-\theta_{yy})dxdy\ .$

We define the new variables $u:=x+y$ and $v:=x-y$, and we verify that

$\int_{\mathbb{R}^{2}}\left|\kappa_{\theta}\right|{\text{dA}}_{\theta}=\int_{\mathbb{R}^{2}}\left|\theta_{xx}-\theta_{yy}\right|dxdy=\int_{\mathbb{R}^{2}}2\left|\theta_{uv}\right|dudv\
.$

Since $\theta_{uv}$ does not change sign,

$\eqalign{\int_{\mathbb{R}^{2}}2\left|\theta_{uv}\right|dudv&=\mathop{{\text{Lim}}}_{R\rightarrow\infty}2\left|\int_{[-R,R]^{2}}\theta_{uv}dudv\right|\cr&=\mathop{{\text{Lim}}}_{R\rightarrow\infty}2\left|\theta(R,R)-\theta(R,-R)-\theta(-R,R)+\theta(-R,-R)\right|\vphantom{\bigg{(}}\cr&\leq
4[\theta]_{0}\ ,\vphantom{\bigg{(}}\cr}$

and the result follows. $\square$

Theorem 3.7.4, Hilbert If $(-k)<{\text{Min}}(0,-c)$, then there exists no
complete CEC surface in $X^{3}_{c}$ of curvature equal to $(-k)$.

Proof: Suppose the contrary. Let $(S,e)$ be a complete CEC surface in
$X^{3}_{c}$ of curvature equal to $(-k)$. We may suppose that $S$ is simply-
connected, and since it is complete, in particular, it is quasicomplete, so
that, by Hazzidaki’s theorem, it has finite total curvature. However, since
$(c-k)<0$, up to rescaling by a constant factor, $(S,{\text{I}}_{e})$ is
isometric to $\mathbb{H}^{2}$, which has infinite total curvature. This is
absurd, and the result follows. $\square$It is worth noting that when $c>0$
and $-c<-k<0$, Hilbert’s theorem still holds, but with a slightly different
proof.

Theorem 3.7.5, Hilbert If $c>0$ and $-c<k<0$, then there exists no complete
CEC surface in $X^{3}_{c}$ of curvature equal to $(-k)$.

Proof: Suppose the contrary, and let $(S,e)$ be such a surface. We may suppose
that $S$ is simply connected. Since it is complete, it is quasicomplete, so
that, by the preceding disussion, it is globally parametrized by
$\mathbb{R}^{2}$. However, since $S$ is complete and of constant, positive
curvature, it is diffeomorphic to the sphere. This is absurd, and the result
follows. $\square$We conclude this section with the following intriguing
corollary of Hilbert’s theorem.

Theorem 3.7.6 For all $\epsilon>0$, there exists no smooth solution
$\theta:\mathbb{R}^{2}\rightarrow[\epsilon,\pi-\epsilon]$ of the hyperbolic
sine-Gordon equation

$\theta_{xx}-\theta_{tt}-{\text{sin}}(\theta)=0\
.\eqnum{\text{{\rm(3.16)}}\hbox{}}$

It is worthwhile reflecting on the physical meaning of this result. The
$1$-dimensional sine-Gordon equation

$\phi_{tt}+{\text{sin}}(\phi)=0$

models the oscillations of a classical pendulum (see, for example, $[18]$).
The corresponding analogue of Theorem 3.7.6 would thus affirm the intuitively
obvious fact that a classical pendulum must eventually approach its axis. The
$2$-dimensional sine-Gordon equation models the motion of an analogous system
which we may view, either as the limit of an infinite row of such pendula
joined to their neighbours by springs, or, more intuitively, as a very long
vertical blind. Theorem 3.7.6 thus states, once again, that such a system must
also eventually approach its axis. It is quite striking that the proof of such
a seemingly reasonable fact should rest on something as abstract as the
infinite total curvature of hyperbolic space!

4 - Bilegendrian surfaces in ${\text{\rm U}}\mathbb{S}^{3}$. 4.1 - Overview.
We conclude this paper by studying bilegendrian surfaces in ${\text{\rm
U}}\mathbb{S}^{3}$, which generalize flat surfaces in $\mathbb{S}^{3}$. The
first known examples of flat surfaces in $\mathbb{S}^{3}$ were probably the
Clifford torus, introduced by Clifford himself in $[5]$ (see also $[50]$),
followed by those constructed by Bianchi in $[2]$, presented in a modern
manner using Hopf fibrations by Spivak in $[48]$ (see also the work $[39]$ of
Sasaki). Spivak also presented a simple construction of compact, embedded flat
surfaces using Hopf projections, which was later studied in greater detail by
Pinkall in $[36]$.

In $[52]$, Yau posed the problem of classifying all compact flat surfaces in
$\mathbb{S}^{3}$. This problem was solved independently by Kitigawa in $[22]$
and by Weiner in $[51]$ using entirely different techniques. Indeed, whilst
Kitigawa builds on Spivak’s work, Weiner uses the factorizations of the Gauss
maps of flat surfaces developed by Enomoto in $[10]$ and $[11]$.

Our approach will contain elements of $[22]$ and $[51]$, but will differ from
both. In particular, the approaches of both these papers break down when the
projection onto the base space degenerates. Instead, we will show in Sections
4.3 and 4.4 how the $J$-invariant condition naturally interacts with the Lie
group structure of $\mathbb{S}^{3}$ to yield in Theorem 4.4.1 canonical
factorizations of complete bilegendrian immersions, allowing us, in
particular, to construct all such surfaces. Furthermore, in Theorem 4.4.3, we
factorize the period lattices of compact bilegendrian immersions, and,
together with Theorem 4.5.5, this will establish a general formula for
constructing all such immersions. We note, in particular, that both of these
factorizations follow in a straightforward manner from a general factorization
criterion for Lie group-valued functions over $\mathbb{R}^{2}$, and are thus,
in a sense, more algebraic in nature than geometric.

Complete bilegendrian immersions in ${\text{\rm U}}\mathbb{S}^{3}$ strike us
as more natural objects than complete flat immersions in $\mathbb{S}^{3}$.
Indeed, as we will see in Theorem 4.6.7, every complete bilegendrian immersion
in ${\text{\rm U}}\mathbb{S}^{3}$ is uniquely defined, up to isometries of
$\mathbb{S}^{3}$, by its angle function
$\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}$, which is a smooth solution of
the classical wave equation

$\partial_{x}^{2}\theta-\partial_{y}^{2}\theta=0\
.\eqnum{\text{{\rm(4.1)}}\hbox{}}$

However, wheras every such solution of (4.1) is the angle function of some
complete, bilegendrian immersion in ${\text{\rm U}}\mathbb{S}^{3}$, it is
still a hard and unsolved problem to fully determine necessary and sufficient
conditions for such a solution of (4.1) to be the angle function of a complete
flat immersion in $\mathbb{S}^{3}$.

In the doubly-periodic case, the distinction is even more striking, as the
family of compact bilegendrian immersions in ${\text{\rm U}}\mathbb{S}^{3}$ is
notably larger than that of compact flat immersions in $\mathbb{S}^{3}$.
Indeed, wheras any pair of closed curves in $\mathbb{S}^{2}$ bounding areas
which are rational multiples of $2\pi$ can arise as the factors of some doubly
periodic bilegendrian immersion in ${\text{\rm U}}\mathbb{S}^{3}$, by Theorem
4.6.8, only those curves which bound areas which are integer multiples of
$2\pi$ are candidates to arise as factors of compact flat immersions in
$\mathbb{S}^{3}$. This has interesting consequences. For example, for any
fixed pair $(q_{1},q_{2})$ of rational numbers, we expect a rigidity result
analogous to the result $[12]$ of Enomoto–Kitigawa–Weiner to hold for compact
bilegendrian immersions with factors that are respectively $q_{1}$\- and
$q_{2}$-quasiperiodic. We propose to address this problem in later work.

4.2 - Quaternions.  Our study of complete bilegendrian immersed surfaces in
${\text{\rm U}}\mathbb{S}^{3}$ will rely heavily on the structure of the
algebra $\mathbb{H}$ of quaternions, whose basic properties we now review.
With the terminology of Section 2.3, $\mathbb{H}$ is nothing other than the
Clifford algebra of a positive-definite, symmetric bilinear form over a
$2$-dimensional real vector space. That is, it is the real algebra generated
by the identity together with elements ${{i}}$, ${{j}}$ and ${{k}}$ satisfying

${{i}}^{2}={{j}}^{2}={{k}}^{2}=-1\ \text{and}\ {{i}}\cdot{{j}}\cdot{{k}}=-1\
.$

Let $\Cal{R}$ and $\Cal{I}$ denote respectively its real and imaginary
subspaces. We define the inner product ${{b}}$ over $\mathbb{H}$ by

${{b}}(x,y):=\Cal{R}(x\cdot\overline{y})\ .$

In this case, ${{b}}$ is positive-definite, so that the set of unit-length
quaternions identifies with the unit sphere $\mathbb{S}^{3}$, that is

$\mathbb{S}^{3}=\\{x\in\mathbb{H}\ |\ \|x\|_{{b}}^{2}=1\\}\ .$

Since length is multiplicative, $\mathbb{S}^{3}$ also inherits from
$\mathbb{H}$ the structure of a Lie group, with Lie algebra $\Cal{I}$. In
particular, the inverse of any element $x$ of $\mathbb{S}^{3}$ is just its
quaternionic conjugate $\overline{x}$.

We view ${\text{\rm U}}\mathbb{S}^{3}$ as a submanifold of
$\mathbb{H}\times\mathbb{H}$ view the identification

${\text{\rm U}}\mathbb{S}^{3}:=\\{(x,y)\in\mathbb{H}\times\mathbb{H}\ |\
{{b}}(x,x)={{b}}(y,y)=1\ ,\ {{b}}(x,y)=0\\}\ .$

In particular, ${\text{\rm U}}\mathbb{S}^{3}$ is a codimension $2$ submanifold
of the manifold $M$ studied in Section 3.3. The subbundle $W$ of $TM$
restricts to a codimension $1$ subbundle of the tangent bundle of ${\text{\rm
U}}\mathbb{S}^{3}$, which we also denote by $W$.

Lemma & Definition 4.2.1 For all $(x,y)\in{\text{\rm U}}\mathbb{S}^{3}$, and
for all $z\in\langle x,y\rangle^{\perp}$,

$-z\cdot\overline{x}\cdot y=y\cdot\overline{x}\cdot z\in\langle
x,y\rangle^{\perp}\ .\eqnum{\text{{\rm(4.2)}}\hbox{}}$

We thus define

$A\cdot z:=A_{(x,y)}\cdot z:=-z\cdot\overline{x}\cdot
y=y\cdot\overline{x}\cdot z\ .\eqnum{\text{{\rm(4.3)}}\hbox{}}$

This map is a rotation by $\pi/2$ radians about the axis $y$ in the positive
direction with respect to the canonical orientation of $\mathbb{S}^{3}$.

Proof: We first show that $z\cdot\overline{x}\cdot y=-y\cdot\overline{x}\cdot
z$. Since $x$, $y$ and $z$ are pairwise orthogonal to one-another, the
elements $\overline{x}\cdot y$ and $\overline{x}\cdot z$ are imaginary, and
also orthogonal to one-another. Since orthogonal, imaginary quaternions
anticommute,

$z\cdot\overline{x}\cdot y=x\cdot(\overline{x}\cdot z)\cdot(\overline{x}\cdot
y)=-x\cdot(\overline{x}\cdot y)\cdot(\overline{x}\cdot
z)=-y\cdot\overline{x}\cdot z\ ,$

as desired.

We now show that this product lies in $\langle x,y\rangle^{\perp}$. Indeed, by
orthogonality,

$\overline{x}\cdot y=-\overline{y}\cdot x\ ,$

so that

${{b}}(z\cdot\overline{x}\cdot y,x)=\Cal{R}(z\cdot\overline{x}\cdot
y\cdot\overline{x})=-\Cal{R}(z\cdot\overline{y}\cdot
x\cdot\overline{x})=-\Cal{R}(z\cdot\overline{y})=-{{b}}(z,y)=0\ .$

Likewise

${{b}}(z\cdot\overline{x}\cdot y,y)=\Cal{R}(z\cdot\overline{x}\cdot
y\cdot\overline{y})=\Cal{R}(z\cdot\overline{x})={{b}}(z,x)=0\ ,$

so that $z\cdot\overline{x}\cdot y$ is indeed an element of $\langle
x,y\rangle^{\perp}$, as asserted.

It remains only to verify the geometric properties of $A$. However,

$A^{2}\cdot z=z\cdot\overline{x}\cdot y\cdot\overline{x}\cdot
y=-z\cdot\overline{x}\cdot x\cdot\overline{y}\cdot y=-z\ ,$

so that $A$ is a positive pseudo-involution, that is, a rotation by $\pi/2$
radians about the axis $y$. Finally, setting $(x,y)=(1,{{k}})$ and $z={{i}}$
yields $A\cdot z=-{{i}}\cdot{{k}}={{j}}$, so that $A$ indeed turns in the
positive direction about $y$ at this point. The result for general $(x,y)$
then holds by continuity, and this completes the proof. $\square$We now set
$\eta=-1$ and will suppress this superscript throughout the sequel. In terms
of $A$, the endomorphism fields $I$, $J$ and $K$ constructed in Section 3.3
are given by

$I=\begin{pmatrix}0\hfill&{\text{Id}}\hfill\cr{\text{Id}}\hfill&0\hfill\cr\end{pmatrix}\
,\ J=\begin{pmatrix}0\hfill&-A\hfill\cr A\hfill&0\hfill\cr\end{pmatrix}\ ,\
\text{and}\ K=\begin{pmatrix}A\hfill&0\hfill\cr
0\hfill&-A\hfill\cr\end{pmatrix}\ .\eqnum{\text{{\rm(4.4)}}\hbox{}}$

In particular,

$J^{2}={\text{Id}}\ .$

Finally, we identify $\mathbb{S}^{2}$ with the set of unit-length imaginary
quaternions, that is

$\mathbb{S}^{2}:=\\{x\in\Cal{I}\ |\ \|x\|_{{b}}^{2}=1\\}\ ,$

and we view ${\text{T}}\mathbb{S}^{2}$ as a submanifold of
$\Cal{I}\times\Cal{I}$ via the identification

${\text{T}}\mathbb{S}^{2}:=\\{(\xi,\mu)\ |\ \|\xi\|=1\ ,\
\langle\xi,\mu\rangle=0\\}\ .$

We recall the adjoint action
${\text{ad}}:\mathbb{S}^{3}\rightarrow{\text{SO}}(\Cal{I})$, given by

${\text{ad}}(x)\xi:=x\cdot\xi\cdot x^{-1}\ .$

For any unit-length imaginary $x$, we define the Hopf projection
$\pi_{x}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{2}$ by

$\pi_{x}(y):={\text{ad}}(y)x\ .$

For all such $x$, the preimage of $x$ under the Hopf projection is the
$1$-parameter subgroup

$\mathbb{S}^{1}_{x}:=\mathbb{S}^{3}\mathop{\cap}\langle 1,x\rangle\ .$

In particular, $\pi_{x}$ projects to a diffeomorphism from
$\mathbb{S}^{3}/\mathbb{S}^{1}_{x}$ to $\mathbb{S}^{2}$, which we also denote
by $\pi_{x}$.

4.3 - Algebraic properties of bilegendrian surfaces.  Let $(S,\phi)$ be a
complete, simply-connected, bilegendrian immersed surface in ${\text{\rm
U}}\mathbb{S}^{3}$, let $C$ and $\hat{C}$ denote respectively its cubic and
complementary cubic forms, and let $X,Y:S\rightarrow\mathbb{S}^{3}$ denote its
two components, that is

$X:=\pi_{1}\circ\phi\ \text{and}\ Y:=\pi_{2}\circ\phi\ .$

By Theorem 3.4.4, we may suppose that $S=\mathbb{R}^{2}$, and that

$\phi^{*}\hat{g}=2(dx_{1}^{2}+dx_{2}^{2})\ .\eqnum{\text{{\rm(4.5)}}\hbox{}}$

We may suppose furthermore that the diagonals $(\partial_{1}+\partial_{2})/2$
and $(\partial_{1}-\partial_{2})/2$ are principal vectors. Since

$J\cdot(\partial_{1}+\partial_{2})/2=(\partial_{1}-\partial_{2})/2\ ,$

it follows that $\partial_{1}$ and $\partial_{2}$ are eigenvectors of $J$ with
respective eigenvalues $(+1)$ and $(-1)$. In particular, since $J$ is
$g$-antisymmetric, for each $i$,

$\phi^{*}g(\partial_{i},\partial_{i})=0\ ,$

so that

${{b}}(\partial_{i}X,\partial_{i}X)=\frac{1}{2}(g+\hat{g})(\partial_{i}\phi,\partial_{i}\phi)=1\
.\eqnum{\text{{\rm(4.6)}}\hbox{}}$

Finally, since $J$ is $C$-symmetric,

$C(\partial_{1},\partial_{2},\partial_{2})=C(\partial_{2},\partial_{1},\partial_{1})=0\
.\eqnum{\text{{\rm(4.7)}}\hbox{}}$

Likewise, by (3.7), for every vector field $\xi$, and for each $i$,

$\hat{C}(\xi,\partial_{i},\partial_{i})=0\ .\eqnum{\text{{\rm(4.8)}}\hbox{}}$

Lemma 4.3.1 The first derivatives of $X$ and $Y$ satisfy the following
algebraic relations.

$\partial_{1}(Y\cdot X^{-1})=\partial_{2}(X^{-1}\cdot Y)=0\
.\eqnum{\text{{\rm(4.9)}}\hbox{}}$

Proof: It suffices to prove the first identity, as the proof of the second is
identical. By the product and chain rules,

$\partial_{1}(Y\cdot X^{-1})=(\partial_{1}Y-Y\cdot
X^{-1}\cdot\partial_{1}X)\cdot X^{-1}\ .$

However, since $\partial_{1}$ is an eigenvector of $J$ with eigenvalue $(+1)$,
by (4.3) and (4.4),

$\partial_{1}Y=A\cdot\partial_{1}X=Y\cdot X^{-1}\cdot\partial_{1}X\ ,$

so that

$\partial_{1}(Y\cdot X^{-1})=0\ ,$

as desired. $\square$

Lemma 4.3.2 The second derivatives of $X$ and $Y$ satisfy the following
algebraic relations.

$\partial_{2}(X^{-1}\cdot(\partial_{1}X))=\partial_{1}((\partial_{2}X)\cdot
X^{-1})=0\ .\eqnum{\text{{\rm(4.10)}}\hbox{}}$

Proof: It suffices to prove the first identity, as the proof of the second is
identical. Since $X$ takes values in $\mathbb{S}^{3}$,
$X^{-1}\cdot(\partial_{1}X)$ takes values in $\Cal{I}$. Since
$(X^{-1}\cdot(\partial_{1}X),X^{-1}\cdot Y,X^{-1}\cdot Y\cdot
X^{-1}\cdot(\partial_{1}X))$ is an orthonormal basis of $\Cal{I}$, it suffices
to show that the inner-products of $\partial_{2}(X^{-1}\cdot(\partial_{1}X))$
with each of these elements vanish.

By (4.6), $\partial_{1}X$ has constant unit length, and therefore so too does
$X^{-1}\cdot(\partial_{1}X)$. It follows that

${{b}}(\partial_{2}(X^{-1}\cdot(\partial_{1}X)),X^{-1}\cdot(\partial_{1}X))=\frac{1}{2}\partial_{2}{{b}}(X^{-1}\cdot(\partial_{1}X),X^{-1}\cdot(\partial_{1}X))=0\
,$

as desired.

By definition, $Y$ is orthogonal to $\partial_{1}X$, so that $X^{-1}\cdot Y$
is orthogonal to $X^{-1}\cdot(\partial_{1}X)$. By (4.9), it follows that

${{b}}(\partial_{2}(X^{-1}\cdot(\partial_{1}X)),X^{-1}\cdot
Y)=-{{b}}(X^{-1}\cdot(\partial_{1}X),\partial_{2}(X^{-1}\cdot Y))=0\ ,$

as desired.

Finally, by the chain and product rules,

$\eqalign{{{b}}(\partial_{2}(X^{-1}\cdot(\partial_{1}X)),X^{-1}\cdot Y\cdot
X^{-1}\cdot(\partial_{1}X))&=-{{b}}(X^{-1}\cdot(\partial_{2}X)\cdot
X^{-1}\cdot(\partial_{1}X),X^{-1}\cdot Y\cdot
X^{-1}\cdot(\partial_{1}X))\cr&\qquad\qquad+{{b}}(X^{-1}\cdot(\partial_{2}\partial_{1}X),X^{-1}\cdot
Y\cdot X^{-1}\cdot(\partial_{1}X))\ .\cr}$

Since $Y$ and $\partial_{2}X$ are orthogonal to one-another, and since
multiplications on the left and on the right by unit quaternions yield
isometries of $\mathbb{H}$, the first term on the right-hand side vanishes.
Likewise, by (4.4), the second term is equal to

${{b}}(D_{\partial_{2}}(\partial_{1}X),A\cdot\partial_{1}X)=\frac{1}{2}(g+\hat{g})(\nabla_{\partial_{2}}(\partial_{1}\phi),K\cdot\partial_{1}\phi)\
.$

Since $K=IJ$, and since $\partial_{1}$ is an eigenvector of $J$ with
eigenvalue $(+1)$, this is equal to

$\frac{1}{2}(g+\hat{g})(\nabla_{\partial_{2}}(\partial_{1}\phi),I\cdot\partial_{1}\phi)=\frac{1}{2}C(\partial_{2},\partial_{1},\partial_{1})+\frac{1}{2}\hat{C}(\partial_{2},\partial_{1},\partial_{1})\
,$

which vanishes by (4.7) and (4.8). This completes the proof. $\square$

4.4 - Factorization.  We now use the algebraic identities of the preceding
section to derive factorizations of complete bilegendrian immersions and, in
the doubly-periodic case, of their period lattices. We first recall some
further elementary definitions from the theory of Lie groups. We define the
left and right translations
$T_{l},T_{r}:\mathbb{S}^{3}\times\mathbb{H}\rightarrow\mathbb{S}^{3}\times\mathbb{H}$
by

$T_{l}(x,\xi):=(x,x\cdot\xi)\ \text{and}\ T_{r}(x,\xi):=(x,\xi\cdot x)\ .$

Both $T_{l}$ and $T_{r}$ restrict to bundle isomorphisms from
$\mathbb{S}^{3}\times\Cal{I}$ into ${\text{T}}\mathbb{S}^{3}$. For every
imaginary quaternion $\xi$, we define its left and right translates
respectively by

$\xi_{l}(x):=T_{l}(x,\xi)\ \text{and}\ \xi_{r}(x):=T_{r}(x,\xi)\ .$

Finally, when $\xi$ is non-zero, we say that a smooth curve
$\gamma:I\rightarrow\mathbb{S}^{3}$ is left $\xi$-horizontal (resp. right
$\xi$-horizontal) whenever $\dot{\gamma}$ is everywhere orthogonal to
$\xi_{l}\circ\gamma$ (resp. $\xi_{r}\circ\gamma$).

Theorem 4.4.1, Factorization of immersions Let
$\phi:\mathbb{R}^{2}\rightarrow{\text{\rm U}}\mathbb{S}^{3}$ be a complete
bilegendrian immersion. There exist a pair of orthogonal elements
$a,b\in\mathbb{S}^{3}$, a right $(\overline{a}\cdot b)$-horizontal arc-length
parametrized curve $\gamma_{1}:\mathbb{R}\rightarrow\mathbb{S}^{3}$, and a
left $(\overline{b}\cdot a)$-horizontal arc-length parametrized curve
$\gamma_{2}:\mathbb{R}\rightarrow\mathbb{S}^{3}$, such that
$\gamma_{1}(0)=\gamma_{2}(0)=1$, and

$\phi(x_{1},x_{2})=(\gamma_{2}(x_{2})\cdot
a\cdot\gamma_{1}(x_{1}),\gamma_{2}(x_{2})\cdot b\cdot\gamma_{1}(x_{1}))\
.\eqnum{\text{{\rm(4.11)}}\hbox{}}$

Furthermore, $a$, $b$, $\gamma_{1}$ and $\gamma_{2}$ are unique. Conversely,
every function of this form is a complete, bilegendrian immersion.

In the case where the bilegendrian surface projects to a flat immersed surface
in $\mathbb{S}^{3}$, two distinct factorizations are already known from the
work $[22]$ of Kitigawa and $[51]$ of Weiner. It is worth reviewing how these
factorizations relate to our own. We first address Kitigawa’s construction.
Here, the first component of (4.11) identifies with the factorization
described at the beginning of Section $4$ of $[22]$. Note, however, that
whilst the factors are constructed in $[22]$ by modifying horizontal lifts of
curves in $\mathbb{S}^{2}$ (see Section $3$ of that paper), in the present
case, the factors are obtained directly as horizontal lifts without
modification upon correctly choosing the Hopf projections, as we will see
presently.

We now discuss Weiner’s construction, which uses the following ideas developed
by Enomoto in $[10]$ and $[11]$. Let ${\text{Gr}}^{+}(2,4)$ denote the
grassmannian of oriented linear planes in $\mathbb{R}^{4}$. In the case where
$X$ is an immersion, Enomoto defines its Gauss map to be the function
$G:\mathbb{R}^{2}\rightarrow{\text{Gr}}^{+}(2,4)$ which maps every point
$(x_{1},x_{2})$ of $\mathbb{R}^{2}$ to the tangent plane of $X$ at this point.
In order to express Enomoto’s Gauss map in terms of the curves
$(\gamma_{1},\gamma_{2})$ of the factorization (4.11), it is necessary to
recall how ${\text{Gr}}^{+}(2,4)$ is described as a quotient of
$\mathbb{S}^{3}\times\mathbb{S}^{3}$. We denote $P_{0}:=\langle
1,{{i}}\rangle$, and we furnish this plane with its canonical orientation. Let
$\rho:\mathbb{S}^{3}\times\mathbb{S}^{3}\rightarrow{\text{SO}}(\mathbb{H})$ be
the homomorphism given by

$\rho(x,y)\cdot z:=x\cdot z\cdot y^{-1}\ .$

This homomorphism defines a transitive action of
$\mathbb{S}^{3}\times\mathbb{S}^{3}$ on ${\text{Gr}}^{+}(2,4)$, and its
stabilizer of $P_{0}$ is $\mathbb{S}^{1}_{{i}}\times\mathbb{S}^{1}_{{i}}$. It
follows that

${\text{Gr}}^{+}(2,4)=(\mathbb{S}^{3}/\mathbb{S}^{1}_{{i}})\times(\mathbb{S}^{3}/\mathbb{S}^{1}_{{i}})=\mathbb{S}^{2}\times\mathbb{S}^{2}\
.\eqnum{\text{{\rm(4.12)}}\hbox{}}$

Let $\pi_{{i}}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{2}$ denote the Hopf
projection, as in Section 4.2. Observe now that the normal plane to $X$ at
each point $(x_{1},x_{2})\in\mathbb{R}^{2}$ is spanned by $X(x_{1},x_{2})$ and
$Y(x_{1},x_{2})$. It follows that, if $m,n\in\mathbb{S}^{3}$ are such that

$a=m\cdot{{j}}\cdot n^{-1}\ \text{and}\ b=m\cdot{{k}}\cdot n^{-1}\ ,$

then, up to reversal of orientation,

$G(x_{1},x_{2})=\langle\gamma_{2}(x_{2})\cdot m\cdot 1\cdot
n^{-1}\cdot\gamma_{1}(x_{1}),\gamma_{2}(x_{2})\cdot m\cdot{{i}}\cdot
n^{-1}\cdot\gamma_{1}(x_{1})\rangle\ ,$

so that, with the identification (4.12),

$G(x_{1},x_{2})=(\pi_{{i}}(\gamma_{2}(x_{2})\cdot
m),\pi_{{i}}(\gamma_{1}(x_{1})^{-1}\cdot n))\ .$

In this manner, we recover Weiner’s and Enomoto’s factorization from that of
Theorem 4.4.1.

Theorem 4.4.1 is a consequence of the following general factorization result
for smooth Lie group valued functions.

Lemma 4.4.2 Let $G$ be a Lie group, and let $M:\mathbb{R}^{2}\rightarrow G$ be
a smooth function. There exist $C\in G$, and smooth functions
$A,B:\mathbb{R}\rightarrow G$ such that $A(0)=B(0)={\text{Id}}$, and

$M(x_{1},x_{2})=B(x_{2})\cdot C\cdot
A(x_{1})\eqnum{\text{{\rm(4.13)}}\hbox{}}$

if and only if

$\partial_{2}(M^{-1}\cdot(\partial_{1}M))=\partial_{1}((\partial_{2}M)\cdot
M^{-1})=0\ .\eqnum{\text{{\rm(4.14)}}\hbox{}}$

Furthermore, the triplet $(A,B,C)$ is unique.

Proof: Suppose first that $M$ is a product of the form (4.13). Then

$\partial_{2}(M^{-1}\cdot(\partial_{1}M))=\partial_{2}(A(x_{1})^{-1}\cdot\dot{A}(x_{1}))=0\
.$

The first identity of (4.14) follows, and the second follows in a similar
manner.

Suppose now that (4.14) holds, and denote

$\alpha(x_{1}):=M^{-1}\cdot(\partial_{1}M)\ \text{and}\
\beta(x_{2}):=(\partial_{2}M)\cdot M^{-1}\ .$

Define $A,B:\mathbb{R}\rightarrow G$ such that $A(0)=B(0)={\text{Id}}$, and

$\dot{A}=A\cdot\alpha\ \text{and}\ \dot{B}=\beta\cdot B\ .$

Since $B$ only depends on $x_{2}$,

$\partial_{1}(B^{-1}\cdot M\cdot A^{-1})=B^{-1}\cdot((\partial_{1}M)-M\cdot
A^{-1}\cdot\dot{A})\cdot A^{-1}=B^{-1}\cdot
M\cdot(\alpha-A^{-1}\cdot\dot{A})\cdot A^{-1}=0\ .$

Likewise

$\partial_{2}(B^{-1}\cdot M\cdot A^{-1})=0\ .$

The product $C:=B^{-1}\cdot M\cdot A^{-1}$ is therefore constant, and the
result follows. $\square$Proof of Theorem 4.4.1: Indeed, let
$X:=\pi_{1}\circ\phi$ and $Y:=\pi_{2}\circ\phi$ denote the two components of
$\phi$. By Lemma 4.3.2, $X$ satisfies the hypotheses of Lemma 4.4.2, and there
therefore exists a unique element $a\in\mathbb{S}^{3}$ and unique smooth
functions $\gamma_{1},\gamma_{2}:\mathbb{R}\rightarrow\mathbb{S}^{3}$ such
that $\gamma_{1}(0)=\gamma_{2}(0)=1$, and

$X(x_{1},x_{2})=\gamma_{2}(x_{2})\cdot a\cdot\gamma_{1}(x_{1})\ .$

Since $Y$ is normal to $X$, by Lemma 4.3.1, $\gamma_{1}$ and $\gamma_{2}$ are
respectively right- and left-horizontal. Furthermore, up to isometries of
$\mathbb{H}$, $\gamma_{1}$ and $\gamma_{2}$ are the respective restrictions of
$X$ to the $x_{1}$\- and $x_{2}$-axes, and it follows by (4.6) that these
curves are arc-length parametrized.

We now determine the form of $Y$. By symmetry, there exist a unique element
$b\in\mathbb{S}^{3}$, and unique arc-length parametrized curves
$\eta_{1},\eta_{2}:\mathbb{R}\rightarrow\mathbb{S}^{3}$ such that
$\eta_{1}(0)=\eta_{2}(0)=1$, and

$Y(x_{1},x_{2})=\eta_{2}(x_{2})\cdot b\cdot\eta_{1}(x_{1})\ .$

Applying (4.9) with $x_{2}=0$ yields

$b\cdot\eta_{1}\cdot\gamma_{1}^{-1}\cdot\overline{a}=b\cdot\overline{a}\ ,$

so that

$\eta_{1}=\gamma_{1}\ .$

Likewise

$\eta_{2}=\gamma_{2}\ ,$

and (4.11) follows.

It remains to verify the geometric properties of $a$, $b$, $\gamma_{1}$ and
$\gamma_{2}$. Upon evaluating (4.11) at $(0,0)$, we see that $a$ is orthogonal
to $b$. Since $\phi$ is an integral curve of $W$, applying (4.9) with
$x_{2}=0$ yields, for all $x_{1}$,

$a\cdot\dot{\gamma}_{1}(x_{1})=\partial_{1}X(x_{1},0)\perp
Y(x_{1},0)=Y(0,0)\cdot X^{-1}(0,0)\cdot X(x_{1},0)=b\cdot\gamma_{1}(x_{1})\ .$

It follows that

$\dot{\gamma}_{1}\cdot\gamma_{1}^{-1}\perp\overline{a}\cdot b\ ,$

so that $\gamma_{1}$ is right $(\overline{a}\cdot b)$-horizontal. In a similar
manner, we show that $\gamma_{2}$ is left $(\overline{b}\cdot a)$-horizontal,
as desired.

Finally, we readily verify that any map of this form defines a bilegendrian
immersion in ${\text{\rm U}}\mathbb{S}^{3}$, and this completes the proof.
$\square$We now show how the period lattice factorizes in the doubly-periodic
case. We will say that a period lattice of a doubly-periodic immersion is
maximal whenever it is not contained in any larger period lattice.

Theorem 4.4.3, Factorization of period lattices Let
$\phi:\mathbb{R}^{2}\rightarrow{\text{\rm U}}\mathbb{S}^{3}$ be a complete
bilegendrian immersion. If $\phi$ is doubly-periodic with maximal period
lattice $\Lambda$, then there exist real numbers $p_{1},p_{2}>0$, and rational
numbers $q_{1},q_{2}>0$ such that

$\Lambda=\left\\{(mp_{1},np_{2})\ |\ (mq_{1}-nq_{2})\in\mathbb{Z}\right\\}\
,\eqnum{\text{{\rm(4.15)}}\hbox{}}$

Furthermore, if $\gamma_{1}$ and $\gamma_{2}$ denote the factors of $\phi$,
then, for all $x$,

$\gamma_{1}(x+p_{1})=e^{-2\pi q_{1}(b\cdot\overline{a})}\cdot\gamma_{1}(x)\
\text{and}\ \gamma_{2}(x+p_{2})=\gamma_{2}(x)\cdot e^{2\pi
q_{2}(\overline{a}\cdot b)}\ .\eqnum{\text{{\rm(4.16)}}\hbox{}}$

In particular, each of these curves is also periodic, and their Hopf
projections in $\mathbb{S}^{2}$ are periodic with respective periods $p_{1}$
and $p_{2}$.

Proof: We use the factorization (4.11). Upon multiplying $\phi$ on the left by
$\overline{a}$, and replacing $\gamma_{2}$ and $b$ respectively by
$\gamma_{2}^{\prime}:=(\overline{a}\cdot\gamma_{2}\cdot a)$ and
$b^{\prime}:=(\overline{a}\cdot b)$, we may suppose that $a=1$ and that $b$ is
a unit imaginary quaternion. Let $\mathbb{S}^{1}_{b}$ denote the subgroup of
$\mathbb{S}^{3}$ generated by $b$.

We first show that, for all $(p_{1},p_{2})\in\Lambda$, there exists an element
$\xi\in\mathbb{S}^{1}_{b}$ such that

$\gamma_{1}(p_{1})=\gamma_{2}(p_{2})^{-1}=\xi\ .$

Indeed, by double-periodicity of the first component,

$\gamma_{2}(p_{2})\cdot\gamma_{1}(p_{1})=X(p_{1},p_{2})=X(0,0)=1\ ,$

so that

$\gamma_{1}(p_{1})=\gamma_{2}(p_{2})^{-1}=:\xi\ .$

Next, by double-periodicity of the second component,

$\gamma_{2}(p_{2})\cdot b\cdot\gamma_{1}(p_{1})=Y(p_{1},p_{2})=Y(0,0)=b\ ,$

so that

$b\cdot\xi=b\cdot\gamma_{1}(p_{1})=\gamma_{2}(p_{2})^{-1}\cdot b=\xi\cdot b\
.$

That is $\xi$ commutes with $b$. Since the centralizer of $b$ in $\mathbb{H}$
is $\langle 1,b\rangle$, it follows that $\xi\in\langle
1,b\rangle\cap\mathbb{S}^{3}=\mathbb{S}^{1}_{b}$, as asserted.

We now claim that, for all $x$,

$\gamma_{1}(x+p_{1})=\xi\cdot\gamma_{1}(x)\ \text{and}\
\gamma_{2}(x+p_{2})=\gamma_{2}(x)\cdot\xi^{-1}\ .$

Indeed, it suffices to prove the first identity, as the proof of the second is
identical. However,

$\gamma_{1}(x+p_{1})^{-1}\cdot\dot{\gamma}_{1}(x+p_{1})=X(x+p_{1},0)^{-1}\cdot(\partial_{1}X)(x+p_{1},0)\
.$

By (4.10), this is equal to

$X(x+p_{1},p_{2})^{-1}\cdot(\partial_{1}X)(x+p_{1},p_{2})=X(x,0)^{-1}\cdot(\partial_{1}X)(x,0)=\gamma_{1}(x)^{-1}\cdot\dot{\gamma}_{1}(x)\
.$

The curves $\gamma_{1}(\cdot+p_{1})$ and $\xi\cdot\gamma_{1}$ therefore solve
the same first-order ordinary differential equation with the same initial
condition, so that, for all $x$,

$\gamma_{1}(x+p_{1})=\xi\cdot\gamma_{1}(x)\ ,$

as asserted.

Let $\Lambda_{1}$ and $\Lambda_{2}$ denote the respective images of $\Lambda$
under projection onto the first and second factors. By the preceding
discussion, there exist homomorphisms
$\xi_{1}:\Lambda_{1}\rightarrow\mathbb{S}^{1}_{b}$ and
$\xi_{2}:\Lambda_{2}\rightarrow\mathbb{S}^{1}_{b}$ such that, for all
$p_{1}\in\Lambda_{1}$, $p_{2}\in\Lambda_{2}$ and $x\in\mathbb{R}$,

$\gamma_{1}(x+p_{1})=\xi_{1}(p_{1})\cdot\gamma_{1}(x)\ \text{and}\
\gamma_{2}(x+p_{2})=\gamma_{2}(x)\cdot\xi_{2}(p_{2})^{-1}\ .$

We claim that $\Lambda_{1}$ and $\Lambda_{2}$ are discrete. Indeed, otherwise,
without loss of generality, $\Lambda_{1}$ would be dense in $\mathbb{R}$, so
that, by continuity, $\gamma_{1}$ would take values in $\mathbb{S}^{1}_{b}$.
In particular, this would yield

$(\partial_{1}X)(0,0)=\dot{\gamma}_{1}(0)=\pm b=\pm Y(0,0)\ ,$

which would be absurd, since $\phi$ is an integral curve of $W$. For each $i$,
let $p_{i}$ be a positive generator of $\Lambda_{i}$, and let $q_{i}$ be such
that

$e^{-2\pi q_{i}b}=\xi_{i}(p_{i})\ .$

By definition, $\Lambda\subseteq\Lambda_{1}\times\Lambda_{2}$. Since $\Lambda$
is maximal, for $m,n\in\mathbb{Z}$, $mp_{1}+np_{2}\in\Lambda$ if and only if

$e^{2\pi nq_{2}b}\cdot e^{-2\pi mq_{1}b}=1\ ,$

which in turn holds if and only if $mq_{1}-nq_{2}\in\mathbb{Z}$. Finally,
since $\Lambda$ is two-dimensional, there exists a non-singular integer-valued
matrix $M$ such that

$M\cdot(q_{1},q_{2})^{t}\in\mathbb{Z}^{2}\ ,$

from which it follows that $q_{1}$ and $q_{2}$ are both rational. This
completes the proof. $\square$

4.5 - Quasiperiodic horizontal curves in $\mathbb{S}^{3}$.  We now describe a
general construction of doubly-periodic bilegendrian immersions in ${\text{\rm
U}}\mathbb{S}^{3}$. Throughout this section, we fix a unit-length quaternion
$\xi$, we denote by $\pi_{\xi}$ its Hopf projection, as defined in Section
4.2, and we denote

$\alpha_{\xi}:=\xi_{l}^{\flat}\ \text{and}\
H_{\xi}:={\text{Ker}}(\alpha_{\xi})\ ,\eqnum{\text{{\rm(4.17)}}\hbox{}}$

where $\flat$ here denotes Berger’s musical isomorphism. We also restrict
attention to left $\xi$-horizontal curves, as right $\xi$-horizontal curves
are treated in an identical manner.

Given a real number $q$, we say that a left $\xi$-horizontal curve $\gamma$ in
$\mathbb{S}^{3}$ is $q$-quasiperiodic whenever there exists $p\in\mathbb{R}$
such that, for all $x$,

$\gamma(x+p)=\gamma(x)\cdot e^{2\pi q\xi}\ .$

By Theorem 4.4.3, the classification of doubly-periodic bilegendrian
immersions in ${\text{\rm U}}\mathbb{S}^{3}$ reduces to the classification of
$q$-quasiperiodic curves in $\mathbb{S}^{3}$ for rational $q$, which we now
address.

Lemma 4.5.1 For all $\xi\in\mathbb{S}^{3}$, and for all real $q$, the image
under $\pi_{\xi}$ of any $q$-quasiperiodic left $\xi$-horizontal curve is a
periodic curve in $\mathbb{S}^{2}$.

Proof: Indeed, let $\gamma$ be a $q$-quasiperiodic left $\xi$-horizontal curve
with period $p$. For all $x\in\mathbb{R}$,

$(\pi_{\xi}\circ\gamma)(x+p)=\gamma(x+p)\cdot\xi\cdot\gamma(x+p)^{-1}=\gamma(x)\cdot
e^{2\pi q\xi}\cdot\xi\cdot e^{-2\pi
q\xi}\cdot\gamma(x)^{-1}=\gamma(x)\cdot\xi\cdot\gamma(x)^{-1}=(\pi_{\xi}\circ\gamma)(x)\
,$

and periodicity follows. $\square$We now examine left $\xi$-horizontal lifts
of periodic curves in $\mathbb{S}^{2}$. We first verify that they exist.

Lemma 4.5.2 The Hopf projection $\pi_{\xi}$ defines a surjective riemannian
submersion from $(\mathbb{S}^{3},{{b}})$ to $(\mathbb{S}^{2},{{b}}/4)$.
Furthermore, the kernel of $D\pi_{\xi}$ is the linear span of $\xi_{l}$, and
its ${{b}}$-orthogonal complement is $H_{\xi}$.

Proof: Since multiplication on the left acts transitively by isometries on
$\mathbb{S}^{3}$, and since ad acts transitively by isometries on
$\mathbb{S}^{2}$, it suffices to prove the result at the identity. However,
for all $\mu\in\Cal{I}$,

$D\pi_{\xi}(1)\cdot\mu=\frac{d}{dt}e^{t\mu}\cdot\xi\cdot
e^{-t\mu}\bigg{|}_{t=0}=[\mu,\xi]\ .$

Suppose now that $\mu\in\langle\xi\rangle^{\perp}$. Since $\mu$ is imaginary,
it anticommutes with $\xi$, so that

$D\pi_{\xi}(1)\cdot\mu=2\xi\cdot\mu\ .$

Hence

$\|D\pi_{\xi}(1)\cdot\mu\|_{{b}}^{2}=\|2\xi\cdot\mu\|_{{b}}^{2}=4\|\mu\|_{{b}}^{2}\
,$

and the riemannian submersion property follows. Finally, observe that
$\xi\in{\text{Ker}}(D\pi_{\xi})$. It follows by the rank nullity theorem that
${\text{Ker}}(D\pi_{\xi})$ coincides with the linear span of $\xi$, and this
completes the proof. $\square$By Lemma 4.5.2, every smooth curve $\gamma$ in
$\mathbb{S}^{2}$ lifts through the Hopf projection to a smooth, left
$\xi$-horizontal curves in $\mathbb{S}^{3}$. We now establish a necessary and
sufficient condition for any Hopf lift to be $q$-quasiperiodic. For this, we
will require the concept of signed area bounded by closed curves in
$\mathbb{S}^{2}$, which we now define. Note first that, since $\mathbb{S}^{2}$
is simply-connected, every piecewise smooth closed curve
$\gamma:\mathbb{S}^{1}\rightarrow\mathbb{S}^{2}$ extends to a piecewise smooth
function $\tilde{\gamma}:\overline{B}_{1}(0)\rightarrow\mathbb{S}^{2}$.

Lemma & Definition 4.5.3 If
$\tilde{\gamma}_{1},\tilde{\gamma}_{2}:\overline{B}_{1}(0)\rightarrow\mathbb{S}^{2}$
are two extensions of $\gamma$, then

$\int_{\overline{B}_{1}(0)}\tilde{\gamma}_{1}^{*}{\text{dArea}}=\int_{\overline{B}_{1}(0)}\tilde{\gamma}_{2}^{*}{\text{dArea}}\
\text{{\rm mod}}\ 4\pi\ ,\eqnum{\text{{\rm(4.18)}}\hbox{}}$

where dArea here denotes the canonical area form of $\mathbb{S}^{2}$. We will
call this integral the signed area bounded by $\gamma$, and we denote it by
${\text{Area}}(\gamma)$.

Proof: Viewing $\mathbb{S}^{1}$ as the equator of $\mathbb{S}^{2}$, we
identify the lower hemisphere of $\mathbb{S}^{2}$ with $B_{1}(0)$, and the
upper hemisphere with $B_{1}(0)$ with its orientation reversed. Let
$\phi:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ be a piecewise smooth map equal
to $\tilde{\gamma}_{1}$ in the lower hemisphere and equal to
$\tilde{\gamma}_{2}$ in the upper hemisphere. Since $\phi^{*}{\text{dArea}}$
is an integral cohomology class over $\mathbb{S}^{2}$,

$\int_{\mathbb{S}^{2}}\phi^{*}{\text{dArea}}=0\ \text{mod}\ 4\pi\ .$

Thus, since

$\int_{\mathbb{S}^{2}}\phi^{*}{\text{dArea}}=\int_{\overline{B}_{1}(0)}\tilde{\gamma}_{1}^{*}{\text{dArea}}-\int_{\overline{B}_{1}(0)}\tilde{\gamma}_{2}^{*}{\text{dArea}}\
,$

the result follows. $\square$It remains only to relate the holonomy of the
$\xi$-horizontal lift to the signed area.

Lemma 4.5.4 The contact form $\alpha_{\xi}$ satisfies

$\|d\alpha_{\xi}\wedge\alpha_{\xi}\|_{{b}}=2\
.\eqnum{\text{{\rm(4.19)}}\hbox{}}$

Proof: Let $\mu$ and $\nu$ be unit-length imaginary quaternions. The exterior
derivative of $\alpha_{\xi}$ is given by

$d\alpha_{\xi}(\mu_{l},\nu_{l})=D_{\mu_{l}}\alpha_{\xi}(\nu_{l})-D_{\nu_{l}}\alpha_{\xi}(\mu_{l})-\alpha_{\xi}([\mu_{l},\nu_{l}])\
.$

Since the first two terms on the right-hand side vanish, and since
$[\mu_{l},\nu_{l}]=[\mu,\nu]_{l}$, it follows that

$d\alpha_{\xi}(\mu_{l},\nu_{l})=-\alpha_{\xi}([\mu,\nu]_{l})=-{{b}}(\xi,[\mu,\nu])\
.$

In particular, if $\mu$, $\nu$ and $\xi$ are pairwise orthogonal, then

$[\mu,\nu]=\pm 2\xi\ ,$

from which it follows that

$\|d\alpha^{\xi}\wedge\alpha^{\xi}\|_{{b}}=2\ ,$

as desired. $\square$

Theorem 4.5.5, Quasiperiodicity of horizontal lifts Let $\gamma$ be a periodic
curve in $\mathbb{S}^{2}$ and with left $\xi$-horizontal lift $\hat{\gamma}$.
If the signed area bounded by $\gamma$ is equal to $(-4\pi q)$ modulo $4\pi$,
then $\hat{\gamma}$ is $q$-quasiperiodic.

Combining Theorem 4.5.5 with Theorems 4.4.1 and 4.4.3, we see that every
doubly-periodic bilegendrian immersion in ${\text{\rm U}}\mathbb{S}^{3}$ may
be constructed by the following ansatz. First choose two orthogonal unit
quaternions $a$ and $b$. Next choose two arc-length parametrized periodic
curves $\gamma_{1}$ and $\gamma_{2}$ in $\mathbb{S}^{2}$, bounding signed
areas equal to rational multiples of $4\pi$, such that
$\gamma_{1}(0)=\overline{a}\cdot b$ and $\gamma_{2}(0)=\overline{b}\cdot a$.
Let $\hat{\gamma}_{1}$ denote the unique right $(\overline{a}\cdot
b)$-horizontal lift of $\gamma_{1}$ such that $\gamma_{1}(0)=1$, and let
$\hat{\gamma}_{2}$ denote the unique left $(\overline{b}\cdot a)$-horizontal
lift of $\gamma_{2}$ such that $\gamma_{2}(0)=1$. The desired doubly-periodic
bilegendrian immersion is now given by (4.11). Note, in particular, that since
the set of area preserving deformations of any closed curve in
$\mathbb{S}^{2}$ is infinite-dimensional, so too is the local deformation
space of any doubly-periodic bilegendrian immersion in ${\text{\rm
U}}\mathbb{S}^{3}$. Proof of Theorem 4.5.5: Without loss of generality, we may
suppose that $\gamma(0)=\gamma(1)=\xi$. Let $q$ be such that
$\hat{\gamma}(1)=\hat{\gamma}(0)\cdot e^{2\pi q\xi}$. By uniqueness of the
horizontal lift, for all $x$,

$\hat{\gamma}(x+1)=\hat{\gamma}(x)\cdot e^{2\pi q\xi}\ .$

We now relate $q$ to the signed area bounded by $\gamma$. We define
$\mu:I\rightarrow\mathbb{S}^{3}$ by

$\mu(t):=e^{2\pi tq\xi}\ .$

Since $\hat{\gamma}$ is horizontal, and since the restriction of
$\alpha_{\xi}$ to $\mathbb{S}^{1}_{\xi}$ has constant unit length,

$2\pi
q=\int_{\mu}\alpha_{\xi}=-\int_{\mu^{-1}}\alpha_{\xi}=-\int_{\mu^{-1}\cdot\hat{\gamma}}\alpha_{\xi}\
.$

We now view $\mu^{-1}\cdot\hat{\gamma}$ as a piecewise smooth function from
$\mathbb{S}^{1}$ to $\mathbb{S}^{3}$. Since $\mathbb{S}^{3}$ is simply-
connected, it extends to a piecewise smooth function
$\phi:\overline{B}_{1}(0)\rightarrow\mathbb{S}^{3}$. By Stokes’ theorem

$2\pi q=-\int_{\overline{B}_{1}(0)}\phi^{*}d\alpha_{\xi}\ .$

However, by Lemmas 4.5.4 and 4.5.2,

$d\alpha_{\xi}=\frac{1}{2}\pi_{\xi}^{*}{\text{dArea}}\ ,$

so that

$4\pi
q=-\int_{\overline{B}_{1}(0)}\phi^{*}\pi_{\xi}^{*}{\text{dArea}}=-\int_{\overline{B}_{1}(0)}(\pi_{\xi}\circ\phi)^{*}{\text{dArea}}\
.$

Since $\pi_{\xi}\circ\phi$ extends $\gamma$, it follows that

$4\pi q=-{\text{Area}}(\gamma)\ \text{mod}\ 4\pi\ ,$

as desired. $\square$

4.6 - Curvature and the angle function.  We now return to the classical case
of complete, flat immersions in $\mathbb{S}^{3}$. Indeed, although Theorems
4.4.1, 4.4.3 and 4.5.5 together yield full classifications of complete and
compact bilegendrian immersions in ${\text{\rm U}}\mathbb{S}^{3}$, it is
clearly of interest to establish the conditions under which such immersions
project to immersions in $\mathbb{S}^{3}$, that is, the conditions under which
they arise as Gauss lifts of complete or compact flat surfaces in the sphere.
Recall that these conditions were established by Kitigawa in $[22]$ and Weiner
in $[51]$. We now show how these results are recovered in the present
framework.

We first introduce the angle function. To this end, it will be useful to
introduce the new coordinate system

$(u_{1},u_{2}):=(x_{1}+x_{2},x_{1}-x_{2})\ .$

We denote the corresponding coordinate vector fields by $\hat{\partial}_{1}$
and $\hat{\partial}_{2}$, so that

$\hat{\partial}_{1}=\frac{1}{2}(\partial_{1}+\partial_{2})\ \text{and}\
\hat{\partial}_{2}=\frac{1}{2}(\partial_{1}-\partial_{2})\ .$

By construction, $\hat{\partial}_{1}$ and $\hat{\partial}_{2}$ are fields of
principal vectors. Let $L_{1}$ and $L_{2}$ denote their corresponding
principal lines and, for each $i$, let
$e_{i}:\mathbb{R}^{2}\rightarrow\mathbb{S}^{3}$ be a global unit section of
$L_{i}$ such that the quadruplet $(X,e_{1},e_{2},Y)$ is positively oriented.
We define the angle function $\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such
that

$\hat{\partial}_{1}\phi=({\text{cos}}(\theta)e_{1},{\text{sin}}(\theta)e_{1})\
.\eqnum{\text{{\rm(4.20)}}\hbox{}}$

Since $\hat{\partial}_{2}\phi=J\cdot\hat{\partial}_{1}\phi$, it follows that

$\hat{\partial}_{2}\phi=(-{\text{sin}}(\theta)e_{2},{\text{cos}}(\theta)e_{2})\
.\eqnum{\text{{\rm(4.21)}}\hbox{}}$

Lemma 4.6.1 For every vector field $\xi$ over $\mathbb{R}^{2}$,

$C(\xi,\hat{\partial}_{1},\hat{\partial}_{1})=C(\xi,\hat{\partial}_{2},\hat{\partial}_{2})=-D_{\xi}\theta\
.\eqnum{\text{{\rm(4.22)}}\hbox{}}$

By symmetry, this suffices to determine every component of $C$. Furthermore,
since $C$ identifies with the second fundamental form of $\phi$, the angle
function $\theta$ thus suffices to determine the immersion $\phi$ up to
postcomposition with isometries of $\mathbb{S}^{3}$. We will examine presently
the necessary and sufficient conditions for $\theta$ to arise as the angle
function of a bilegendrian immersion.

Corollary 4.6.2 For every vector field $\xi$ over $\mathbb{R}^{2}$,

$C(\partial_{1},\partial_{1},\partial_{1})=-4\partial_{1}\theta\ \text{and}\
C(\partial_{2},\partial_{2},\partial_{2})=-4\partial_{2}\theta\
.\eqnum{\text{{\rm(4.23)}}\hbox{}}$

Proof of Lemma 4.6.1:  Since $e_{1}$ has constant unit length, for every
vector field $\xi$ over $\mathbb{R}^{2}$,

${{b}}(D_{\xi}e_{1},e_{1})=\frac{1}{2}D_{\xi}{{b}}(e_{1},e_{1})=0.$

It follows that

$\eqalign{C(\xi,\hat{\partial}_{1},\hat{\partial}_{1})&=g(D_{\xi}({\text{cos}}(\theta)e_{1},{\text{sin}}(\theta)e_{1}),I\cdot({\text{cos}}(\theta)e_{1},{\text{sin}}(\theta)e_{1}))\cr&={{b}}(D_{\xi}({\text{cos}}(\theta)e_{1}),{\text{sin}}(\theta)e_{1})-{{b}}(D_{\xi}({\text{sin}}(\theta)e_{1}),{\text{cos}}(\theta)e_{1})\cr&=-{\text{sin}}^{2}(\theta)D_{\xi}\theta-{\text{cos}}^{2}(\theta)D_{\xi}\theta\cr&=-D_{\xi}\theta\
,\cr}$

as desired. Next, since $J$ is $C$-symmetric,

$C(\xi,\hat{\partial}_{2},\hat{\partial}_{2})=C(\xi,J\cdot\hat{\partial}_{1},J\cdot\hat{\partial}_{1})=C(\xi,\hat{\partial}_{1},J^{2}\cdot\hat{\partial}_{1})=C(\xi,\hat{\partial}_{1},\hat{\partial}_{1})=-D_{\xi}\theta\
,$

and this completes the proof. $\square$We now relate the above formulae to the
geometry of asymptotic curves. To this end, we recall some elementary
definitions of classical curve theory. We define a framed curve in
$\mathbb{S}^{3}$ to be a smooth function
$\gamma:I\rightarrow{\text{SO}}({{b}})$ such that, for all $t$, the matrix
$\gamma(t)^{-1}\cdot\dot{\gamma}(t)$ is tridiagonal. Recall that a matrix is
said to be tridiagonal whenever its only non-vanishing elements are those
lying just above, on, and just below the diagonal. Given a framed curve
$\gamma$, abusing notation, we denote

$\gamma=:(\gamma,T,N,B)\ ,$

and we call $T$, $N$ and $B$ respectively its tangent, normal and binormal. We
will say that the framed curve is arc-length parametrized whenever its first
component has this property. When this holds, we will suppose, furthermore,
that $T=\dot{\gamma}$. Given an arc-length parametrized framed curve, we
define its curvature $\kappa$ and torsion $\tau$ by

$\kappa:={{b}}(\dot{T},N)\ \text{and}\ \tau:={{b}}(\dot{N},B)\ .$

For each $i$, we now define

$\gamma_{i}(t):=X(t\partial_{i})\ ,\ T_{i}(t):=(\partial_{i}X)(t\partial_{i})\
,\ N_{i}(t):=A\cdot T_{i}(t)\ ,\ \text{and}\ B_{i}(t):=Y(t\partial_{i})\ .$

Lemma 4.6.3 $\gamma_{i}:=(\gamma_{i},T_{i},N_{i},B_{i})$ is a framed curve in
$\mathbb{S}^{3}$.

Proof: By definition of $A$, this quadruplet is positively oriented. We now
show that $\gamma_{i}^{-1}\cdot\gamma_{i}$ is tridiagonal. Since this matrix
is antisymmetric, and since $\dot{\gamma}_{i}=T_{i}$, it suffices to show that
${{b}}(\dot{T}_{i},B_{i})$ vanishes. However, since $Y$ is normal to $X$,

${{b}}(\dot{T}_{i},B_{i})={{b}}(D_{\partial_{i}}(\partial_{i}X),Y)=-{{b}}(\partial_{i}X,\partial_{i}Y)\
.$

An elementary calculation involving (4.20) and (4.21) shows that this
vanishes, and it follows that $\gamma_{i}^{-1}\cdot\gamma_{i}$ is tridiagonal,
as desired. $\square$

Lemma 4.6.4 For each $i$, the curvature of $\gamma_{i}$ is given by

$\kappa_{i}=-2\epsilon_{i}\theta_{i}^{\prime}\
.\eqnum{\text{{\rm(4.24)}}\hbox{}}$

where $\epsilon_{i}$ here denotes the $J$-eigenvalue of $\partial_{i}$.

Proof: Indeed,

$\kappa_{i}={{b}}(\dot{T}_{i},N_{i})={{b}}(D_{\partial_{i}}\partial_{i}X,A\cdot\partial_{i}X)=\frac{1}{2}(g+\hat{g})(\nabla_{\partial_{i}}\partial_{i}\phi,K\cdot\partial_{i}\phi)\
.$

Since $K=IJ$, this yields

$\kappa_{i}=\frac{\epsilon_{i}}{2}(g+\hat{g})(\nabla_{\partial_{i}}\partial_{i}\phi,I\cdot\partial_{i}\phi)=\frac{\epsilon_{i}}{2}C(\partial_{i},\partial_{i},\partial_{i})+\frac{\epsilon_{i}}{2}\hat{C}(\partial_{i},\partial_{i},\partial_{i})\
,$

so that, by (4.8) and (4.23),

$\kappa_{i}=-2\epsilon_{i}\partial_{i}\theta\ ,$

as desired. $\square$

Lemma 4.6.5 For each $i$, the torsion of $\gamma_{i}$ is given by

$\tau_{i}=-\epsilon_{i}\ .\eqnum{\text{{\rm(4.25)}}\hbox{}}$

where $\epsilon_{i}$ here denotes the $J$-eigenvalue of $\partial_{i}$.

Remark 4.6.1. In fact, this follows immediately from Lemma 4.3.1. Indeed, it
is an interesting exercise to show that a smooth framed curve in
$\mathbb{S}^{3}$ is left (resp. right) horizontal if and only if it has unit
positive (resp. negative) torsion (see, for example, Lemma $44$ of $[48]$).
Proof: Indeed, since $Y$ is normal to $X$,

$\tau_{i}={{b}}(\dot{N}_{i}(t),B_{i}(t))={{b}}(D_{\partial_{i}}(A\cdot\partial_{i}X),Y)=-{{b}}(A\cdot
DX\cdot\partial_{i},DY\cdot\partial_{i})\ ,$

and the result now follows by an elementary calculation using (4.4), (4.20)
and (4.21). $\square$

Lemma 4.6.6 The angle function $\theta$ satisfies the wave equation

$\partial_{1}\partial_{2}\theta=0\ .\eqnum{\text{{\rm(4.26)}}\hbox{}}$

In particular, there exist smooth functions
$\theta_{1},\theta_{2}:\mathbb{R}\rightarrow\mathbb{R}$ such that

$\theta(x_{1},x_{2})=\theta_{1}(x_{1})+\theta_{2}(x_{2})\
.\eqnum{\text{{\rm(4.27)}}\hbox{}}$

Remark 4.6.2. In fact, this also follows immediately from (4.11) and (4.23).
Indeed, since left and right multiplication by unit quaternions yield
isometries of $\mathbb{H}$, the curvatures of horizontal (resp. vertical)
lines are independent of the vertical (resp. horizontal) coordinate. However,
the following, much longer, proof is of interest because it is also valid for
more general bilegendrian surfaces. Proof: It will be easier to prove the
equivalent identity

$\hat{\partial}_{1}^{2}\theta-\hat{\partial}_{2}^{2}\theta=0\ .$

By (4.22) and symmetry of $C$,

$\hat{\partial}_{1}^{2}\theta=-D_{\hat{\partial}_{1}}C(\hat{\partial}_{1},\hat{\partial}_{1},\hat{\partial}_{2})=-D_{\hat{\partial}_{1}}\omega_{i}(\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),(\hat{\partial}_{2}\phi))\
.$

Since $\omega_{i}$ is covariant constant, this yields

$\hat{\partial}_{1}^{2}\theta=-\omega_{i}(\nabla_{\hat{\partial}_{1}}\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),(\hat{\partial}_{2}\phi))-\omega_{i}(\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),\nabla_{\hat{\partial}_{1}}(\hat{\partial}_{2}\phi))\
.$

However,

$\omega_{i}(\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),\nabla_{\hat{\partial}_{1}}(\hat{\partial}_{2}\phi))=\hat{g}(\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),\hat{I}\cdot\nabla_{\hat{\partial}_{1}}(\hat{\partial}_{2}\phi))\
.$

Since $\phi^{*}\hat{g}$ is the euclidean metric, the coordinate vector fields
are $\phi^{*}\hat{g}$-covariant constant. The first argument of $\hat{g}$ on
the right-hand side of the above identity is therefore $\hat{g}$-normal to
$TS$, whilst the second argument is tangential. Their inner product therefore
vanishes, and we conclude that

$\hat{\partial}_{1}^{2}\theta=-\omega_{i}(\nabla_{\hat{\partial}_{1}}\nabla_{\hat{\partial}_{2}}(\hat{\partial}_{1}\phi),(\hat{\partial}_{2}\phi))\
.$

Likewise,

$\hat{\partial}_{2}^{2}\theta=-\hat{\partial}_{2}C(\hat{\partial}_{1},\hat{\partial}_{1},\hat{\partial}_{2})=-\omega_{i}(\nabla_{\hat{\partial}_{2}}\nabla_{\hat{\partial}_{1}}(\hat{\partial}_{1}\phi),(\hat{\partial}_{2}\phi))\
,$

and since $[\hat{\partial}_{1},\hat{\partial}_{2}]=0$,

$\hat{\partial}_{1}^{2}\theta-\hat{\partial}_{2}^{2}\theta=-\omega_{i}(R_{\hat{\partial}_{1}\hat{\partial}_{2}}\hat{\partial}_{1},\hat{\partial}_{2})\
.$

By (A.1), this vanishes, and the result follows. $\square$We now conclude the
classification of complete bilegendrian immersions in ${\text{\rm
U}}\mathbb{S}^{3}$. Let $\hat{\Cal{B}}$ denote the space of smooth
bilegendrian immersions $\phi:\mathbb{R}^{2}\rightarrow{\text{\rm
U}}\mathbb{S}^{3}$ satisfying (4.5), furnished with the
$C^{\infty}_{\text{loc}}$ topology. Let $\Cal{B}$ denote the quotient of this
space under the action of postcomposition by elements of
${\text{SO}}(\mathbb{H})$. Let $C^{\infty}(\mathbb{R})$ denote the space of
smooth, real-valued functions over $\mathbb{R}$, also furnished with the
$C^{\infty}_{\text{loc}}$ topology, and let $\mathbb{S}^{1}$ denote the unit
circle in $\mathbb{C}$. We construct the function
$\Theta:\hat{\Cal{B}}\rightarrow\mathbb{S}^{1}\times
C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})$ as follows. Let
$\phi:\mathbb{R}^{2}\rightarrow{\text{\rm U}}\mathbb{S}^{3}$ be a smooth,
bilegendrian immersion satisfying (4.5), and consider the factorization
(4.11). Upon post-composing with an element of ${\text{SO}}(\mathbb{H})$, we
may suppose that $a=1$, $b={{k}}$, and $\dot{\gamma}_{1}(0)=2{{i}}$, from
which it follows that

$\dot{\gamma}_{2}(0)=2e^{\theta_{0}{{k}}}{{i}}\ ,$

for some $\theta_{0}\in[0,2\pi[$. We now define $\Theta(\phi)$ by

$\Theta(\phi):=(e^{i\theta_{0}},\partial_{1}\theta,\partial_{2}\theta)\ ,$

where $\theta$ here denotes the angle function of $\phi$. We now define the
set-valued function I over $C^{\infty}(\mathbb{R})$ by

${\text{I}}(f):={\text{Im}}(F)\ ,$

where

$F(x):=\int_{0}^{x}f(y)dy\ .$

Theorem 4.6.7, Classification of complete immersions $\Theta$ descends to a
homeomorphism from $\Cal{B}$ into $\mathbb{S}^{1}\times
C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})$. Furthermore,
$\phi\in\hat{\Cal{B}}$ projects to a flat immersed surface in $\mathbb{S}^{3}$
if and only if

$(\theta_{0}/2+{\text{I}}(\theta_{x})+{\text{I}}(\theta_{y}))\mathop{\cap}(\pi/2)\mathbb{Z}=\emptyset\
.\eqnum{\text{{\rm(4.28)}}\hbox{}}$

Proof: We first show that $\Theta$ is invertible. Note first that every
element of ${\text{SO}}(\mathbb{H})$ are uniquely determined by an orthonormal
triple in $\mathbb{H}$, corresponding to the image of $(1,{{k}},{{i}})$. Given
a framed curve $\gamma$, let $\kappa(\gamma)$ denote its curvature, and define
$\hat{\Theta}:\hat{\Cal{B}}\rightarrow{\text{SO}}(\mathbb{H})\times\mathbb{S}^{1}\times
C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})$ by

$\hat{\Theta}(\phi):=((a,b,\dot{\gamma}_{1}(0)/2),e^{i\theta_{0}},-\kappa(\gamma_{1})/2,\kappa(\gamma_{2})/2)\
,$

where $(a,b,\gamma_{1},\gamma_{2})$ are as in the factorization (4.11), and
$\theta_{0}$ denotes the angle between $\dot{\gamma}_{1}(0)$ and
$\dot{\gamma}_{2}(0)$, measured in the positive direction. Since left and
right horizontal curves in $\mathbb{S}^{2}$ are uniquely defined by their
initial values, their initial directions, and their curvatures,
$\hat{\Cal{B}}$ is a homeomorphism, and therefore so too is $\Cal{B}$.

It remains only to establish when an element of $\hat{\Cal{B}}$ projects to an
immersion in $\mathbb{S}^{3}$. However, by (4.20) and (4.21), this holds
whenever ${\text{cos}}(\theta){\text{sin}}(\theta)$ never vanishes, that is,
when

${\text{Im}}(\theta)\cap(\pi/2)\mathbb{Z}=\emptyset\ .$

Thus, since

${\text{Im}}(\theta)=\theta_{0}/2+{\text{I}}(f)+{\text{I}}(g)\ ,$

the result follows. $\square$The classification of doubly-periodic immersed
bilegendrian surfaces in ${\text{\rm U}}\mathbb{S}^{3}$ does not lend itself
to such a simple formulation. We will thus limit ourselves to the following
result, which recovers Theorem $3.9$ of $[22]$.

Theorem 4.6.8 Let $\phi:\mathbb{R}^{2}\rightarrow{\text{\rm U}}\mathbb{S}^{3}$
be a smooth, doubly-periodic bilegendrian immersion, and let
$(a,b,\gamma_{1},\gamma_{2})$ denote its factorization as in (4.11). If
$(\pi_{\xi}\circ\phi)$ is also an immersion, then each of $\gamma_{1}$ and
$\gamma_{2}$ is either periodic, or $(1/2)$-quasiperiodic.

Proof: It suffices to prove the result for $\gamma_{1}$, as the proof for
$\gamma_{2}$ is identical. Let $q\in\mathbb{Q}$ and $p\in\mathbb{R}$ be such
that $\gamma_{1}$ is $q$-quasiperiodic with period $p$. Let $\theta$ denote
the angle function of $\phi$, and let $\kappa_{1}$ denote the geodesic
curvature of $(\pi_{\xi}\circ\gamma_{1})$. By (4.24),

$\kappa_{1}=-2(\partial_{1}\theta)\ .$

In particular,

$\int_{0}^{p}\kappa_{1}(x)dx=-2\int_{0}^{p}(\partial_{1}\theta)(x)dx=-2(\theta(p,0)-\theta(0,0))\
.$

This vanishes, for otherwise, by quasiperiodicity, $\theta$ would be
unbounded, contradicting (4.28). By the Gauss–Bonnet Theorem, the curve
$(\pi_{\xi}\circ\gamma_{1})$ therefore bounds a signed area equal to an
integer multiple of $2\pi$ modulo $4\pi$, and the result now follows by
Theorem 4.5.5. $\square$

A - Calculating the curvature. Let $W$ denote the bundle defined in Section
3.3. We now determine certain components of its curvature. The calculation is
rather technical and uninformative, and for this reason we have chosen to
relegate it to this appendix.

Lemma A.0.9 At every point $(x,y)$ of $M$, and for every vector $\xi\in
W_{(x,y)}$ such that $\pi_{1}(\xi)$ and $\pi_{2}(\xi)$ are colinear,

$\omega_{i}(R_{X,J^{\eta}X}X,J^{\eta}X)=\frac{-\epsilon\|(x,y)\|_{g}^{2}}{2\|x\|_{{{b}}}^{2}\|y\|_{{{b}}}^{2}}\hat{g}(X,I^{\eta}X)\hat{g}(X,X)\
.\eqnum{\text{{\rm(A.1)}}\hbox{}}$

Proof: Indeed, let $\lambda_{x}$ and $\lambda_{y}$ denote respectively the
${{b}}$-lengths of $x$ and $y$. The normal bundle of $W$ is generated by the
vector fields

$N_{1}:=\frac{1}{\lambda_{x}}(x,0),\ N_{2}:=\frac{1}{\lambda_{y}}(y,0),\
N_{3}:=\frac{1}{\lambda_{x}}(0,x),\ \text{and}\
N_{4}:=\frac{1}{\lambda_{y}}(0,y)\ .$

For $X$ and $Y$ vector fields taking values in $W$, the respective shape
operators of these normals are

$\eqalign{B_{1}(X,Y)&=-\frac{1}{\lambda_{x}}{{b}}(\pi_{1}(X),\pi_{1}(Y))\ ,\cr
B_{2}(X,Y)&=-\frac{1}{\lambda_{y}}{{b}}(\pi_{2}(X),\pi_{1}(Y))\ ,\cr
B_{3}(X,Y)&=-\frac{\eta}{\lambda_{x}}{{b}}(\pi_{1}(X),\pi_{2}(Y))\ ,\
\text{and}\cr
B_{4}(X,Y)&=-\frac{\eta}{\lambda_{y}}{{b}}(\pi_{2}(X),\pi_{2}(Y))\ .\cr}$

We now suppose that $\pi_{1}(X)$ and $\pi_{2}(X)$ are colinear. Furthermore,
we recall that

$\eqalign{\pi_{1}\circ J^{\eta}&=\eta A\circ\pi_{2}\ ,\cr\pi_{1}\circ
K&=A\pi_{1}\ ,\cr\pi_{2}\circ J^{\eta}&=A\circ\pi_{1}\ ,\
\text{and}\cr\pi_{2}\circ K&=-A\pi_{2}\ .\cr}$

Recall also that

$A^{2}=-\epsilon{\text{Id}}\ ,$

where $\epsilon$ here denotes the sign of ${{b}}$. For each $i$, let
$\sigma_{i}$ denote the component of curvature arising from $B_{i}$. We obtain

$\eqalign{\sigma_{1}(X,J^{\eta}X,X,KX)&=-\frac{1}{\|x\|_{{b}}^{2}}{{b}}(\pi_{1}(X),\pi_{1}(X)){{b}}(\pi_{1}(J^{\eta}X),\pi_{1}(KX))\cr&=-\frac{\eta}{\|x\|_{{b}}^{2}}{{b}}(\pi_{1}(X),\pi_{1}(X)){{b}}(A\pi_{2}(X),A\pi_{1}(X))\cr&=-\frac{\epsilon\eta}{\|x\|_{{b}}^{2}}{{b}}(\pi_{1}(X),\pi_{1}(X)){{b}}(\pi_{1}(X),\pi_{2}(X))\
.\cr}$

In a similar manner, we obtain

$\eqalign{\sigma_{2}(X,J^{\eta},X,KX)&=-\frac{\epsilon}{\|y\|_{{b}}^{2}}{{b}}(\pi_{1}(X),\pi_{1}(X)){{b}}(\pi_{2}(X),\pi_{1}(X))\
,\cr\sigma_{3}(X,J^{\eta},X,KX)&=\frac{\epsilon}{\|x\|_{{b}}^{2}}{{b}}(\pi_{1}(X),\pi_{2}(X)){{b}}(\pi_{2}(X),\pi_{2}(X))\
,\
\text{and}\cr\sigma_{4}(X,J^{\eta},X,KX)&=\frac{\epsilon\eta}{\|y\|_{{b}}^{2}}{{b}}(\pi_{2}(X),\pi_{2}(X)){{b}}(\pi_{1}(X),\pi_{1}(X))\
.\cr}$

Combining these identities yields

$\eqalign{g(R_{X,J^{\eta}X}X,KX)&={{b}}(\pi_{1}(X),\pi_{2}(X))\bigg{(}-\frac{\epsilon\eta}{\|x\|_{{b}}^{2}}\hat{g}(X,X)-\frac{\epsilon}{\|y\|_{{b}}^{2}}\hat{g}(X,X)\bigg{)}\cr&=\frac{-\epsilon\|(x,y)\|_{g}^{2}}{2\|x\|_{{b}}^{2}\|y\|_{{b}}^{2}}\hat{g}(X,I^{\eta}X)\hat{g}(X,X)\
,\cr}$

as desired. $\square$

B - Bibliography. $[1]$ Barbot T., Béguin F., Zeghib A., Prescribing Gauss
curvature of surfaces in $3$-dimensional spacetimes: application to the
Minkowski problem in the Minkowski space, Ann. Inst. Fourier, 61, no. 2,
(2011), 511–-591  $[2]$ Bianchi L., Sulle superficie a curvatura nulla in
geometria ellittica, Ann. Mat. Pura Appl., 24, 93–129, (1896)  $[3]$ Bonsante
F., Mondello G., Schlenker J. M., A cyclic extension of the earthquake flow I,
Geom. Topol., 17, no. 1, (2013), 157–234  $[4]$ Bonsante F., Mondello G.,
Schlenker J. M., A cyclic extension of the earthquake flow II, Ann. Sci. Ec.
Norm. Supér., 48, no. 4, (2015), 811–-859  $[5]$ Clifford W. K., Preliminary
sketch of biquaternons, Proc. London Math. Soc., IV, 381–395, (1873)  $[6]$
Clifford W. K., Applications of Grassmann’s extensive algebra, Amer. J. Math.,
1, 350–358, (1878)  $[7]$ Clifford W. K., On the classification of geometric
algebras, in Mathematical papers of William Kingdon Clifford, Tucker R. (ed),
MacMillan, London, (1882)  $[8]$ Cruceanu V., Fortuny P., Gadea P. M., A
survey on paracomplex geometry, Rocky Mt. J. Math., 26, no. 1, 83–115, (1996)
$[9]$ Dajczer M., Nomizu K., On flat surfaces in $\mathbb{S}^{3}$ and
$\mathbb{H}^{3}$, in Manifolds and Lie Groups, Hano J., Morimoto A., Murakami
S., Okamoto K., Ozeki H. (eds.), Progress in Mathematics, 14, Birkhäuser-
Verlag, (1981)  $[10]$ Enomoto K., The Gauss image of flat surfaces in
$\mathbb{R}^{4}$, Kodai Math. J., 9, 19–32, (1986)  $[11]$ Enomoto K., Global
properties of the Gauss image of flat surfaces in $\mathbb{R}^{4}$, Kodai
Math. J., 10, 272–284, (1987)  $[12]$ Enomoto K., Kitigawa Y., Weiner J. L., A
rigidity theorem for the Clifford tori in $\mathbb{S}^{3}$, Proc. Amer. Math.
Soc., 124, 265–268, (1996)  $[13]$ Fillastre F., Smith G., Group actions and
scattering problems in Teichmüller theory, in Handbook of group actions IV,
Advanced Lectures in Mathematics, 40, (2018), 359–417  $[14]$ Gálvez J. A.,
Mártinez A., Milán F., Flat surfaces in the hyperbolic $3$-space, Math. Ann.,
316, no. 3, 419–435, (2000)  $[15]$ Gálvez J. A., Mira P., Embedded isolated
singularities of flat surfaces in hyperbolic $3$-space, Cal. Var. PDE., 24,
239–260, (2005)  $[16]$ Gálvez J. A., Mira P., Isometric immersions of
$\mathbb{R}^{2}$ into $\mathbb{R}^{4}$ and perturbation of Hopf tori, Math.
Zeit., bf 266, 207–227, (2010)  $[17]$ Garling D. J. H., Clifford algebras: an
introduction, Cambridge University Press, (2011)  $[18]$ Goldstein H. P.,
Classical mechanics, Pearson, (2001)  $[19]$ Harvey F. R., Lawson H. B.,
Pseudoconvexity for the special lagrangian potential equation, Calc.Var.
PDEs., 60, (2021)  $[20]$ Hitchin N. J., The moduli space of special
lagrangian submanifolds, , Annali della Scuola Normale Superiore de Pisa,