# On Some Local Geometry of Bi-Contact Structures

Taylor J. Klotz and George R. Wilkens  Department of Mathematics, Universty of
Hawai‘i at Mānoa, Keller Hall 401A, Honolulu, Hawai‘i 96822<EMAIL_ADDRESS>
###### Abstract.

We investigate the local geometry of a pair of independent contact structures
on 3-manifolds under maps that independently preserve each contact structure.
We discover that such maps are homotheties on the contact 1-forms and we
discover differential invariants associated to such structures under these
equivalences. This allows us to generalize the notion of contact circles and
(equilateral) hyperbolas to contact ellipses and hyperbolas. Moreover, these
invariants may sometimes be used to define a complete local normal form and in
at least one case are related to symplectic structures through a natural
prolongation in the Cartan equivalence method. Finally, there is a type of
natural Riemannian metric (but not necessarily the well-known associated
contact metric) and we discover certain curvatures may be written in terms of
the bi-contact differential invariants.

###### Key words and phrases:

Contact Geometry, Exterior Differential Systems, Moving Frames, Differential
Geometry

###### 2010 Mathematics Subject Classification:

53D22, 53B20, 58A15

###### Contents

  1. 1 Introduction
    1. 1.1 Contact Ellipses and Hyperbolas
  2. 2 The Equivalence Problem
    1. 2.1 Bi-contact Equivalence
    2. 2.2 The invariant function $C$
    3. 2.3 Non-Constant C
    4. 2.4 Constant C
    5. 2.5 $(-\epsilon)$-Cartan Structures
  3. 3 A Local Normal Form
  4. 4 An Invariant Metric
    1. 4.1 Bi-contact Equivalence $C_{3}=0$ Connection
    2. 4.2 Bi-contact Equivalence $C_{3}\neq 0$ Connection
    3. 4.3 Minimal Contact Hypersurfaces in a Symplectic Manifold
  5. 5 Conclusion and Further Work

##  1\. Introduction

Contact geometry and topology has become a large industry of mathematical
research in recent decades, especially in the context of additional compatible
structures [2]. Most results that do not involve additional structures are
topological in nature due to the wonderful theorem of Pfaff, which says that
there exists a local coordinate system $(x^{i},u,p_{i})$ about any point $p$
in a contact manifold $M$ such that the contact 1-form $\theta$ has the
expression $\theta=du-p_{i}\,dx^{i}$. In other words, all (co-orientable)
contact 1-forms have the same local geometry and geometric questions become
global ones concerning a contact structure’s associated Legendrian
submanifolds, or other interesting embedding properties such as overtwisted
disks.

However, if one were to simultaneously consider two linearly independent
contact structures on the same manifold $M$, known as bi-contact
structures[15], then the Pfaff/Darboux normal form doesn’t necessarily apply
to both contact structures simultaneously. It is then worth asking if such a
normal form exists or if instead local invariant arise out of equivalences
determined by maps which simultaneously preserve both contact structures as
contactomorphisms.

######  Definition 1.

Let $M$ be an orientable 3-manifold with two contact distributions
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ such that
$\mathcal{D}_{1}(p)\neq\mathcal{D}_{2}(p)$ for every $p\in M$. Then we call
$(\mathcal{D}_{1},\mathcal{D}_{2})$ a bi-contact structure. Whenever, for
$i=1,2$, $\mathcal{D}_{i}=\ker\omega^{i}$ for 1-forms $\omega^{i}$, then we
may refer to $(\omega^{1},\omega^{2})$ as the bi-contact structure.

Bi-contact structures are not to be confused with so-called contact pairs,
which were introduced with such terminologoy in [1], but were once called bi-
contact structures in [3]. However, we will discover that there is some
relation between certain bi-contact structures on 3-manifolds and symplectic
couples/pairs on an associated 4-manifold, see 6. The notion of symplectic
couple was introduced in the work of Geiges in [7].

Many of our results will be local, and when they are not local, we will assume
our bi-contact structures are both co-orientable. As such, it may be assumed
that all bi-contact structures mentioned here are co-orientable. In fact, we
will study bi-contact structures via differential forms generally.

######  Definition 2.

A map $\Phi:M\to M$ is called a bicontactomorphism if $\Phi$ if a
contactomorphism for both $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$
simultaneously.

Bi-contact structures have been studied since at least the early 90’s. Indeed,
as far as the author’s have found, it was Mitsumatsu who introduced the
concept in [15], but with an additional requirement that the two contact forms
induce opposite orientations. We dispense with this requirement as we are
interested in studying the equivalence problem with respect to the above
mentioned bi-contactomorphisms. It turns out that the opposing orientations
will actually define a particular subclass of bi-contact structures that is
detected through the first step of torsion normalization in Cartan’s method of
equivalence.

Among the bi-contact structures, we will introduce three general cases in
Definition 5 in Section 2.2: elliptic bi-contact structures, hyperbolic bi-
contact structures, and linear bi-contact structures. The linear case will not
be explored in this work but merits additional study. Moreover, as special
cases, we will naturally discover bi-contact circles and bi-contact
equilateral hyperbolas. Bi-contact circles were first introduced by Geiges and
Gonzalo in [8]. Perrone studied bi-contact circles further by adding an
additional compatible Riemannian metric structure known as an associated
contact metric [17]. More recently, Perrone also introduced the notion of bi-
contact equilateral hyperbolas in [18] (which Perrone simply called contact
hyperbolas) and again studied such structures with associated contact metrics.
Our work does not invoke any special notions of compatibility with other
structures; however, we do recover an invariant Riemannian metric.

Moreover, we detect the special case of $(-\epsilon)$-Cartan structures. Which
are a further specialization among bi-contact circles and bi-contact
hyperbolas.

###  1.1. Contact Ellipses and Hyperbolas

######  Definition 3.

A bi-contact structure is called a contact ellipse if

$\omega_{a}=a_{1}\omega^{1}+a_{2}\omega^{2},$

is a contact form for every $a=(a_{1},a_{2})$ belonging to an ellipse
$E_{S^{1}}$ in $\mathbb{R}^{2}$. Similarly, we say that a bi-contact structure
is a contact hyperbola if

$\omega_{a}=a_{1}\omega^{1}+a_{2}\omega^{2},$

is a contact form for every $a=(a_{1},a_{2})$ belonging to a hyperbola
$H^{1}_{r}$ in $\mathbb{R}^{2}$ where $r=\pm 1$ determines the branch of the
hyperbola. In addition, if a contact ellipse $\omega_{a}$ generates the same
volume form $\omega_{a}\wedge d\omega_{a}$ independent of $a\in E_{S^{1}}$
then $\omega_{a}$ is called a taut contact ellipse. If a contact hyperbola
$\omega_{a}$ generates the same volume form up to sign $\omega_{a}\wedge
d\omega_{a}=r\omega^{1}\wedge d\omega^{1}=-r\omega^{2}\wedge d\omega^{2}$
independent of $a\in H^{1}_{r}$ then $\omega_{a}$ is called a taut contact
hyperbola.

The special case of taut contact circles and taut contact hyperbolas (for the
equilateral hyperbolas) have been studied before. Indeed, 3-manifolds
admitting taut contact circles have been completely classified in the compact
case [8] and taut contact equilateral hyperbolas have been well-studied by
Perrone and Zessin [18, 20], but are not completely understood. Moreover,
there is a beautiful theory of contact circles up to conformal/homothety class
again by Geiges and Gonzalo in [10],[11]. As such, bi-contactmorphism has in
this way been studied before, but in a way specific to taut contact circles.

######  Definition 4.

A bi-contact structure $(\omega^{1},\omega^{2})$ is called a
$(-\epsilon)-$Cartan Structure if

$\omega^{1}\wedge d\omega^{2}=\omega^{2}\wedge d\omega^{1}=0.$

The $(-\epsilon)-$Cartan Structures are significantly more rigid than taut
contact circles and hyperbolas.

Since we study the general problem of equivalence under bi-contactomorphisms,
we will naturally discover both previous results and new geometry that is
independent of additional structure, allowing the study of bi-contact
structures to stand on its own. Moreover, there are natural generalizations,
not only in the way of considering contact spheres and hyperboloids in the
sense of Perrone and Zessin, but also in the sense of curves of higher degree
polynomials on odd dimensional manifold of dimension 5 and higher.

Lastly, we mention that there is some work to study the global topology of
Anosov flows on 3-manifolds, which are intimately related to bi-contact
structures [19, 12, 15, 14], including a development of Floer theory applied
to such bi-contact structures [5]. Although we are restricting ourselves to a
geometry of bi-contact equivalence, we hope that perhaps some of the structure
equations found here are helpful for such efforts. Indeed, in a forthcoming
work, we will study applications of these structure equations to Anosov flows
and Beltrami fields.

The paper is organized as follows: in Section 2 we setup the equivalence
problem and prove that bi-contactomorphisms are in fact homotheties of the
contact 1-forms. Moreover, we discover a local differential invariant $C$ and
classify bi-contact structures as elliptic, hyperbolic, or linear based on a
quadratic form defined by $C$ and an orientation condition. This quadratic
form appears naturally from the equivalence problem and was found by Geiges
and Gonzalo in [8] and [9], but only closely studied when it represented a
circle of contact structures. We then apply the equivalence method to three
cases based on $dC$. In each case, we can either find an $e$-estructure
locally, with complicated torsion, or we can prolong to $e$-structures with
simpler torsion or to an involutive system. In Secton 3 we provide a local
normal form in a special case. In Section 4 we explore some curvature
calculations of natural Riemannian metrics from our structure equations. We
discover that some submanifolds have mean curvature in terms of differential
invariants of bi-cintact structures, and in one case, we discover that if an
associated symplectic 4-manifold to a special type of bi-contact structure is
closed, then it must have positive Euler characteristic.

##  2\. The Equivalence Problem

Here we implement E. Cartan’s method of equivalence to study the problem of
equivalence of bi-contact structures under bi-contactomorphisms. The
uninitiated may consult [13, 6, 16] for an introduction to this technique.

###  2.1. Bi-contact Equivalence

Let $M$ be a three dimensional manifold with coframing
$\\{\omega^{1},\omega^{2},\omega^{3}\\}$ such that $\omega^{1}$ and
$\omega^{2}$ are contact forms that are linearly independent and $\omega^{3}$
any 1-form such that $\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\neq 0$. Thus
we should expect that a change of coframe under a bi-contactomorphism should
be as follows

$\displaystyle\begin{bmatrix}\tilde{\omega}^{1}\\\ \tilde{\omega}^{2}\\\
\tilde{\omega}^{3}\\\ \end{bmatrix}=\begin{bmatrix}a_{1}&0&0\\\ 0&a_{2}&0\\\
b_{1}&b_{2}&b_{3}\\\ \end{bmatrix}^{-1}\begin{bmatrix}\omega^{1}\\\
\omega^{2}\\\ \omega^{3}\end{bmatrix}.$

Let $G_{0}<GL(3,\mathbb{R})$ be the group given by invertible matrices of the
above form. Then the structure equations on the principal $G_{0}$ bundle
$\mathcal{B}_{0}\subset\mathcal{F}_{GL(3,\mathbb{R})}$ are

(1) $d\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{bmatrix}=-\begin{bmatrix}\alpha_{1}&0&0\\\ 0&\alpha_{2}&0\\\
\beta_{1}&\beta_{2}&\beta_{3}\\\
\end{bmatrix}\wedge\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{bmatrix}+\begin{bmatrix}T^{1}_{23}\,\omega^{2}\wedge\omega^{3}\\\
T^{2}_{13}\,\omega^{1}\wedge\omega^{3}\\\ 0\end{bmatrix}$

. Notice that from these structure equations

$\omega^{1}\wedge
d\omega^{1}=T^{1}_{23}\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\text{ and
}\omega^{2}\wedge
d\omega^{2}=-T^{2}_{13}\omega^{1}\wedge\omega^{2}\wedge\omega^{3}$

so that the non-vanishing intrinsic torsion for this $0$-adapted coframe
encodes that $\omega^{1}$ and $\omega^{2}$ ought to descend to contact 1-forms
on $M$. We can find a normalization of this torsion by considering the induced
action of $G_{0}$ on the intrinsic torsion. Indeed, let us rewrite the
structure equations (1) as $d\omega=-\theta\wedge\omega+T(\omega\wedge\omega)$
where $\theta$ is the (pseudo)connection form. Now we can check what happens
to the intrinsic torsion under the change of coframe defined by $G_{0}$.
Indeed we get that $d(g\,\tilde{\omega})=-\theta
g\wedge\tilde{\omega}+T(g\tilde{\omega}\wedge g\tilde{\omega})$ which may be
rewritten again as
$d\tilde{\omega}=-\tilde{\theta}\wedge\tilde{\omega}+g^{-1}\,T(g\tilde{\omega}\wedge
g\tilde{\omega})$ where
$\tilde{\theta}=g^{-1}dg+g^{-1}\theta\,g+t(\tilde{\omega})$ where
$t(\tilde{\omega})$ is the absorbable torsion of $\tilde{T}$. However,
$d\tilde{\omega}=-\tilde{\theta}\wedge\tilde{\omega}+\tilde{T}(\tilde{\omega}\wedge\tilde{\omega})$
so upon comparing intrinsic torsion terms one can finds that
$\tilde{T}^{1}_{23}=\frac{a_{1}}{a_{2}b_{3}}\,T^{1}_{23}$ and
$\tilde{T}^{2}_{13}=\frac{a_{2}}{a_{1}b_{3}}T^{2}_{13}$. Hence
$\tilde{T}^{1}_{23}\tilde{T}^{2}_{13}=\frac{1}{b_{3}^{2}}T^{1}_{23}T^{2}_{13}$
meaning that the sign of $T^{1}_{23}T^{2}_{13}$ is fixed with $b_{3}=\delta$,
$a_{2}=\delta a_{1}$, where $\delta=\pm 1$. Thus, we can reduce to a new
structure group $G_{1}$.

Let $T^{1}_{23}T^{2}_{13}=\epsilon$ where $\epsilon=\pm 1$ and in the
following we take $T^{1}_{23}=1$ and $T^{2}_{13}=\epsilon$. This immediately
yields

(2) $\omega^{1}\wedge d\omega^{1}=-\epsilon\omega^{2}\wedge d\omega^{2}.$

Thus, we begin to see the relation to contact circles and contact hyperbolas.
Let us label $a=a_{1}$ so that $a_{2}=\delta a$. This yields a restriction to
a subbundle $\mathcal{B}_{1}\hookrightarrow\mathcal{B}_{0}$ with group
$G_{1}<G_{0}$ described by the change of coframe

(3) $\begin{bmatrix}\tilde{\omega}^{1}\\\ \tilde{\omega}^{2}\\\
\tilde{\omega}^{3}\\\ \end{bmatrix}=\begin{bmatrix}a&0&0\\\ 0&\delta a&0\\\
b_{1}&b_{2}&\delta\\\ \end{bmatrix}^{-1}\begin{bmatrix}\omega^{1}\\\
\omega^{2}\\\ \omega^{3}\end{bmatrix}.$

Any coframe that transforms according to (3) will be called 1-adapted. Notice
also that $\mathcal{B}_{1}$ is a double cover of $M$ from the presence of the
sign ambiguity in (3). The 1-adapted structure equations are given by

$\displaystyle d\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{bmatrix}=-\begin{bmatrix}\alpha&0&0\\\ 0&\alpha&0\\\
\beta_{1}&\beta_{2}&0\\\ \end{bmatrix}\wedge\begin{bmatrix}\omega^{1}\\\
\omega^{2}\\\
\omega^{3}\end{bmatrix}+\begin{bmatrix}T^{1}_{13}\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3}\\\
T^{2}_{23}\omega^{2}\wedge\omega^{3}+\epsilon\omega^{1}\wedge\omega^{3}\\\
0\end{bmatrix}.$

We now have new torsion and notice that
$\alpha\mapsto\alpha-\frac{1}{2}(T^{1}_{13}+T^{2}_{23})\omega^{3}$ gives for
the torsion components
$\frac{1}{2}(T^{1}_{13}-T^{2}_{23})\omega^{1}\wedge\omega^{3}$ and
$-\frac{1}{2}(T^{1}_{13}-T^{2}_{23})\omega^{2}\wedge\omega^{3}$. Calling
$C=-\frac{1}{2}(T^{1}_{13}-T^{2}_{23})\omega^{2}\wedge\omega^{3}$ we rewrite
the structure equations as

(4) $d\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{bmatrix}=-\begin{bmatrix}\alpha&0&0\\\ 0&\alpha&0\\\
\beta_{1}&\beta_{2}&0\\\ \end{bmatrix}\wedge\begin{bmatrix}\omega^{1}\\\
\omega^{2}\\\
\omega^{3}\end{bmatrix}+\begin{bmatrix}-C\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3}\\\
\epsilon\omega^{1}\wedge\omega^{3}+C\omega^{2}\wedge\omega^{3}\\\
0\end{bmatrix}.$

We discover that bi-contactomorphisms are homotheties of the bi-contact
structure. These homotheties have been mentioned before in [8] and [18], but
only in the context of the torsion function $C=0$.

Notice that (4) have not arisen from a reduction of the structure group
$G_{1}$. Rather, we eliminated the ambiguity in the torsion via torsion
absorption. The resulting intrinsic torsion is fundamental to bi-contact
structures.

######  Proposition 1.

$C$ is an invariant function on $M$. Moreover, $C$ is the local obstruction
for the bi-contact structure to be taut.

###### Proof.

Notice that

$\displaystyle(d^{2}\omega^{1})\wedge\omega^{2}+(d^{2}\omega^{2})\wedge\omega^{1}=-2\,dC\wedge\omega^{1}\wedge\omega^{2}\wedge\omega^{3},$

and therefore there are functions $C_{i}$, $i=1,2,3$ on $M$ such that
$dC=C_{i}\omega^{i}$ and so $C$ is an invariant function on $M$. It is then
immediate what role the invariant function $C$ plays. Indeed,

(5) $\omega^{1}\wedge d\omega^{2}+\omega^{2}\wedge
d\omega^{1}=2C\,\omega^{1}\wedge\omega^{2}\wedge\omega^{3},$

and so by easy theorems in [8, 9] and [17, 18] we have that the invariant
function $C$ is an obstruction to being a contact circle of equilateral
hyperbola. ∎

That $\omega^{1}\wedge d\omega^{2}+\omega^{2}\wedge d\omega^{1}$ vanishing
defines contact circles and equilateral hyperbolas is a part of the definition
of such structures. What is interesting is that this arises naturally when
considering equivalence under bi-contactomorphisms.

###  2.2. The invariant function $C$

We investigate some of the implications of the invariant function $C$ and its
exterior derivative $dC$. It naturally leads us to introduce the more general
notion of elliptic bi-contact structures and hyperbolic bi-contact structures.
The key behind properly interpreting the $C$ invariant is to think carefully
about the following. Let $(\omega^{1},\omega^{2})$ be a bi-contact structure
on a 3-manifold $M$. Now let $\omega_{a}=a_{1}\omega^{1}+a_{2}\omega^{2}$ for
some real numbers $a_{1}$ and $a_{2}$. In order for $\omega_{a}$ to be a
contact form, we need it to induce a volume form on $M$. From the structure
equations (4) we have

$\omega_{a}\wedge d\omega_{a}=(a_{1}^{2}-\epsilon
a_{2}^{2}+2Ca_{1}a_{2})\Omega,$

where $\Omega=\omega^{1}\wedge\omega^{2}\wedge\omega^{3}$. Therefore, in order
that $\omega_{a}$ be a contact form, we need simply avoid the zero locus of
the quadratic form $\mathcal{P}_{C}(a):=a_{1}^{2}-\epsilon
a_{2}^{2}+2Ca_{1}a_{2}$.

######  Definition 5.

Given a bi-contact structure $(\omega^{1},\omega^{2})$ on a 3-manifold $M$, if
the $(a_{1},a_{2})$ for $\omega_{a}$ are chosen in such a way that
$\mathcal{P}_{C}(a)=r$ for $r=\pm 1$ then we say the bi-contact structure is
taut. Moreover, if $\mathcal{P}_{C}(a)$ is a definite quadratic form then we
call the bi-contact structure elliptic. If $\mathcal{P}_{C}(a)$ is indefinite
then we say the bi-contact structure is hyperbolic. Finally, if
$\mathcal{P}_{C}(a)$ is degenerate then we say that the bi-contact structure
is linear.

Thus the following proposition is immediate.

######  Proposition 2.

Let $(\omega^{1},\omega^{2})$ be a bi-contact structure on 3-manifold $M$.

  1. (1)

If $\epsilon=-1$ and $|C|<1$ then the bi-contact structure is elliptic and
admits a taut contact ellipse.

  2. (2)

If $\epsilon=-1$ and $|C|=1$ then the bi-contact structure is linear and
admits a taut contact line.

  3. (3)

If $\epsilon=1$ or $|C|>1$ with $\epsilon=-1$ then the bi-contact structure is
hyperbolic and admits a taut contact hyperbola $\omega_{a}$. Additionally, if
the taut contact hyperbola is induced by the case $\epsilon=1$ then
$\omega^{1}\wedge d\omega^{1}=-\omega^{2}\wedge d\omega^{2}$ and if the taut
contact hyperbola is induced by the case $|C|>1$ and $\epsilon=-1$ then
$\omega^{1}\wedge d\omega^{1}=\omega^{2}\wedge d\omega^{2}$.

Moreover, it is interesting to note that we need not restrict ourselves
entirely to the case that $(a_{1},a_{2})$ are constant on $M$.

######  Proposition 3.

Let $\omega^{3}$ be a 1-form completing a bi-contact structure
$(\omega^{1},\omega^{2})$ on a 3-manifold $M$ to a local co-framing. Then
contact circles and contact hyperbolas may be parameterized by two functions
$a_{1},a_{2}\in C^{2}(M)$ such that

$a_{1}a_{2,3}-a_{2}a_{1,3}=0$

where $a_{i,3}$ are the covariant derivatives of $a_{i}$ in the direction
$e_{3}$ which is dual to $\omega^{3}$. Moreover, if $\omega^{3}$ locally
defines a foliation of $M$ then the ratio $a_{1}/a_{2}$ is a first integral of
$e_{3}$.

###### Proof.

Let $\omega_{a}=a_{1}\omega^{1}+a_{2}\omega^{2}$. Then

$\omega_{a}\wedge
d\omega_{a}=\mathcal{P}_{C}(a)\Omega+\omega_{a}\wedge(da_{1}\wedge\omega^{1}+da_{2}\wedge\omega^{2}),$

where $\Omega=\omega^{1}\wedge\omega^{2}\wedge\omega^{3}$. Expanding the
$da_{i}$ terms and computing the wedge product immediately yields the claimed
differential equation for $a_{1}$ and $a_{2}$. Note also that

$\left(\frac{a_{1}}{a_{2}}\right)_{3}=\frac{a_{1}a_{2,3}-a_{2}a_{1,3}}{a_{2}^{2}},$

and so when $\omega^{3}$ defines a foliation we have that $a_{1}/a_{2}$ is
constant along integral curves of $e_{3}$. ∎

The invariant function $C$ is the first obstruction of a given 1-adapted (i.e.
the volume forms are fixed) bi-contact structure to admit taut contact circles
or taut equilateral hyperbolas. However, as Theorem 4 below demonstrates, the
derivative of $C$ is the true obstruction to the existence of taut contact
circles and taut contact equilateral hyperbolas outside of the homothety/bi-
contactomorphism class of a given bi-contact structure.

######  Proposition 4.

Let $(\omega^{1},\omega^{2})$ be a bi-contact structure on 3-manifold $M$ such
that $dC\wedge\omega^{1}\wedge\omega^{2}=0$ on $M$. If the bi-contact
structure is elliptic then $M$ admits a taut contact circle that is not bi-
contactomorphic to $(\omega^{1},\omega^{2})$. If the bi-contact structure is
hyperbolic then $M$ admits a taut contact equilateral hyperbola that is not
bi-contactomorphic to $(\omega^{1},\omega^{2})$.

###### Proof.

We construct such taut bi-contact structures directly in terms of the given
bi-contact structure as though we were diagonalizing the quadratic form
$\mathcal{P}_{C}(a)$. Let us first consider the case of $\epsilon=-1$. Indeed,
let

(6) $\displaystyle\eta^{1}$
$\displaystyle=\frac{1}{\sqrt{|1+C|}}\frac{(\omega^{1}+\omega^{2})}{\sqrt{2}},$
$\displaystyle\eta^{2}$
$\displaystyle=\frac{1}{\sqrt{|1-C|}}\frac{(\omega^{1}-\omega^{2})}{\sqrt{2}}.$

First note that

$\eta^{1}\wedge\eta^{2}=-\frac{1}{\sqrt{|1-C^{2}|}}\omega^{1}\wedge\omega^{2},$

and therefore to have $(\eta^{1},\eta^{2})$ to be a 1-adapted bi-contact
structure, we need to define $\eta^{3}=-\sqrt{(|1-C^{2}|}\,\,\omega^{3}$ so
that the volume forms from $(\eta^{1},\eta^{2})$ and $(\omega^{1},\omega^{2})$
agree. Now we need simply check that the 1-forms $\eta^{1}$ and $\eta^{2}$
define taut contact circles for $|C|<1$ and taut contact equilateral
hyperbolas when $|C|>1$. Routine calculation yields the following

$\displaystyle d\eta^{1}$
$\displaystyle=\frac{1}{\sqrt{|1+C|}}\frac{d\omega^{1}+d\omega^{2}}{\sqrt{2}}+\frac{\text{sgn}(1+C)}{2}\frac{dC}{|1+C|^{3/2}}\wedge\frac{(\omega^{1}+\omega^{2})}{\sqrt{2}},$
$\displaystyle d\eta^{2}$
$\displaystyle=\frac{1}{\sqrt{|1-C|}}\frac{d\omega^{1}-d\omega^{2}}{\sqrt{2}}-\frac{\text{sgn}(1-C)}{2}\frac{dC}{|1-C|^{3/2}}\wedge\frac{(\omega^{1}-\omega^{2})}{\sqrt{2}},$
$\displaystyle\eta^{1}\wedge d\eta^{1}$
$\displaystyle=\frac{1}{2|1+C|}(\omega^{1}\wedge d\omega^{1}+\omega^{1}\wedge
d\omega^{2}+\omega^{2}\wedge d\omega^{1}+\omega^{2}\wedge d\omega^{2}),$
$\displaystyle=\frac{1+C}{|1+C|}\Omega=\text{sgn}(1+C)\Omega,$
$\displaystyle\eta^{2}\wedge d\eta^{2}$
$\displaystyle=\frac{1}{2|1-C|}(\omega^{1}\wedge d\omega^{1}-\omega^{1}\wedge
d\omega^{2}-\omega^{2}\wedge d\omega^{1}+\omega^{2}\wedge d\omega^{2}),$
$\displaystyle=\frac{1-C}{|1-C|}\Omega=\text{sgn}(1-C)\Omega,$
$\displaystyle\eta^{1}\wedge d\eta^{2}$
$\displaystyle=-\frac{1}{2|1-C^{2}|}\left(\omega^{1}\wedge
d\omega^{2}-\omega^{2}\wedge
d\omega^{1}\right)+\frac{1}{4}\frac{\text{sgn}(1-C)dC}{|1-C|\sqrt{|1-C^{2}|}}\wedge\omega^{1}\wedge\omega^{2},$
$\displaystyle\eta^{2}\wedge d\eta^{1}$
$\displaystyle=\frac{1}{2|1-C^{2}|}\left(\omega^{1}\wedge
d\omega^{2}-\omega^{2}\wedge
d\omega^{1}\right)+\frac{1}{4}\frac{\text{sgn}(1+C)dC}{|1+C|\sqrt{|1-C^{2}|}}\wedge\omega^{1}\wedge\omega^{2}.$

As such, we find that

$\eta^{1}\wedge d\eta^{2}+\eta^{2}\wedge
d\eta^{1}=C_{3}\,\frac{\text{sgn}(1-C)|1+C|+\text{sgn}(1+C)|1-C|}{2(|1-C^{2}|)^{3/2}}\,\Omega$

and hence if $(a_{1},a_{2})\in S^{1}\subset\mathbb{R}^{2}$ with $|C|<1$ for
$\eta_{a}=a_{1}\eta^{1}+a_{2}\eta^{2}$ then

$\displaystyle\eta_{a}\wedge
d\eta_{a}=\left(1+\frac{C_{3}}{(1-C^{2})^{3/2}}\right)\Omega.$

In case $(a_{1},a_{2})\in H^{1}_{r}$ with $|C|>1$ then

$\displaystyle\eta_{a}\wedge
d\eta_{a}=\left(1-\frac{C_{3}}{(C^{2}-1)^{3/2}}\right)\Omega.$

In both cases we see that if $C_{3}=0$ on $M$, equivalently
$dC\wedge\omega^{1}\wedge\omega^{2}=0$, then the contact forms $\eta^{1}$ and
$\eta^{2}$ form a contact circle or a contact equilateral hyperbola. To handle
the remaining hyperbolic case, we will apply the same argument, but with
different 1-forms $\eta^{1}$ and $\eta^{2}$. In what follows, it is slightly
more convenient to express $C=\sinh(\theta)$ and work with
hyperbolic/exponential functions of some $\theta\in C^{\infty}(M)$. Let

$\displaystyle\eta^{1}$
$\displaystyle=\frac{\cosh\left(\frac{\theta}{2}\right)\omega^{1}+\sinh\left(\frac{\theta}{2}\right)\omega^{2}}{\cosh(\theta)},$
$\displaystyle\eta^{2}$
$\displaystyle=\frac{-\sinh\left(\frac{\theta}{2}\right)\omega^{1}+\cosh\left(\frac{\theta}{2}\right)\omega^{2}}{\cosh(\theta)}.$

Then straightforward calculations yield

$\displaystyle d\eta^{1}$
$\displaystyle=\frac{\cosh\left(\frac{\theta}{2}\right)d\omega^{1}+\sinh\left(\frac{\theta}{2}\right)d\omega^{2}}{\cosh(\theta)}-\frac{\left((\cosh(\theta)+2)\sinh\left(\frac{\theta}{2}\right)\omega^{1}+(\cosh(\theta)-2)\cosh\left(\frac{\theta}{2}\right)\omega^{2}\right)\wedge
d\theta}{2\cosh^{2}(\theta)},$ $\displaystyle d\eta^{2}$
$\displaystyle=\frac{-\sinh\left(\frac{\theta}{2}\right)d\omega^{1}+\cosh\left(\frac{\theta}{2}\right)d\omega^{2}}{\cosh(\theta)}+\frac{\left((\cosh(\theta)-2)\cosh\left(\frac{\theta}{2}\right)\omega^{1}-(\cosh(\theta)+2)\sinh\left(\frac{\theta}{2}\right)\omega^{2}\right)\wedge
d\theta}{2\cosh^{2}(\theta)},$

and therefore

$\displaystyle\eta^{1}\wedge d\eta^{1}$
$\displaystyle=\Omega-\frac{\omega^{1}\wedge\omega^{2}\wedge
d\theta}{2\cosh^{2}(\theta)},$ $\displaystyle\eta^{2}\wedge d\eta^{2}$
$\displaystyle=-\Omega-\frac{\omega^{1}\wedge\omega^{2}\wedge
d\theta}{2\cosh^{2}(\theta)},$ $\displaystyle\eta^{1}\wedge d\eta^{2}$
$\displaystyle=\frac{\omega^{2}\wedge d\omega^{1}-\omega^{1}\wedge
d\omega^{2}}{2\cosh(\theta)}-\frac{1}{2}\text{sech}(\theta)\text{tanh}(\theta)\omega^{1}\wedge\omega^{2}\wedge
d\theta,$ $\displaystyle\eta^{2}\wedge d\eta^{1}$
$\displaystyle=-\frac{\omega^{2}\wedge d\omega^{1}-\omega^{1}\wedge
d\omega^{2}}{2\cosh(\theta)}+\frac{1}{2}\text{sech}(\theta)\text{tanh}(\theta)\omega^{1}\wedge\omega^{2}\wedge
d\theta.$

Interestingly $\eta^{1}$ and $\eta^{2}$ can be used to define a contact
equilateral hyperbola, but not generally taut. Since $dC=\cosh(\theta)d\theta$
it is then immediately clear that $\eta^{1}$ and $\eta^{2}$ form a taut
contact equilateral hyperbola if and only if
$dC\wedge\omega^{1}\wedge\omega^{2}=0$ on all of $M$. ∎

######  Corollary of the Proof 1.

Let $M$ be a compact orientable 3-manifold and let $(M,\omega^{1},\omega^{2})$
be a bi-contact structure such that $dC\wedge\omega^{1}\wedge\omega^{2}=0$
everywhere on $M$. If $(\omega^{1},\omega^{2})$ is elliptic then $M$ is
diffeomorphic to the quotient of a Lie group G by a discrete subgroup acting
by left multiplication where $G=SU(2),\widetilde{SL}(2,\mathbb{R}),$ or
$\widetilde{E}(2)$ where $\tilde{\cdot}$ denotes the universal cover.
Moreover, any such 3-manifold $M$ realizes a distinct elliptic bi-
contactomorphism class for every $C\in C^{1}(M)$ such that
$dC\wedge\omega^{1}\wedge\omega^{2}=0$ everywhere on $M$.

###### Proof.

Note that the transformation (6) is well defined on all of $M$ and is of full
rank for any $C\in C^{1}(M)$ such that $dC\wedge\omega^{1}\wedge\omega^{2}=0$.
So we may always induce a taut contact circle on $M$. Then by the main result
of [8] -i.e. the classification of compact 3-manifolds admitting taut contact
circles- in conjunction with $C$ being an invariant under bicontactomorphism,
the Corollary follows. ∎

Before returning to the equivalence problem, we provide a local example on
some $\mathbb{R}^{3}$ of a bicontact structure (both hyperbolic and elliptic
depeneding on the value of $z$ and $\epsilon$) with
$dC\wedge\omega^{1}\wedge\omega^{2}\neq 0$.

######  Example 1.

Let $M=\mathbb{R}^{3}$ with coordinates $(x,y,z)$ and let $C=z$ with
$C_{3}=C_{3}(z)$ arbitrary as a non-zero function of $z$. Note that this means
$dC\wedge dC_{3}=0$.

(7) $\displaystyle\omega^{1}$
$\displaystyle=dx+dy-\frac{q_{1}(x,y,z)}{C_{3}(z)}dz,$
$\displaystyle\omega^{2}$ $\displaystyle=dx-
dy+\frac{q_{2}(x,y,z)}{C_{3}(z)}dz,$ $\displaystyle\omega^{3}$
$\displaystyle=\frac{dz}{C_{3}(z)},$

where

(8) $\displaystyle q_{1}(x,y,z)$ $\displaystyle=(x+y)z-(x-y),$ $\displaystyle
q_{2}(x,y,z)$ $\displaystyle=\epsilon(x+y)+(x-y)z.$

It is the case that

$\omega^{1}\wedge d\omega^{2}+\omega^{2}\wedge
d\omega^{1}=2z\omega^{1}\wedge\omega^{2}\wedge\omega^{3}.$

Matters being so, we find

$dC\wedge\omega^{1}\wedge\omega^{2}=\frac{1}{C_{3}(z)}\omega^{1}\wedge\omega^{2}\wedge\omega^{3}.$

which is nonzero. Curiously, this gives an example of what one might call a
mixed bi-contact structure in that for $\epsilon=-1$ and $|z|<1$ it is a
contact ellipse but when $|z|>1$ it is a contact hyperbola. Such structures
warrant additional investigation which we do not pursue here.

Returning to the equivalence problem, it is easy to see that the 1-form
$\alpha$ is unique since any other choice $\alpha^{\prime}$ would give
$(\alpha-\alpha^{\prime})\wedge\omega^{1}=0$ and
$(\alpha-\alpha^{\prime})\wedge\omega^{2}=0$. However, Cartan’s lemma gives

(9) $\begin{bmatrix}\beta_{1}\\\
\beta_{2}\end{bmatrix}=\begin{bmatrix}\beta^{\prime}_{1}\\\
\beta^{\prime}_{2}\end{bmatrix}+\begin{bmatrix}h_{11}&h_{12}\\\
h_{12}&h_{22}\end{bmatrix}\begin{bmatrix}\omega_{1}\\\
\omega_{2}\end{bmatrix}.$

It is then immediate that the tableaux (call it $A$) is not involutive since
$\dim({A}^{(1)})=3$ but the nonzero Cartan characters are $s_{1}=2$ and
$s_{2}=1$.

At this stage, we may either prolong to a larger bundle where the tautological
1-form includes the Lie algebra 1-forms or we may seek additional reduction of
the structure group by investigating the action on the 1-jet of the invariant
function $C$. The second option turns out to save us some effort for non-
constant $C$ and is more in spirit with the work of E. Cartan’s 1908 paper
[4]. Moreover, thanks to Theorem 4, we have direct geometric interpretations
for at least $C_{3}$. We cannot entirely avoid prolongation. We analyze the
case of constant $C$ later in Section 2.4, which will also require
prolongation.

###  2.3. Non-Constant C

We now check the action of the group $G_{1}$ on $dC$, where

$dC=C_{1}\tilde{\omega}^{1}+C_{2}\tilde{\omega}^{2}+C_{3}\tilde{\omega}^{3}.$

We can easily see that under a change of coframe

$dC=\left(\frac{C_{1}}{a}-\frac{b_{1}C_{3}}{a}\right)\omega^{1}+\left(\frac{C_{2}}{\delta
a}-\frac{b_{2}C_{3}}{\delta a}\right)\omega^{2}+C_{3}\omega^{3}.$

We now consider the different possibilities of each $C_{i}$ for $1\leq i\leq
3$ and their associated reductions of the structure group.

  1. (1)

Case 1: $dC\wedge\omega^{1}\wedge\omega^{2}=0$ ($C_{3}=0$). In this scenario,
we are left with the action of $G_{1}$ on $dC$ as

$dC=\frac{C_{1}}{a}\omega^{1}+\frac{C_{2}}{\delta a}\omega^{2}.$

In addition to reducing the group parameter $a$, we also discover that we may
normalize

$C_{1}^{2}+C_{2}^{2}=1,$

and as such we may take $C_{1}=\cos(\xi)$ and $C_{2}=\delta\sin(\xi)$ for some
function $\xi$ on $M$. However, the group parameters $b_{1}$ and $b_{2}$ are
still undetermined. In fact, the tableaux for the new structure equations is
easily seen to be involutive. That said, we have introduced additional torsion
into the structure equations. Indeed,

$\alpha=A_{1}\omega^{1}+A_{2}\omega^{2}+A_{3}\omega^{3},$

so that our new structure equations take the form

$\displaystyle d\omega^{1}$
$\displaystyle=A_{2}\omega^{1}\wedge\omega^{2}+(A_{3}-C)\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=-A_{1}\omega^{1}\wedge\omega^{2}+\epsilon\omega^{1}\wedge\omega^{3}+(A_{3}+C)\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=-\beta_{1}\wedge\omega^{1}-\beta_{2}\wedge\omega^{2}.$

Now we look to reduce again by normalizing the new torsion. The action of the
current group on this torsion is

$\displaystyle A_{1}$ $\displaystyle\mapsto A_{1}+b_{2}-(A_{3}+C)b_{1},$
$\displaystyle A_{2}$ $\displaystyle\mapsto A_{2}+b_{1}-(A_{3}-C)b_{2},$

Thus, we can translate away the torsion terms $A_{1}$ and $A_{2}$ and reduce
the group again to achieve an $e$-structure as long as $A_{3}^{2}-C^{2}\neq 1$
(we do not explore the alternative case). Upon this reduction the forms
$\beta_{1}$ and $\beta_{2}$ become basic so that

$\displaystyle\beta_{1}$
$\displaystyle=B_{5}\omega^{1}+\frac{1}{2}(B_{3}+B_{4})\omega^{2}+B_{2}\omega^{3},$
$\displaystyle\beta_{2}$
$\displaystyle=\frac{1}{2}(B_{4}-B_{3})\omega^{1}+B_{6}\omega^{2}+B_{1}\omega^{3},$

where we have chosen the labeling of the $B$ coefficients for simplicity of
presentation in the structure equations below

(10) $\displaystyle d\omega^{1}$
$\displaystyle=(A_{3}-C)\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=\epsilon\omega^{1}\wedge\omega^{3}+(A_{3}+C)\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=B_{3}\omega^{1}\wedge\omega^{2}+B_{2}\omega^{1}\wedge\omega^{3}+B_{1}\omega^{2}\wedge\omega^{3}.$

We have now achieved an $e$-structure, but with complicated orbits, and
equivalences determined by all the remaining torsion.

  2. (2)

Case 2: $dC\wedge\omega^{1}\wedge\omega^{2}\neq 0$ and $dC\wedge dC_{3}\neq
0$. In this scenario, we are left with the action of $G_{1}$ on $dC$ as

$dC=-\frac{C_{1}-b_{1}C_{3}}{a}\omega^{1}-\frac{C_{2}-b_{2}C_{3}}{\delta
a}\omega^{2}+C_{3}\omega^{3}.$

We see immediately that we can adapt our coframe so that we have the group
reduction $b_{1}=b_{2}=0$ by normalizing the $C_{1}=C_{2}=0$. We are still
left with one group parameter and the 1-forms $\beta_{1}$ and $\beta_{2}$
become basic, so that

$\displaystyle\beta_{1}$
$\displaystyle=B_{5}\omega^{1}+\frac{1}{2}(B_{3}+B_{4})\omega^{2}+B_{2}\omega^{3},$
$\displaystyle\beta_{2}$
$\displaystyle=\frac{1}{2}(B_{4}-B_{3})\omega^{1}+B_{6}\omega^{2}+B_{1}\omega^{3},$

where we have chosen the labeling of the $B$ coefficients for simplicity of
presentation in the structure equations below,

$\displaystyle d\omega^{1}$
$\displaystyle=-\alpha\wedge\omega^{1}-C\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=-\alpha\wedge\omega^{2}+\epsilon\omega^{1}\wedge\omega^{3}+C\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=B_{3}\omega^{1}\wedge\omega^{2}+B_{2}\omega^{1}\wedge\omega^{3}+B_{1}\omega^{2}\wedge\omega^{3}.$

As such, we have reduced the group $G_{1}$ to $G_{2}$ and we are now
restricted to a subbundle $\mathcal{B}_{2}\subset\mathcal{B}_{1}$. The
tableaux is still not involutive since $\dim(A^{(1)})=0$ and the first Cartan
character is nonzero. Thus, we seek an additional reduction of the structure
group. First, however, we may notice that $B_{3}=0$. Indeed, since
$dC=C_{3}\omega^{3}$, we can check the relations determined by $d^{2}C=0$.
Doing so, we find that

(11) $0=dC_{3}\wedge\omega^{3}+C_{3}d\omega^{3},$

meaning that $B_{3}$ must be zero. Since we are assuming that $dC\wedge
dC_{3}\neq 0$ on $M$ then $B_{1}$ and $B_{2}$ cannot both simultaneously
vanish. As such, we now check the action of the group on the remaining
torsion.

$\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}\mapsto\begin{bmatrix}a&0\\\
0&\delta a\end{bmatrix}\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}.$

We recognize that $a^{2}(B_{1}^{2}+B_{2}^{2})$ may be normalized to $1$ and so
we may reduce the structure group again so that $a=1$. This means there is a
function $\zeta$ on $M$ such that $B_{2}=\delta\sin(\zeta)$ and
$B_{1}=\cos(\zeta)$. We also observe the following relations,

$C_{31}=-\delta C_{3}\sin(\zeta)\text{ and }C_{32}=-C_{3}\cos(\zeta).$

The group reduction introduces additional torsion since $\alpha$ is now basic.
Indeed, let

$\alpha=A_{1}\omega^{1}+A_{2}\omega^{2}+A_{3}\omega^{3}.$

Then the structure equations become

(12) $\displaystyle d\omega^{1}$
$\displaystyle=A_{2}\omega^{1}\wedge\omega^{2}+(A_{3}-C)\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=-A_{1}\omega^{1}\wedge\omega^{2}+\epsilon\omega^{1}\wedge\omega^{3}+(A_{3}+C)\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=\delta\sin(\zeta)\omega^{1}\wedge\omega^{3}+\cos(\zeta)\omega^{2}\wedge\omega^{3}.$

We have now reduced to an $e$-structure with complicated torsion.

  3. (3)

Case 3: $dC\wedge\omega^{1}\wedge\omega^{2}\neq 0$ and $dC\wedge dC_{3}=0$
($C_{3}\neq 0$ and $B_{1}=B_{2}=0$). We pick up from Case 2 in the situation
where $B_{1}=B_{2}=0$ and therefore we no longer have remaining torsion to
normalize using the scaling factor $a$. We remark again that the tableaux is
not involutive and as such we must prolong to a new bundle
$\mathcal{B}_{2}^{(1)}\mapsto\mathcal{B}_{2}$. Let $\alpha=\omega^{4}$. Then
it is immediate from $d^{2}\omega^{1}=0,d^{2}\omega^{2}=0$ that

$d\omega^{4}=E\omega^{1}\wedge\omega^{2},$

for some $E$ on $\mathcal{B}_{2}^{(1)}$. Moreover, the condition that
$d^{2}\omega^{4}=0$ yields

$dE=E_{1}\omega^{1}+E_{2}\omega^{2}+2E\omega_{4}.$

Thus, on the bundle $\mathcal{B}_{2}^{(1)}$ we have the following structure
equations

(13) $\displaystyle d\omega^{1}$
$\displaystyle=\omega^{1}\wedge\omega^{4}-C\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=\omega^{2}\wedge\omega^{4}+C\omega^{2}\wedge\omega^{3}+\epsilon\omega^{1}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$ $\displaystyle=0,$ $\displaystyle d\omega^{4}$
$\displaystyle=E\omega^{1}\wedge\omega^{2}.$

We summarize the two non-prolonged cases above as a single Theorem.

######  Theorem 1.

Let $(\omega^{1},\omega^{2},\omega^{3})$ be a coframing of a 3-manifold $M$
such that $\omega^{1}$ and $\omega^{2}$ form a bicontact structure. Away from
the critical set of $C$ we may adapt the coframing to one of the following
structure equations:

  1. (1)

If $C_{3}=0$ then

(14) $\displaystyle d\omega^{1}$
$\displaystyle=(A_{3}-C)\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=\epsilon\omega^{1}\wedge\omega^{3}+(A_{3}+C)\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=B_{3}\omega^{1}\wedge\omega^{2}+B_{2}\omega^{1}\wedge\omega^{3}+B_{1}\omega^{2}\wedge\omega^{3},$
$\displaystyle dC$
$\displaystyle=\cos(\xi)\omega^{1}+\delta\sin(\xi)\omega^{2}.$

  2. (2)

If $C_{3}\neq 0$ then $C_{1}=C_{2}=0$ and

(15) $\displaystyle d\omega^{1}$
$\displaystyle=A_{2}\omega^{1}\wedge\omega^{2}+(A_{3}-C)\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=-A_{1}\omega^{1}\wedge\omega^{2}+\epsilon\omega^{1}\wedge\omega^{3}+(A_{3}+C)\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{3}$
$\displaystyle=\delta\sin(\zeta)\omega^{1}\wedge\omega^{3}+\cos(\zeta)\omega^{2}\wedge\omega^{3},$
$\displaystyle dC$ $\displaystyle=C_{3}\omega^{3}.$

###### Proof.

The structure equations themselves follow from the preceding moving frame
calculations. Equations (15) are not valid at critical points of $C$ by
hypothesis that $C_{3}\neq 0$. Moreover, the equations (14) never allow for a
critical point of $C$ by virtue of the fact that $C_{1}^{2}+C_{2}^{2}=1$. ∎

We also include this proposition that extends the structure equations from
Theorem 1.

######  Proposition 5.

Let $(\omega^{1},\omega^{2},\omega^{3})$ be a bi-contact structure as in
Theorem 1.

  1. (1)

If $C_{3}=0$ and $B_{3}=0$ everywhere on $U\subset M$ then

(16) $\displaystyle d\xi$
$\displaystyle=-\rho\sin(\xi)\omega^{1}+\rho\cos(\xi)\omega^{2}+\xi_{3}\omega^{3},$
$\displaystyle\xi_{3}$
$\displaystyle=\frac{1}{2}(\epsilon+1)\cos(2\xi)+C\sin(2\xi)+\frac{1}{2}(1-\epsilon),$
$\displaystyle A_{3}$
$\displaystyle=\frac{1}{2}\frac{\partial\xi_{3}}{\partial\xi},$
$\displaystyle\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}C-A_{3}&1\\\
-\epsilon&C+A_{3}\end{bmatrix}^{-1}\begin{bmatrix}\cos(\xi)&-\sin(\xi)\\\
\sin(\xi)&\cos(\xi)\end{bmatrix}\begin{bmatrix}\displaystyle\frac{\rho}{2}\displaystyle\frac{\partial
A_{3}}{\partial\xi}-\sin(2\xi)\\\
-\displaystyle\frac{1}{2}\displaystyle\frac{\partial A_{3}}{\partial
C}-\cos(2\xi)\end{bmatrix},$ $\displaystyle\rho$
$\displaystyle=\frac{\sin(2\xi)}{\xi_{3}+\epsilon-1}.$

  2. (2)

If $C_{3}\neq 0$ on $U\subset M$ then

(17) $d\zeta=(W\cos(\zeta)+\delta A_{2})\omega^{1}-(\delta W\sin(\zeta)+\delta
A_{1})\omega^{2}+\zeta_{3}\omega^{3}$

for some functions $W$ and $\zeta_{3}$ on $M$.

###### Proof.

Let us prove part $(1)$ first. Starting from equations (14), we compute
$d^{2}C$ and let $d\xi=\xi_{1}\omega^{1}+\xi_{2}\omega^{2}+\xi_{3}\omega^{3}$.
Then

$\displaystyle d^{2}C$
$\displaystyle=(\xi_{1}\cos(\xi)+\xi_{2}\sin(\xi))\,\omega^{1}\wedge\omega^{2}+((A_{3}-C)\cos(\xi)-\sin(\xi)(-\epsilon-\xi_{3}))\,\omega^{1}\wedge\omega^{3}$
$\displaystyle+((A_{3}+C)\sin(\xi)+\cos(\xi)(1-\xi_{3}))\,\omega^{2}\wedge\omega^{3}.$

Thus there is some function $\rho$ on $M$ such that

$\displaystyle\xi_{1}$ $\displaystyle=-\rho\sin(\xi),$ $\displaystyle\xi_{2}$
$\displaystyle=\rho\cos(\xi).$

Moreover, rearranging the remaining 2-form coefficients into a matrix equation
yields

$\displaystyle\begin{bmatrix}A_{3}\\\ -\xi_{3}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos(\xi)&-\sin(\xi)\\\
\sin(\xi)&\cos(\xi)\end{bmatrix}^{-1}\begin{bmatrix}\cos(\xi)&-\epsilon\sin(\xi)\\\
-\sin(\xi)&\cos(\xi)\end{bmatrix}\begin{bmatrix}C\\\ 1\end{bmatrix}.$

As such

$\displaystyle\xi_{3}$
$\displaystyle=C\sin(2\xi)+\frac{1}{2}(1+\epsilon)\cos(2\xi)+\frac{1}{2}(1-\epsilon),$
$\displaystyle A_{3}$
$\displaystyle=C\cos(2\xi)-\frac{1}{2}(1+\epsilon)\sin(2\xi).$

It is easy to observe the slightly simpler expression that is
$A_{3}=\displaystyle{\frac{1}{2}}\displaystyle\frac{\partial\xi_{3}}{\partial\xi}$
with $A_{3}$ thought of as a function of $C$ and $\xi$. We remark as well that

$\displaystyle\frac{\partial A_{3}}{\partial\xi}=1-\epsilon-2\xi_{3}$

as it is sometimes helpful later on. Now we need to inspect $d^{2}\omega^{1}$
and $d^{2}\omega^{2}$. Doing so we find

$\displaystyle d^{2}\omega^{1}$
$\displaystyle=((C-A_{3})B_{1}+B_{2}+\sin(\xi))\Omega-\omega^{1}\wedge
dA_{3}\wedge\omega^{3},$ $\displaystyle d^{2}\omega^{2}$
$\displaystyle=(-\epsilon
B_{1}+(C+A_{3})B_{2}+\cos(\xi))\Omega+dA_{3}\wedge\omega^{2}\wedge\omega^{3},$

so that we need only $(A_{3})_{1}$ and $(A_{3})_{2}$. Of course,

$dA_{3}=\frac{1}{2}\left(\displaystyle\frac{\partial^{2}\xi_{3}}{\partial\xi\partial
C}dC+\displaystyle\frac{\partial^{2}\xi_{3}}{\partial\xi^{2}}d\xi\right),$

so

$\displaystyle-(A_{3})_{2}$
$\displaystyle=\frac{1}{2}\left(-\frac{\partial^{2}\xi_{3}}{\partial\xi^{2}}\rho\cos(\xi)-\frac{\partial^{2}\xi_{3}}{\partial\xi\partial
C}\sin(\xi)\right),$ $\displaystyle(A_{3})_{1}$
$\displaystyle=\frac{1}{2}\left(-\frac{\partial^{2}\xi_{3}}{\partial\xi^{2}}\rho\sin(\xi)+\frac{\partial^{2}\xi_{3}}{\partial\xi\partial
C}\cos(\xi)\right).$

The vanishing of $d^{2}\omega^{1}$ and $d^{2}\omega^{2}$ may now be recast as
the equations

$\begin{bmatrix}C-A_{3}&1\\\
-\epsilon&C+A_{3}\end{bmatrix}\begin{bmatrix}B_{1}\\\
B_{2}\end{bmatrix}+\begin{bmatrix}\sin(\xi)\\\
\cos(\xi)\end{bmatrix}+\frac{1}{2}\begin{bmatrix}\cos(\xi)&-\sin(\xi)\\\
\sin(\xi)&\cos(\xi)\end{bmatrix}\begin{bmatrix}-\rho\displaystyle\frac{\partial^{2}\xi_{3}}{\partial\xi^{2}}\\\
\displaystyle\frac{\partial^{2}\xi_{3}}{\partial\xi\partial
C}\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$

Thus

$\begin{bmatrix}B_{1}\\\ B_{2}\end{bmatrix}=\begin{bmatrix}C-A_{3}&1\\\
-\epsilon&C+A_{3}\end{bmatrix}^{-1}\begin{bmatrix}\cos(\xi)&-\sin(\xi)\\\
\sin(\xi)&\cos(\xi)\end{bmatrix}\begin{bmatrix}\displaystyle\frac{\rho}{2}\displaystyle\frac{\partial
A_{3}}{\partial\xi}-\sin(2\xi)\\\
-\displaystyle\frac{1}{2}\displaystyle\frac{\partial A_{3}}{\partial
C}-\cos(2\xi)\end{bmatrix}.$

Next we investigate a $d^{2}\xi=0$ conditions. Let
$d\rho=\rho_{1}\omega^{1}+\rho_{2}\omega^{2}+\rho_{3}\omega^{3}$. Then

$d^{2}\xi\equiv\left(\rho_{1}\cos(\xi)+\rho_{2}\sin(\xi)+\rho^{2}\right)\omega^{1}\wedge\omega^{2}\mod\\{\omega^{3}\\}$

so we conclude that there exists a function $W$ such that

$\displaystyle\rho_{1}$ $\displaystyle=-\rho^{2}\cos(\xi)-W\sin(\xi),$
$\displaystyle\rho_{2}$ $\displaystyle=-\rho^{2}\sin(\xi)+W\cos(\xi).$

To determine $\rho_{3}$ it is simplest to check that $d(dC\wedge d\xi)=0$.
Indeed,

$\displaystyle d(dC\wedge d\xi)$
$\displaystyle=d(\rho\,\omega^{1}\wedge\omega^{2}+\xi_{3}dC\wedge\omega^{3}),$
$\displaystyle 0$ $\displaystyle=(\rho_{3}-2A_{3}\rho)\Omega+d\xi_{3}\wedge
dC\wedge\omega^{3}-\xi_{3}dC\wedge d\omega^{3}.$

However,

$d\xi_{3}=2A_{3}d\xi+\displaystyle\frac{\partial\xi_{3}}{\partial C}dC$

and therefore

$0=(\rho_{3}-4A_{3}\rho+\xi_{3}B_{2}\sin(\xi)-\xi_{3}B_{1}\cos(\xi))\Omega$

so we conclude that

$\rho_{3}=4A_{3}\rho+\xi_{3}B_{1}\cos(\xi)-\xi_{3}B_{2}\sin(\xi).$

This completely determines $d\rho$. Next we investigate $d^{2}\omega^{3}=0$.
Indeed,

$\displaystyle d^{2}\omega^{3}$ $\displaystyle=\frac{\partial B_{1}}{\partial
C}dC\wedge\omega^{2}\wedge\omega^{3}+\frac{\partial
B_{1}}{\partial\xi}d\xi\wedge\omega^{2}\wedge\omega^{3}+\frac{\partial
B_{1}}{\partial\rho}d\rho\wedge\omega^{2}\wedge\omega^{3}$
$\displaystyle+\frac{\partial B_{2}}{\partial
C}dC\wedge\omega^{1}\wedge\omega^{3}+\frac{\partial
B_{2}}{\partial\xi}d\xi\wedge\omega^{1}\wedge\omega^{3}+\frac{\partial
B_{2}}{\partial\rho}d\rho\wedge\omega^{1}\wedge\omega^{3}.$

Conveniently, this results in a linear equation in $W$

$\displaystyle 0$ $\displaystyle=\cos(\xi)\left(\frac{\partial B_{1}}{\partial
C}+\rho\frac{\partial B_{2}}{\partial\xi}-\rho^{2}\frac{\partial
B_{1}}{\partial\rho}\right)-\sin(\xi)\left(\frac{\partial B_{2}}{\partial
C}+\rho\frac{\partial B_{1}}{\partial\xi}-\rho^{2}\frac{\partial
B_{2}}{\partial\rho}\right)$ $\displaystyle-W\left(\cos(\xi)\frac{\partial
B_{2}}{\partial\rho}+\sin(\xi)\frac{\partial B_{1}}{\partial\rho}\right).$

We can now determine $W$ explicitly in terms of $C,\xi$ and $\rho$.

$W=\frac{\cos(\xi)\left(\displaystyle\frac{\partial B_{1}}{\partial
C}+\rho\displaystyle\frac{\partial
B_{2}}{\partial\xi}-\rho^{2}\displaystyle\frac{\partial
B_{1}}{\partial\rho}\right)-\sin(\xi)\left(\displaystyle\frac{\partial
B_{2}}{\partial C}+\rho\displaystyle\frac{\partial
B_{1}}{\partial\xi}-\rho^{2}\displaystyle\frac{\partial
B_{2}}{\partial\rho}\right)}{\cos(\xi)\displaystyle\frac{\partial
B_{2}}{\partial\rho}+\sin(\xi)\displaystyle\frac{\partial
B_{1}}{\partial\rho}}.$

It is important to remark that $W$ is quadratic in $\rho$ since each $B_{1}$
and $B_{2}$ are linear in $\rho$.

To determine $\rho$, we proceed by checking $d(d\rho\wedge d\xi)=0$. This
calculation becomes difficult to complete by hand. As such, we relegate the
calculation to the computer algebra software MAPLE and use the Cartan Package
(which was written by Yunliang Yu, and now improved and maintained by Jeanne
Clelland and Tom Ivey). In part because of the presence of $W$, we are left
with a cubic polynomial in $\rho$. Solving, and checking additional
consistencies with $d(d\rho\wedge dC)=0$, we arrive at the unique expression

$\rho=\frac{\sin(2\xi)}{\xi_{3}+\epsilon-1}.$

For the case of $C_{3}\neq 0$, let

$d\zeta=\zeta_{1}\omega^{1}+\zeta_{2}\omega^{2}+\zeta_{3}\omega^{3}.$

Then

(18) $\displaystyle d^{2}C_{3}$ $\displaystyle\equiv
C_{3}(\delta(A_{1}\cos(\zeta)+\zeta_{1}\sin(\zeta))-A_{2}\sin(\zeta)+\zeta_{2}\cos(\zeta))\omega^{1}\wedge\omega^{2}\mod\\{\omega^{3}\\}$

meaning there is a function $W$ on $M$ such that

(19) $\displaystyle\zeta_{1}$ $\displaystyle=W\cos(\zeta)+\delta A_{2},$
$\displaystyle\zeta_{2}$ $\displaystyle=-\delta W\sin(\zeta)-\delta A_{1},$

as claimed. ∎

We end with this section with two propositions and two examples of the third
case of $C$ non-constant. The proposition below was noted in [18] but we see
this arise naturally in the context of studying bicontactomorphisms.
Additionally, the construction in [18] only yields the case of $E=0$. Let us
also briefly give a convenient definition. A symplectic quadratic on a
symplectic manifold is simply a pair of symplectic forms
$\theta^{1},\theta^{2}$ such that $\theta_{a}=a_{1}\theta^{1}+a_{2}\theta^{2}$
has the property that $\theta_{a}\wedge\theta_{a}=\mathcal{P}_{C}(a)\Omega$
where $\mathcal{P}_{C}(a)=1$ for some function $C$. As with bi-contact
structures, it may be that a hyperbola defined by $\mathcal{P}_{C}(a)$ may
force the orientation of the volume form for $\theta^{1}$ and $\theta^{2}$ to
oppose one another.

######  Proposition 6.

Given a bi-contact structure $(\omega^{1},\omega^{2})$ on some $M$, if
$dC\wedge\omega^{1}\wedge\omega^{2}=C_{3}\Omega\neq 0$ but
$dC_{3}\wedge\omega^{3}=0$ on $M$ then there is a manifold $\widetilde{M}$
containing $M$ as a submanifold such that $\widetilde{M}$ admits a symplectic
quadratic.

###### Proof.

From the structure equations (13) we see that

(20) $\displaystyle d\omega^{1}\wedge d\omega^{1}$
$\displaystyle=2\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\wedge\omega^{4},$
$\displaystyle d\omega^{2}\wedge d\omega^{2}$
$\displaystyle=-2\epsilon\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\wedge\omega^{4}.$

In fact, if $\theta^{1}=d\omega^{1}$ and $\theta^{2}=d\omega^{2}$ then the
2-form $\theta_{a}=a_{1}\theta^{1}+a_{2}\theta^{2}$ is a symplectic quadratic.
∎

Once again, we find that the obstruction to the existence of a taut contact
circle or equilateral hyperbola is also an obstruction for $\theta^{1}$ and
$\theta^{2}$ to form a symplectic couple or symplectic pair [7]. In particular
the invariant $C$ arises in the form of
$\theta^{1}\wedge\theta^{2}=2C\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\wedge\omega^{4}$.
Let us give an example of the case $E=0$. Reconsider example 1. Relabel the
1-forms $\omega^{1}$ and $\omega^{2}$ as
$e^{-s}\omega^{1}+e^{-s}(ds+s(1-z)C(z)^{-1}dz)$ and
$e^{-s}\omega^{2}+e^{-s}(ds+s(z+\epsilon)C(z)^{-1}dz)$ and introduce
$\omega^{4}=ds$. Then

(21) $\displaystyle d\omega^{1}$
$\displaystyle=\omega^{1}\wedge\omega^{4}-z\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=\omega^{2}\wedge\omega^{4}+\omega^{1}\wedge\omega^{3}-z\omega^{2}\wedge\omega^{4},$
$\displaystyle d\omega^{3}$ $\displaystyle=0,$ $\displaystyle d\omega^{4}$
$\displaystyle=0.$

Notice that each symplectic form $\theta^{i}=d\omega^{i}$, $i=1,2$ share the
family of Lagrangian submanifolds determined by $(x,y,z_{0},s_{0})$ for
$z_{0}$ and $s_{0}$ constants.

Now we consider an example where the invariant $E\neq 0$. In general, such
examples are locally foliated by contact 3-manifolds that are transverse to
$M$.

######  Example 2.

Let $M$ and $\widetilde{M}$ be as in Proposition 6 and let $(x,y,z,w)$ be
local coordinates for $\widetilde{M}$. Moreover, we have taken $C(z)=\tan(z)$
and we assume that $-\pi/2<z<\pi/2$. Matters being so, the following 1-forms
satisfy the structure equations (13) with $E=y^{-1}e^{2w-y(x+z)}$.

(22) $\displaystyle\omega^{1}$ $\displaystyle=ze^{-w}(dx+dz)+ye^{y(x+z)-w}dy,$
$\displaystyle\omega^{2}$
$\displaystyle=-(z\tan(z)+1)e^{-w}(dx+dz)-y\tan(z)e^{y(x+z)-w}dy,$
$\displaystyle\omega^{3}$ $\displaystyle=dz,$ $\displaystyle\omega^{4}$
$\displaystyle=dw-y(dx+dz).$

It is easy to verify that $E$ is as claimed since
$d\omega^{4}=-dy\wedge(dx+dz)$.

######  Proposition 7.

If $E$ is constant then $E=0$.

###### Proof.

If $E$ is constant then
$d^{2}\omega^{4}=E(d\omega^{1}\wedge\omega^{2}-\omega^{1}\wedge d\omega^{2})$
but $d\omega^{1}\wedge\omega^{2}-\omega^{1}\wedge
d\omega^{2}=2\omega^{1}\wedge\omega^{2}\wedge\omega^{4}$ thus forcing $E=0$ to
satisfy $d^{2}\omega^{4}=0$. ∎

###  2.4. Constant C

The contactomorphism group for a contact structure on a 3-manifold is infinite
dimensional, but parameterized by a single function of 3 variables. That is,
the space of contactomorphisms is in one-to-one correspondence with smooth
functions on $M$. We now prove the following theorem which yields a similar
result in case the invariant function $C$ is constant.

######  Theorem 2.

All bi-contact structures with the same non-zero constant $C$ and $\epsilon$
are locally equivalent under bi-contactomorphisms. Moreover, the group of bi-
contactomorphisms for a given bi-contact structure is locally parameterized by
two functions of one variable.

###### Proof.

Assume $C\neq 0$. We will prolong the principal bundle associated to the non-
involutive structure equations (4) to a new principal bundle which we denote
$\mathcal{B}^{(1)}_{1}$. Relabeling $\alpha=\omega^{4},\beta_{1}=\omega^{5}$,
and $\beta_{2}=\omega^{6}$, we take ${\omega}^{(1)}$ as the new tautological
1-form with components $\omega^{i}$ for $1\leq i\leq 6$. Transformations of
${\omega}^{(1)}$ are given by

(23) ${\tilde{\omega}}^{(1)}=\left[\begin{array}[]{c|c}I_{3\times
3}&0_{3\times 3}\\\ \hline\cr H_{0}&I_{3\times
3}\end{array}\right]^{-1}{\omega}^{(1)},$

where $H_{0}$ is a $3\times 3$ matrix of the form:

(24) $H_{0}=\begin{bmatrix}0&0&0\\\ h_{11}&h_{12}&0\\\
h_{12}&h_{22}&0\end{bmatrix}.$

The structure equations then have the form:

(25) $d{\omega}^{(1)}=-\left[\begin{array}[]{c|c}0_{3\times 3}&0_{3\times
3}\\\ \hline\cr\Phi_{0}&0_{3\times
3}\end{array}\right]\wedge{\omega}^{(1)}+T({\omega}^{(1)}\wedge{\omega}^{(1)}),$

where

(26) $\Phi_{0}=\begin{bmatrix}0&0&0\\\ \phi_{11}&\phi_{12}&0\\\
\phi_{12}&\phi_{22}&0\end{bmatrix},$

and

(27)
$T({\omega}^{(1)}\wedge{\omega}^{(1)})=\begin{bmatrix}-C\omega^{1}\wedge\omega^{3}+\omega^{1}\wedge\omega^{4}+\omega^{2}\wedge\omega^{3}\\\
C\omega^{2}\wedge\omega^{3}+\epsilon\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{4}\\\
\omega^{1}\wedge\omega^{5}+\omega^{2}\wedge\omega^{6}\\\
E\omega^{1}\wedge\omega^{2}+\pi_{1}+\pi_{2}\\\ T^{5}\\\ T^{6}\end{bmatrix},$

where

(28)
$\pi_{1}+\pi_{2}=\epsilon\omega^{1}\wedge\omega^{6}+\omega^{2}\wedge\omega^{5}+C\omega^{2}\wedge\omega^{6}-C\omega^{1}\wedge\omega^{5}.$

The torsion term $E\omega^{1}\wedge\omega^{2}$ results from the fact that
$(d\alpha-\pi^{1})\wedge\omega^{1}=(d\alpha-\pi^{2})\wedge\omega^{2}=0$
(derived from $d^{2}\omega^{1}$ and $d^{2}\omega^{2}$) implies the existence
of a semi-basic 2-form with no $\omega^{3}$ dependence. To understand the
torsion terms $T^{5}$ and $T^{6}$, we need to consider $d^{2}\omega^{3}$.
Indeed, doing so results in

(29) $\displaystyle d^{2}\omega^{3}$
$\displaystyle=\left(-d\beta_{1}+\alpha\wedge\beta_{1}-C\omega^{3}\wedge\beta_{1}+\epsilon\omega^{3}\wedge\beta_{2}\right)\wedge\omega^{1}$
$\displaystyle+\left(-d\beta_{2}+\alpha\wedge\beta_{2}+\omega^{3}\wedge\beta_{1}+C\omega^{3}\wedge\beta_{2}\right)\wedge\omega^{2}.$

An application of Cartan’s lemma implies the existence of a symmetric $2\times
2$ matrix $\Phi^{\prime}_{0}=[\phi^{\prime}_{ij}]$ which is 1-form valued so
that

(30) $\begin{bmatrix}d\beta_{1}\\\
d\beta_{2}\end{bmatrix}=-\begin{bmatrix}\phi^{\prime}_{11}&\phi^{\prime}_{12}\\\
\phi^{\prime}_{12}&\phi^{\prime}_{22}\end{bmatrix}\wedge\begin{bmatrix}\omega^{1}\\\
\omega^{2}\end{bmatrix}+\begin{bmatrix}\alpha\wedge\beta_{1}-C\omega^{3}\wedge\beta_{1}+\epsilon\omega^{3}\wedge\beta_{2}\\\
\alpha\wedge\beta_{2}+\omega^{3}\wedge\beta_{1}+C\omega^{3}\wedge\beta_{2}\end{bmatrix}.$

Therefore we can conclude that

(31) $\displaystyle T^{5}$
$\displaystyle=\omega^{4}\wedge\omega^{5}-C\omega^{3}\wedge\omega^{5}+\epsilon\omega^{3}\wedge\omega^{6},$
$\displaystyle T^{6}$
$\displaystyle=\omega^{4}\wedge\omega^{6}+\omega^{3}\wedge\omega^{5}+C\omega^{3}\wedge\omega^{6}.$

We next wonder how the function $E$ varies on the fibers, since it is the only
non-trivial torsion term that is new for this prolongation. As such, we note
that $g^{-1}T(g{\omega}^{(1)}\wedge g{\omega}^{(1)})$ yields

(32) $E\mapsto E-(h_{11}-\epsilon h_{22}+2Ch_{12}).$

Or, infinitesimally,

(33)
$d^{2}\omega^{4}=(dE-(\phi_{11}-\epsilon\phi_{22}+2C\phi_{12}))\wedge\omega^{1}\wedge\omega^{1}.$

Thus, we can always normalize $E=0$, and this further implies we may reduce
the structure group using $h_{12}=(\epsilon h_{22}-h_{11})/(2C)$.
Infinitesimally, this means that $\phi_{11}-\epsilon\phi_{22}+2C\phi_{12}$
become semi-basic and as such
$\phi_{12}=\frac{1}{2C}(\epsilon\phi_{22}-\phi_{11})+I_{i}\omega^{i}$ where we
have summed over $1\leq i\leq 6$. However, it turns out that the $I_{1}$ and
$I_{2}$ terms will be absorbable while $I_{j}=0$ for $3\leq j\leq 6$ by (33).
Let us relabel so that $h=h_{11}$, $k=h_{22}$, $\phi=\phi_{11}$, and
$\kappa=\phi_{22}$. We now have an additional reduction of the structure group
yielding a new principal bundle
$\left(\mathcal{B}^{(1)}\right)_{1}\hookrightarrow\mathcal{B}^{(1)}_{1}$. The
tautological 1-form now transforms as

(34) ${\tilde{\omega}}^{(1)}=\left[\begin{array}[]{c|c}I_{3\times
3}&0_{3\times 3}\\\ \hline\cr H_{1}&I_{3\times
3}\end{array}\right]^{-1}{\omega}^{(1)},$

where $H_{1}$ is a $3\times 3$ matrix of the form:

(35) $H_{1}=\begin{bmatrix}0&0&0\\\ h&\frac{h-\epsilon k}{2C}&0\\\
\frac{h-\epsilon k}{2C}&\epsilon k&0\end{bmatrix}.$

Currently,

(36) $\displaystyle T^{5}$
$\displaystyle=\omega^{4}\wedge\omega^{5}-C\omega^{3}\wedge\omega^{5}+\epsilon\omega^{3}\wedge\omega^{6},$
$\displaystyle T^{6}$
$\displaystyle=\omega^{4}\wedge\omega^{6}+\omega^{3}\wedge\omega^{5}+C\omega^{3}\wedge\omega^{6},$

and the structure equations are now

(37) $d{\omega}^{(1)}=-\left[\begin{array}[]{c|c}0_{3\times 3}&0_{3\times
3}\\\ \hline\cr\Phi_{1}&0_{3\times
3}\end{array}\right]\wedge{\omega}^{(1)}+\begin{bmatrix}-C\omega^{1}\wedge\omega^{3}+\omega^{1}\wedge\omega^{4}+\omega^{2}\wedge\omega^{3}\\\
C\omega^{2}\wedge\omega^{3}+\epsilon\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{4}\\\
\omega^{1}\wedge\omega^{5}+\omega^{2}\wedge\omega^{6}\\\ \pi_{1}+\pi_{2}\\\
T^{5}\\\ T^{6}\end{bmatrix},$

where

(38) $\Phi_{1}=\begin{bmatrix}0&0&0\\\
\phi&\frac{\epsilon\kappa-\phi}{2C}&0\\\
\frac{\epsilon\kappa-\phi}{2C}&\epsilon\kappa&0\end{bmatrix}.$

At this point, $d^{2}\omega^{i}\mod\\{\omega^{1},\omega^{2}\\}$ imposes no new
conditions for $i=5,6$. Moreover, the only non-zero Cartan character is
$s_{1}=2$ and the freedom in the connection has dimension exactly $2$ by an
application of Cartan’s lemma to $d^{2}\omega^{5}$ and $d^{2}\omega^{6}$.

Now assume that $C=0$. In this situation, the analysis is no different up to
the structure equations for ${\omega}^{(1)}$ but with $C=0$. The only change
is in the additional reduction step for the torsion term $E$. In this case, we
can still normalize $E$ to be zero, however we have no restriction on the
$h_{12}$ which we will now denote as $k$. As such, we recover the following
change of coframe for ${\omega}^{(1)}$

(39) ${\tilde{\omega}}^{(1)}=\left[\begin{array}[]{c|c}I_{3\times
3}&0_{3\times 3}\\\ \hline\cr K_{1}&I_{3\times
3}\end{array}\right]^{-1}{\omega}^{(1)},$

where $K_{1}$ is a $3\times 3$ matrix of the form:

(40) $K_{1}=\begin{bmatrix}0&0&0\\\ h&\epsilon k&0\\\ \epsilon
k&h&0\end{bmatrix}.$

The structure equations then immediately follow

(41) $d{\omega}^{(1)}=-\left[\begin{array}[]{c|c}0_{3\times 3}&0_{3\times
3}\\\ \hline\cr\Psi_{1}&0_{3\times
3}\end{array}\right]\wedge{\omega}^{(1)}+\begin{bmatrix}\omega^{1}\wedge\omega^{4}+\omega^{2}\wedge\omega^{3}\\\
\epsilon\omega^{1}\wedge\omega^{3}+\omega^{2}\wedge\omega^{4}\\\
\omega^{1}\wedge\omega^{5}+\omega^{2}\wedge\omega^{6}\\\
\epsilon\omega^{1}\wedge\omega^{6}+\omega^{2}\wedge\omega^{5}\\\
\omega^{4}\wedge\omega^{5}+\epsilon\omega^{3}\wedge\omega^{6}\\\
\omega^{4}\wedge\omega^{6}+\omega^{3}\wedge\omega^{5}\end{bmatrix},$

where

(42) $\Psi_{1}=\begin{bmatrix}0&0&0\\\ \phi&\epsilon\psi&0\\\
\epsilon\psi&\phi&0\end{bmatrix}.$

We now check for involutivity. We immediately see that there is only one non-
zero Cartan character $s_{1}=2$. This we need only demonstrate that
$\dim({A}^{(2)})=2$. Consider a change of pseudo-connection
$\Psi^{\prime}_{1}$. Since all torsion has been normalized, we immediately
understand that
$(\phi-\phi^{\prime})\wedge\omega^{1}+\epsilon(\psi-\psi^{\prime})\wedge\omega^{2}=0$
and
$\epsilon(\psi-\psi^{\prime})\wedge\omega^{1}+(\phi-\phi^{\prime})\wedge\omega^{2}=0$.
Thus, Cartan’s lemma gives

(43) $\displaystyle\begin{bmatrix}\phi\\\ \epsilon\psi\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\phi^{\prime}\\\
\epsilon\psi^{\prime}\end{bmatrix}+\begin{bmatrix}s_{11}&s_{12}\\\
s_{12}&s_{22}\end{bmatrix}\begin{bmatrix}\omega^{1}\\\
\omega^{2}\end{bmatrix},$ $\displaystyle\begin{bmatrix}\phi\\\
\epsilon\psi\end{bmatrix}$ $\displaystyle=\begin{bmatrix}\phi^{\prime}\\\
\epsilon\psi^{\prime}\end{bmatrix}+\begin{bmatrix}t_{12}&t_{11}\\\
t_{22}&t_{12}\end{bmatrix}\begin{bmatrix}\omega^{1}\\\
\omega^{2}\end{bmatrix},$

which together imply that

(44) $\begin{bmatrix}\phi\\\
\epsilon\psi\end{bmatrix}=\begin{bmatrix}\phi^{\prime}\\\
\epsilon\psi^{\prime}\end{bmatrix}+\begin{bmatrix}s_{11}&s_{12}\\\
s_{12}&s_{11}\end{bmatrix}\begin{bmatrix}\omega^{1}\\\
\omega^{2}\end{bmatrix}.$

Thus we have that $\dim({A}^{(2)})=2$ and we have that our structure equations
are indeed involutive. ∎

######  Example 3.

A simple example of the constant $C$ case for hyperbolic bi-contact structures
is the following: let $M=\mathbb{T}^{2}\times\mathbb{R}$ with coordinates
$(\theta,\varphi,z)$ and consider

$\displaystyle\omega^{1}$
$\displaystyle=f(z)d\theta+g(z)d\varphi+(g(z)-C\,f(z))dz,$
$\displaystyle\omega^{2}$
$\displaystyle=(C\,f(z)-g(z))d\theta-f(z)d\varphi+g(z)dz,$
$\displaystyle\omega^{3}$
$\displaystyle=(f(z)g^{\prime}(z)-f^{\prime}(z)g(z))dz,$

for any $f,g\in C^{1}(\mathbb{R})$ such that $f(z)^{2}-g(z)^{2}+2Cf(z)g(z)=1$.

It is straightforward to check that the constant $C$ is indeed the invariant
$C$ and that $\omega^{2}\wedge d\omega^{2}=-\omega^{1}\wedge d\omega^{1}$.
This demonstrates that $M$ admits infinitely many bi-contactomorphism classes
parameterized by $\mathbb{R}$.

###  2.5. $(-\epsilon)$-Cartan Structures

There is an additional special case introduced in [8] and investigated some in
[18].

######  Definition 6.

A $(-\epsilon)$-Cartan structure is a bi-contact structure such that

$\omega^{1}\wedge d\omega^{2}=\omega^{2}\wedge d\omega^{1}=0.$

Clearly, such structures must also have $C=0$. However, we have additional
rigidity imposed on bi-contactomorphism equivalence. We attribute the
following theorem to Perrone for his structure equations in item (iii) of
theorem 7.1 from [18] and also to Geiges and Gonzalo who also made the
observation in [8]. Though we have additionally uncovered the equivalence
under bi-contactomorphisms.

######  Theorem 3.

Bi-contactomorphism classes of $(-\epsilon)$-Cartan structures are in one-to-
one correspondence with isometry classes of Riemannian surfaces
($\epsilon=-1$) or isometry classes of Lorenztian surfaces ($\epsilon=1$).
Moreover, if $\underline{\omega}^{1}$ and $\underline{\omega}^{2}$ are the
projections of the bi-contact structure from the coframe bundle to the
Riemannian (resp. Lorentzian) surface $\Sigma$ then the metric on $\Sigma$ is
exactly $(\underline{\omega}^{1})^{2}-\epsilon(\underline{\omega}^{2})^{2}$.

###### Proof.

The proof follows by continuing Cartan’s method of equivalence. We start from
the structure equations (4) where we have that $C=0$. The $(-\epsilon)$-Cartan
structure requirement means that we must impose

$\omega^{1}\wedge d\omega^{2}=\omega^{2}\wedge d\omega^{1}=0$

on the structure equations. Doing so yields

$\alpha\wedge\omega^{1}\wedge\omega^{2}=0$

meaning that $\alpha$ must in fact be basic and depend soley on $\omega^{1}$
and $\omega^{2}$. As such we set $\alpha=A_{1}\omega^{1}+A_{2}\omega^{2}$ for
some $A_{1}$ and $A_{2}$ on the coframe bundle. We now have the following
structure equations

(45) $d\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{bmatrix}=-\begin{bmatrix}0&0&0\\\ 0&0&0\\\
\beta_{1}&\beta_{2}&0\\\ \end{bmatrix}\wedge\begin{bmatrix}\omega^{1}\\\
\omega^{2}\\\
\omega^{3}\end{bmatrix}+\begin{bmatrix}A_{2}\omega^{1}\wedge\omega^{2}+\omega^{2}\wedge\omega^{3}\\\
-A_{1}\omega^{1}\wedge\omega^{2}+\epsilon\omega^{1}\wedge\omega^{3}\\\
0\end{bmatrix}.$

Notice that

$\displaystyle d^{2}\omega^{1}$
$\displaystyle=(dA_{2}-A_{1}\omega^{3}+\beta_{1})\wedge\omega^{1}\wedge\omega^{2},$
$\displaystyle d^{2}\omega^{2}$ $\displaystyle=-(dA_{1}-\epsilon
A_{2}\omega^{3}+\epsilon\beta_{2})\wedge\omega^{1}\wedge\omega^{2}.$

Thus we can normalize the torsion terms $A_{1}$ and $A_{2}$ to zero. As such,
$\beta_{1}$ and $\beta_{2}$ become basic, but in such way that
$\beta_{i}\equiv 0\mod\\{\omega^{1},\omega^{2}\\}$ so that
$d^{2}\omega^{1}=d^{2}\omega^{2}=0$. As such, if we take

$\displaystyle\beta_{1}$
$\displaystyle=B_{1}\omega^{1}+\frac{1}{2}\left(B_{3}+K\right)\omega^{2},$
$\displaystyle\beta_{2}$
$\displaystyle=\frac{1}{2}\left(K-B_{3}\right)\omega^{1}+B_{2}\omega^{2},$

where $B_{i}$, $1\leq i\leq k$ and $K$ are new torsion terms. This results in
the following structure equations

$\displaystyle d\omega^{1}$ $\displaystyle=\omega^{2}\wedge\omega^{3},$
$\displaystyle d\omega^{2}$
$\displaystyle=\epsilon\omega^{1}\wedge\omega^{3},$ $\displaystyle
d\omega^{3}$ $\displaystyle=K\omega^{1}\wedge\omega^{2}.$

Moreover, in order that $d^{2}\omega^{3}=0$ we must have
$dK\wedge\omega^{1}\wedge\omega^{2}=0$ everywhere. These are precisely the
structure equations for the coframe bundle of a Riemannian or Lorentzian
surface (depending on the sign of $\epsilon$) where $\omega^{1}$ and
$\omega^{2}$ make up the coframe for a surface $\Sigma$ and $\omega^{3}$ plays
the role of the connection 1-form and $K$ is the Gaussian curvature. ∎

In [18] Perrone mentions this relation but only in the context of Webster
curvature (which reduces to Gaussian curvature in special cases), yielding
some nice results about associated contact metrics. Moreover, the structure
equations Perrone presents have a factor of two that can be eliminated to
resemble the above structure equations by scaling the contact 1-forms by one-
half.

##  3\. A Local Normal Form

In this section we present some special circumstances of the non-zero $C_{3}$
case. First, we remark that several combinations of the invariant functions
may be used to form local invariant coordinate systems, implying the existence
of local normal forms. We prove the existence of at least one invariant
coordinate system for a bi-contact structure but are unable to find a normal
form due to the complexity of the associated PDEs; however, when we consider a
further special case ($A_{1}=A_{2}=0$ identically on $M$) we are able recover
a local normal form.

######  Proposition 8.

Let $(\omega^{1},\omega^{2},\omega^{3})$ constitute a coframing on a
3-manifold $M$ that has the structure equations (15). On any open $U\subset M$
such that $C_{3}\neq 0$ there is an invariant coordinate system defined by
$(x,y,z)=(C,C_{3},C_{33})$ for at least one of the possibilities $\epsilon=\pm
1$. Generically, such an invariant coordinate system exists for both cases of
$\epsilon=\pm 1$.

###### Proof.

We need to find an expression for $dC_{33}$, which we achieve by checking the
conditions imposed by $d^{2}C_{3}=0$. By the structure equations from Theorem
1 and Proposition 5 we find that

(46) $\displaystyle dC_{33}$
$\displaystyle=(C_{3}(\zeta_{3}-\delta\epsilon)\cos(\zeta)+(2C_{33}+C_{3}C-C_{3}A_{3})\sin(\zeta))\omega^{1}$
$\displaystyle+(-C_{3}(\zeta_{3}+\delta)\sin(\zeta)+\delta(2C_{33}-C_{3}C-C_{3}A_{3})\cos(\zeta))\omega^{2}+C_{333}\omega^{3}.$

We also have

(47) $\displaystyle dC\wedge dC_{3}$
$\displaystyle=C_{3}^{2}(\sin(\zeta)\omega^{1}\wedge\omega^{3}+\delta\cos(\zeta)\omega^{2}\wedge\omega^{3}),$

and it now follows that

(48) $\displaystyle\Omega_{C}^{\epsilon}$
$\displaystyle=-C_{3}^{3}(1-(1+\epsilon)\cos^{2}(\zeta)+2\delta
C\sin(\zeta)\cos(\zeta)+\delta\zeta_{3})\omega^{1}\wedge\omega^{2}\wedge\omega^{3},$

where $\Omega_{C}^{\epsilon}=dC\wedge dC_{3}\wedge dC_{33}$. Thus, aside from
$C_{3}\neq 0$, the function $\zeta$ must not solve the following PDE

(49)
$\zeta_{3}=-\delta+\delta(1+\epsilon)\cos^{2}(\zeta)-2C\sin(\zeta)\cos(\zeta),$

otherwise, the 3-form $\Omega_{C}^{\epsilon}$ is zero when this is the case.
Assume $\zeta$ solves (49) on some open $U\subset M$. Then the 3-form
$\Omega_{C}^{-\epsilon}$ has the expression

(50)
$\Omega_{C}^{-\epsilon}=-2C_{3}^{3}\cos^{2}(\zeta)\omega^{1}\wedge\omega^{2}\wedge\omega^{3}.$

Thus, so long as $C_{3}\neq 0$ and $\zeta\neq\frac{(2k+1)\pi}{2}$ the volume
form $\Omega^{-\epsilon}_{C}$ is nonzero. In fact, we can go further. If
$\zeta$ satisfies the PDE (49), and $p\in M$ any point where
$\zeta_{3}\big{|}_{p}=0$, then $\zeta_{p}=\zeta\big{|}_{p}$ and
$C_{p}=C\big{|}_{p}$ must satisfy

(51)
$-\delta+\delta(1+\epsilon)\cos^{2}(\zeta_{p})=2C_{p}\sin(\zeta_{p})\cos(\zeta_{p}),$

equivalently

(52) $\epsilon\cos^{2}(\zeta_{p})-\sin^{2}(\zeta_{p})=\delta
C_{p}\sin(2\zeta_{p}).$

For $\epsilon=1$ this yields that

(53) $\zeta_{p}=\frac{1}{2}\text{arccot}(\delta C_{p}),$

which means that $\zeta_{p}$ can only ever take values in the open intervals
$(n\pi/2,(n+1)\pi/2)$ for $n\in\mathbb{Z}$. In case $\epsilon=-1$, we have
instead

(54) $\zeta_{p}=\frac{1}{2}\text{arccsc}(-\delta C_{p}).$

Thus, $\zeta_{p}$ can only take values in the intervals
$[n\pi/2-\pi/4,n\pi/2)\cup(n\pi/2,n\pi/2+\pi/4]$. Therefore, for $C_{3}\neq 0$
on any open $U\subset M$, whenever $\Omega_{C}^{\epsilon}=0$ then
$\Omega_{C}^{-\epsilon}$ is never zero. ∎

######  Remark 1.

The PDE (49) may be transformed into the equivalent PDEs

(55) $\displaystyle\frac{\partial H}{\partial x}$
$\displaystyle=\frac{1}{y}(1-x\sinh(2H)),\quad\epsilon=1$
$\displaystyle\frac{\partial G}{\partial x}$
$\displaystyle=-\frac{1}{y}(1+x\sin(2G)),\quad\epsilon=-1,$

where $x=C$ and $y=C_{3}$. In particular, the PDE for function $H$ may be
transformed to almost the PDE for $G$. Indeed, let $\widehat{H}=i\,H$ and
$x\mapsto-i\,\widehat{x}$. Then the PDE for $H$ becomes

$\displaystyle\frac{\partial\widehat{H}}{\partial\widehat{x}}=\frac{1}{y}(1+\widehat{x}\sin(2\widehat{H})).$

It is interesting to note that, although the PDEs (55) are nearly related by
complex rotations, the square of the derivatives may be transformed to one
another, at the cost of swapping branches.

Now we restrict to a special case and find a local normal form. As a
consequence, we manage to uncover an additional bi-contact structure. First,
an important lemma.

######  Lemma 1.

Let $M$ be a 3-manifold with bi-contact structure satisfying the structure
equations (15). If $A_{1}=A_{2}=0$ on an open subset $U$ of $M$ then $dC\wedge
d\zeta=0$ on $U$. Moreover,

  1. (1)

If $\epsilon=-1$ then $C=-\csc(2\zeta)$ and

  2. (2)

if $\epsilon=1$ then $C=\cot(2\zeta)$.

###### Proof.

The proof is an exercise in checking $d^{2}=0$ conditions. Moving forward with
our computation, we find that since $A_{1}=A_{2}=0$

$\zeta_{1}\sin(\zeta)+\zeta_{2}\cos(\zeta)=0,$

where subscripts on $\zeta$ denote covariant derivatives. This implies the
existence of a function $W$ such that

$\zeta_{1}=-W\cos(\zeta)\text{ and }\zeta_{2}=W\sin(\zeta).$

Next, we assign some covariant derivatives of $A_{3}$ by inspecting
$d^{2}\omega^{1}$ and $d^{2}\omega^{2}$. This yields

$\displaystyle(A_{3})_{1}$
$\displaystyle=\epsilon\cos(\zeta)-(A_{3}+C)\sin(\zeta),$
$\displaystyle(A_{3})_{2}$ $\displaystyle=\sin(\zeta)-(A_{3}-C)\cos(\zeta).$

We need not assign a covariant derivative for $A_{3}$ in the direction of
$\textbf{e}_{3}$ at this time. Indeed, it is sufficient to compute
$d^{2}A_{3}\mod\\{\omega^{3}\\}$ and so we find

$d^{2}A_{3}\equiv(2(W+1)\cos(\zeta)\sin(\zeta)C-W((\epsilon+1)\cos^{2}(\zeta)-\epsilon)-\epsilon\cos^{2}(\zeta)+\sin^{2}(\zeta)))\omega^{1}\wedge\omega^{2}\,\text{mod}\\{\omega^{3}\\}.$

This forces the above coefficient of $\omega^{1}\wedge\omega^{2}$ to vanish.
We may use this to solve for the invariant function $C$ and indeed this yields

(56)
$C=\frac{W((\epsilon+1)\cos^{2}(\zeta)-\epsilon)+\epsilon\cos^{2}(\zeta)-\sin^{2}(\zeta))}{2(W+1)\cos(\zeta)\sin(\zeta)}.$

At this point, it is now prudent to split into cases. Let us first handle the
case that $\epsilon=1$. Then our expression for $C$ given by equation (56)
becomes

$C=\cot(2\zeta).$

We now have the form claimed in the Lemma. The case that $\epsilon=-1$
requires marginally more effort. Indeed, we find that

$C=\frac{W-1}{(W+1)\sin(2\zeta)},$

and so to prove our claim we must demonstrate that $W=0$ on $M$. We calculate
$dC$ and find that

$dC\equiv\frac{2\sin(\zeta)W_{1}+W(W^{2}-1)\cos(2\zeta)}{(W+1)^{2}\sin(2\zeta)\sin(\zeta)}\omega^{1}+\frac{2\cos(\zeta)W_{2}-W(W^{2}-1)\cos(2\zeta)}{(W+1)^{2}\sin(2\zeta)\cos(\zeta)}\omega^{2}\,\text{mod}\\{\omega^{3}\\},$

where $dW\equiv W_{1}\omega^{1}+W_{2}\omega^{2}\mod\\{\omega^{3}\\}$. We
recall that in this adapted coframing $dC=C_{3}\omega^{3}$ and therefore it
must be the case the the above coefficients of $\omega^{1}$ and $\omega^{2}$
vanish. Indeed, this yields

$\displaystyle W_{1}$
$\displaystyle=-\frac{W(W^{2}-1)\cos(2\zeta)}{2\sin(\zeta)},$ $\displaystyle
W_{2}$ $\displaystyle=\frac{W(W^{2}-1)\cos(2\zeta)}{2\cos(\zeta)}.$

Now we investigate the conditions forced upon us by $d^{2}\zeta=0$. Indeed,
since we now have that

$d\zeta\equiv-W\cos(\zeta)\omega^{1}+W\sin(\zeta)\omega^{2}\mod\\{\omega^{3}\\}$

then

$\displaystyle d^{2}\zeta$
$\displaystyle\equiv(\cos(\zeta)W_{2}+\sin(\zeta)W_{1}-W\sin(\zeta)\zeta_{2}+W\cos(\zeta)\zeta_{1})\omega^{1}\wedge\omega^{2}\mod\\{\omega^{3}\\},$
$\displaystyle\equiv-W^{2}\omega^{1}\wedge\omega^{2}\mod\\{\omega^{3}\\}.$

Thus, we must deduce that $W=0$ on $M$. Hence when $\epsilon=-1$ we have that
$C=-\csc(2\zeta)$ as claimed. ∎

There is an immediate corollary of Lemma 1.

######  Corollary 1.

If $(\omega^{1},\omega^{2})$ is a bi-contact structure a 3-manifold $M$ with
$C_{3}\neq 0$ and $A_{1}=A_{2}=0$ identically on $M$ then
$(\omega^{1},\omega^{2})$ is never elliptic.

######  Theorem 4.

Let $(\omega^{1},\omega^{2},\omega^{3})$ be any coframing on a 3-manifold $M$
that satisfies the structure equations (15) and let $\zeta_{3},$ and
$\zeta_{33}$ be the derivatives of $\zeta$ and $\zeta_{3}$ respectively in the
direction dual to $\omega^{3}$. If $A_{1}=A_{2}=0$ identically on an open set
$U$ of $M$ where $\zeta\in(0,\pi/4)$ and $\zeta_{3}\neq 0$, then there exists
a local invariant coordinate system $(x,y,z)=(\zeta,\zeta_{3},\zeta_{33})$ and
arbitrary functions $f(x)\in C^{1}(\mathbb{R})$ and $g(x)\in C(\mathbb{R})$
such that the coframing $(\omega^{1},\omega^{2},\omega^{3})$ has the form

$\displaystyle\omega^{1}$ $\displaystyle=\cos(x)\eta^{1}+\sin(x)\eta^{2},$
$\displaystyle\omega^{2}$ $\displaystyle=-\sin(x)\eta^{1}+\cos(x)\eta^{2},$
$\displaystyle\omega^{3}$ $\displaystyle=\frac{1}{y}dx,$

where $\eta^{1}$ and $\eta^{2}$ are 1-forms given by

$\displaystyle\eta^{1}$ $\displaystyle=\frac{1}{y^{2}}(N_{1}dx+N_{2}dy-dz),$
$\displaystyle\eta^{2}$ $\displaystyle=\frac{z}{y^{2}}\,dx-\frac{1}{y}dy.$

The functions $N_{1}$ and $N_{2}$ are depend on $\epsilon$.

  1. (1)

If $\epsilon=1$ then

$\displaystyle N_{1}$
$\displaystyle=y^{2}g(x)+2y(f(x)-2\cot(2x))\csc(2x)-(f(x)^{2}+f^{\prime}(x)+1)y^{2}\ln(y)-zf(x),$
$\displaystyle N_{2}$ $\displaystyle=\frac{2z-y^{2}f(x)-2y\csc(2x)}{y},$

where $f^{\prime}(x)$ denotes the derivative of $f(x)$.

  2. (2)

else if $\epsilon=-1$ then

$\displaystyle N_{1}$
$\displaystyle=y^{2}g(x)-2y\cot(2x)f(x)+2y+4y\csc^{2}(2x)-(f(x)^{2}+f^{\prime}(x)+1)y^{2}\ln(y)-zf(x),$
$\displaystyle N_{2}$ $\displaystyle=\frac{2z-y^{2}f(x)+y\cot(2x)}{y}.$

###### Proof.

We first demonstrate that $\zeta,\zeta_{3}$ and $\zeta_{33}$ yield a local
invariant coordinate system. Now by Lemma 1 and the structure equations (15)
we find that $d\zeta=\zeta_{3}\,\omega^{3}$ regardless of either $\epsilon=\pm
1$ case from Lemma 1. Indeed, this turns out to be true for the following
computations below, and we find that only $\epsilon$ itself contributes.
Differentiating again we find

$d\zeta_{3}=-\zeta_{3}\sin(\zeta)\omega^{1}-\zeta_{3}\cos(\zeta)\omega^{2}+\zeta_{33}\omega^{3},$

and differentiating yet again we discover

$\displaystyle d\zeta_{33}$
$\displaystyle=\left(-\zeta_{3}\left(\zeta_{3}+\frac{1-\epsilon}{2}\right)\cos(\zeta)+(A_{3}\zeta_{3}-2\zeta_{33})\sin(\zeta)+\frac{1}{2}\zeta_{3}\sec(\zeta)\right)\omega^{1}$
$\displaystyle+\left((A_{3}\zeta_{3}-2\zeta_{33})\cos(\zeta)+\zeta_{3}\left(\zeta_{3}+\frac{1-\epsilon}{2}\right)\sin(\zeta)+\epsilon\frac{1}{2}\zeta_{3}\csc(\zeta)\right)\omega^{2}+\zeta_{333}\omega^{3}.$

Thus, we now see that

$d\zeta\wedge d\zeta_{3}\wedge
d\zeta_{33}=-\zeta_{3}^{4}\,\omega^{1}\wedge\omega^{2}\wedge\omega^{3}.$

Therefore, away from subsets of $M$ where $\zeta_{3}=0$, we find that
$(\zeta,\zeta_{3},\zeta_{33})$ forms a local coordinate system. Let us label
such a coordinate system as $(x,y,z)$ where $x=\zeta,y=\zeta_{3},$ and
$z=\zeta_{33}$. We now wish to find expressions of $\omega^{1},\omega^{2}$ and
$\omega^{3}$ in terms of $dx,dy,$ and $dz$. Let us now specialize to the case
of $\epsilon=-1$ and invert the relations for $dx$, $dy$ and $dz$ above.

$\displaystyle\omega^{1}$
$\displaystyle=\frac{1}{2y^{4}}\left((2yz\varphi+2y\psi-4z^{2})\cos(x)+2y^{2}z\sin(x)-\frac{yz\cos(2x)}{\sin(x)}\right)dx$
$\displaystyle-\frac{1}{2y^{3}}\left((2y\varphi-4z)\cos(x)+2y^{2}\sin(x)-\frac{y\cos(2x)}{\sin(x)}\right)dy-\frac{\cos(x)}{y^{2}}dz,$
$\displaystyle\omega^{2}$
$\displaystyle=-\frac{1}{2y^{4}}\left((2yz\varphi+2y\psi-4z^{2})\sin(x)-2y^{2}z\cos(x)-\frac{yz\cos(2x)}{\cos(x)}\right)dx$
$\displaystyle+\frac{1}{2y^{3}}\left((2y\varphi-4z)\sin(x)-2y^{2}\cos(x)-\frac{y\cos(2x)}{\cos(x)}\right)dy+\frac{\sin(x)}{y^{2}}dz,$
$\displaystyle\omega^{3}$ $\displaystyle=\frac{1}{y}dx,$

where $\varphi(x,y,z)=A_{3}$ and $\psi(x,y,z)=\zeta_{333}$. We now arrange
$\omega^{1}$ and $\omega^{2}$ in terms of $\eta^{1}$ and $\eta^{2}$ as claimed
and then we will recover the functions $\varphi$ and $\psi$. Notice that

$\displaystyle\omega^{1}$
$\displaystyle=\left(\frac{yz\varphi+y\psi-2z^{2}-yz\cot(2x)}{y^{2}}dx-\frac{y\varphi-2z-y\cot(2x)}{y}dy-
dz\right)\frac{\cos(x)}{y^{2}}$ $\displaystyle+\left(zdx-
ydy\right)\frac{\sin(x)}{y^{2}}$ $\displaystyle\omega^{2}$
$\displaystyle=-\left(\frac{yz\varphi+y\psi-2z^{2}-yz\cot(2x)}{y^{2}}dx-\frac{y\varphi-2z-y\cot(2x)}{y}dy-
dz\right)\frac{\sin(x)}{y^{2}}$ $\displaystyle+\left(zdx-
ydy\right)\frac{\cos(x)}{y^{2}}$

Let

$\displaystyle\eta^{1}$
$\displaystyle=\frac{yz\varphi+y\psi-2z^{2}-yz\cot(2x)}{y^{4}}dx-\frac{y\varphi-2z+y\cot(2x)}{y^{3}}dy-\frac{1}{y^{2}}dz,$
$\displaystyle\eta^{2}$ $\displaystyle=\frac{z}{y^{2}}dx-\frac{1}{y}dy.$

Finally, we need to recover the functions $\varphi$ and $\psi$ in order to
acquire the claimed form in the Theorem. Indeed, we may do so by checking
$d^{2}=0$ conditions and solving the resulting PDEs. The $\omega^{3}$ 1-form
does not contribute to this effort. Suppressing some calculation details, we
find that $\sin(x)d^{2}\omega^{1}+\cos(x)d^{2}\omega^{2}=-\varphi_{z}/y^{2}$
and as such $\varphi$ must have no dependence on $z$. Using this fact,
$\sin(x)d^{2}\omega^{1}$ reduces to

$\sin(x)d^{2}\omega^{1}=\cos(2x)+(\varphi-y\varphi_{y})\sin(2x).$

It follows that $\varphi(x,y)$ has the form

$\varphi(x,y)=yf(x)-\cot(2x),$

where $f(x)$ is an arbitrary differentiable function. To determine
$\psi(x,y,z)$, we simply compute $d\omega^{1}$ and $d\omega^{2}$ in terms of
$\omega^{1},\omega^{2},$ and $\omega^{3}$ and solve the PDE conditions that
are forced upon $\psi(x,y,z)$ in order to satisfy the structure equations.
Again suppressing details, we find that

$\omega^{2}\wedge d\omega^{1}-\omega^{1}\wedge
d\omega^{2}=\psi_{z}-\frac{4z}{y}$

and therefore $\psi(x,y,z)=-2z\varphi(x,y)+2z^{2}/y+\xi(x,y)$ for some
$\xi(x,y)$ since $\omega^{2}\wedge d\omega^{1}-\omega^{1}\wedge
d\omega^{2}=-2\varphi(x,y)\Omega$ by the structure equations. The structure
equations also yield the identity $\omega^{1}\wedge
d\omega^{1}-\omega^{2}\wedge d\omega^{2}=0$ which imposes the condition

$\displaystyle 0$
$\displaystyle=\frac{3\xi(x,y)\cos(2x)}{y^{2}}-\frac{\xi_{y}(x,y)\cos(2x)}{y}+\cot(2x)(f(x)\cos(2x)-2(1+\csc(2x))$
$\displaystyle-(f(x)^{2}+f^{\prime}(x)+1)y\cos(2x)$

with solution given by

$\xi(x,y)=y^{3}g(x)-(f(x)^{2}+f^{\prime}(x)+1)y^{3}\ln(y)+2(1+\csc^{2}(2x)-f(x)\cot(2x))y^{2}$

for some $g(x)$. At this point, the structure equations are completely
satisfied, despite the arbitrary functions $f(x)$ and $g(x)$. The proof of the
case that $\epsilon=1$ is the same, but with slightly different PDE conditions
in coordinates. Indeed, a similar analysis yields that

$\displaystyle\varphi(x,y,z)$ $\displaystyle=yf(x)+\csc(2x),$
$\displaystyle\psi(x,y,z)$ $\displaystyle=-2z\varphi(x,y)+2z^{2}/y+\xi(x,y)$

for some $f(x)$ and for some $\xi(x,y)$ that satisfies $\omega^{1}\wedge
d\omega^{1}+\omega^{2}\wedge d\omega^{2}=0$. ∎

######  Proposition 9.

The 1-forms $\eta^{1}$ and $\eta^{2}$ in Theorem 4 form mixed bi-contact
structures when $\epsilon=1$ i.e. each $U\subset M$ for which Theorem 4
applies may be separated into at least two regions where $(\eta^{1},\eta^{2})$
is elliptic on one of the regions and $(\eta^{1},\eta^{2})$ is hyperbolic on
the other.

###### Proof.

We differentiate the relation between $\eta^{1},\eta^{2}$ and $\omega^{1}$ and
$\omega^{2}$. Let us first write

$\bm{\omega}=R(x)\bm{\eta},$

where $\bm{\omega}=[\omega^{1}\quad\omega^{2}]^{T}$,
$\bm{\eta}=[\eta^{1}\quad\eta^{2}]^{T}$, and

$R(x)=\begin{bmatrix}\cos(x)&\sin(x)\\\ -\sin(x)&\cos(x)\end{bmatrix}.$

It follows that

$d\bm{\eta}=R^{-1}d\bm{\omega}-R^{-1}dR\wedge\bm{\eta}.$

Moreover, if we let

$S=\begin{bmatrix}C-A_{3}&-1\\\ -\epsilon&-A_{3}-C\end{bmatrix}$

then

$d\bm{\omega}=S\omega^{3}\wedge\bm{\omega}$

and therefore

$d\bm{\eta}=(R^{-1}SR-yR^{-1}R^{\prime})\omega^{3}\wedge\bm{\eta},$

where $R^{\prime}$ denotes the derivative of $R$ with respect to $x$. As such,
when $\epsilon=1$ we get

$\displaystyle d\eta^{1}$
$\displaystyle=yf(x)\eta^{1}\wedge\omega^{3}+y\eta^{2}\wedge\omega^{3},$
$\displaystyle d\eta^{2}$
$\displaystyle=-y\eta^{1}\wedge\omega^{3}-(yf(x)+2\csc(2x))\eta^{2}\wedge\omega^{3},$

and so

$\displaystyle\eta^{1}\wedge d\eta^{1}$
$\displaystyle=y\eta^{1}\wedge\eta^{2}\wedge\omega^{3},$
$\displaystyle\eta^{1}\wedge d\eta^{2}$
$\displaystyle=-(yf(x)+2\csc(2x))\eta^{1}\wedge\eta^{2}\wedge\omega^{3},$
$\displaystyle\eta^{2}\wedge d\eta^{1}$
$\displaystyle=-yf(x)\eta^{1}\wedge\eta^{2}\wedge\omega^{3},$
$\displaystyle\eta^{2}\wedge d\eta^{2}$
$\displaystyle=y\eta^{1}\wedge\eta^{2}\wedge\omega^{3}.$

Thus, if $\eta_{a}=a_{1}\eta^{1}+a_{2}\eta^{2}$ we find that

$\displaystyle\eta_{a}\wedge d\eta_{a}$ $\displaystyle=a_{1}^{2}\eta^{1}\wedge
d\eta^{1}+a_{1}a_{2}\eta^{1}\wedge d\eta^{2}+a_{1}a_{2}\eta^{2}\wedge
d\eta^{1}+a_{2}^{2}\eta^{2}\wedge d\eta^{2},$
$\displaystyle=(ya_{1}^{2}+ya_{2}^{2}-2a_{1}a_{2}(yf(x)+\csc(2x)))\eta^{1}\wedge\eta^{2}\wedge\omega^{3}.$

Notice that if we set $\eta^{3}=y\omega^{3}$ then the we have that

$\displaystyle\eta^{1}\wedge d\eta^{1}$
$\displaystyle=\eta^{1}\wedge\eta^{2}\wedge\eta^{3},$
$\displaystyle\eta^{1}\wedge d\eta^{2}$
$\displaystyle=-\left(f(x)+\frac{2\csc(2x)}{y}\right)\eta^{1}\wedge\eta^{2}\wedge\eta^{3},$
$\displaystyle\eta^{2}\wedge d\eta^{1}$
$\displaystyle=-f(x)\eta^{1}\wedge\eta^{2}\wedge\eta^{3},$
$\displaystyle\eta^{2}\wedge d\eta^{2}$
$\displaystyle=\eta^{1}\wedge\eta^{2}\wedge\eta^{3}.$

It then follows that

$\displaystyle\eta_{a}\wedge d\eta_{a}$ $\displaystyle=a_{1}^{2}\eta^{1}\wedge
d\eta^{1}+a_{1}a_{2}\eta^{1}\wedge d\eta^{2}+a_{1}a_{2}\eta^{2}\wedge
d\eta^{1}+a_{2}^{2}\eta^{2}\wedge d\eta^{2},$
$\displaystyle=\left(a_{1}^{2}+a_{2}^{2}-2a_{1}a_{2}\left(f(x)+\frac{\csc(2x)}{y}\right)\right)\eta^{1}\wedge\eta^{2}\wedge\eta^{3}.$

So $(\eta^{1},\eta^{2})$ forms a bi-contact ellipse or hyperbola depending on
when $|f(x)+\frac{\csc(2x)}{y}|<1$ or $|f(x)+\frac{\csc(2x)}{y}|>1$
respectively. ∎

If one tries to extend Proposition 9 to the case of $\epsilon=-1$, it becomes
clear that there is no way to normalize the induced volume forms from
$\eta^{1}$ and $\eta^{2}$ to be consistent with a bi-contact ellipse. It is
interesting to note however, that the function $f(x)$ essentially controls the
mixed type of the bi-contact structure $(\eta^{1},\eta^{2})$.

##  4\. An Invariant Metric

In all of the bi-contact structures studied in this paper so far, each of them
admits a natural invariant Riemannian metric $M$ given by

$g=(\omega^{1})^{2}+(\omega^{2})^{2}+(\omega^{3})^{2}.$

We dedicate this section to some curvature calculations associated to this
metric for each of the bi-contactomorphism cases. Notice that this metric need
not be compatible with the bi-contact structure as in [17, 18].

######  Remark 2.

We note that there is another natural invariant metric defined by

$g_{\mathcal{P}_{C}}=(\omega^{1})^{2}+2C\omega^{1}\omega^{2}-\epsilon(\omega^{2})^{2}+(\omega^{3})^{2}.$

In this case the dual frame $(\textbf{e}_{1},\textbf{e}_{2},\textbf{e}_{3})$
fails to be orthonormal since
$g_{\mathcal{P}_{C}}(\textbf{e}_{1},\textbf{e}_{2})=C$. A related version of
this metric will appear in forthcoming work.

In each case, we will be able to rewrite the invariant structure equations as
the first Cartan structure equations for the Levi-Civita connection of the
invariant metric.

$\omega=\begin{bmatrix}\omega^{1}\\\ \omega^{2}\\\ \omega^{3}\end{bmatrix}$

so that each connection 1-form $\theta$ may be written as

$\theta=\begin{bmatrix}0&\omega^{1}_{2}&\omega^{1}_{3}\\\
\omega^{2}_{1}&0&\omega^{2}_{3}\\\
\omega^{3}_{1}&\omega^{3}_{2}&0\end{bmatrix}$

where $\omega^{i}_{j}=-\omega^{j}_{i}$ for all $1\leq i,j\leq 3$. so that the
Cartan structure equations

$d\omega=-\theta\wedge\omega$

hold. In each subsection, we present the connection 1-form coming from the
invariant structure equations for bi-contact equivalence. Then we’ll present
some curvature calculations of certain submanifolds via the curvature two form
$\Theta=d\theta+\theta\wedge\theta$. We find in particular, that the invariant
$C$ and $A_{3}$ has special significance in different cases.

Assume that $\omega^{3}$ is integrable and let
$\omega^{i}_{j}=\Gamma^{i}_{jk}\omega^{k}$ so that if $\Sigma$ is an integral
manifold of $\omega^{3}$, then the second fundamental form for $\Sigma$ is

(57) $\displaystyle II_{\Sigma}$
$\displaystyle=\bar{\omega}^{T}\begin{bmatrix}\Gamma^{3}_{11}&\Gamma^{3}_{12}\\\
\Gamma^{3}_{21}&\Gamma^{3}_{22}\end{bmatrix}\bar{\omega},$
$\displaystyle=\bar{\omega}^{T}S_{\Sigma}\bar{\omega}$

where $S_{\Sigma}$ is the fundamental form matrix and $\bar{\omega}$ is the
pullback of $\omega$ to leaves of the foliation of $\omega^{3}$. Thus, the
mean curvature for each leaf of the $\omega^{3}$ foliation is
$H=\Gamma^{3}_{11}+\Gamma^{3}_{22}$. Recall also, that

$\Theta^{1}_{2}=d\omega^{1}_{2}+\omega^{3}_{1}\wedge\omega^{3}_{2}$

meaning the curvature 2-form of a leaf $\iota:\Sigma\hookrightarrow M$ of
$\omega^{3}$ is

(58)
$\iota^{*}(d\omega^{1}_{2})=\iota^{*}(\Theta^{1}_{2})-\det(S_{\Sigma})\bar{\omega}^{1}\wedge\bar{\omega}^{2}.$

###  4.1. Bi-contact Equivalence $C_{3}=0$ Connection

Let $\theta^{Bi_{1}}$ be the Levi-Civita connection 1-form for the $C_{3}=0$
case. Then $\theta^{Bi_{1}}$ has entries given by

$\displaystyle\omega^{1}_{2}$ $\displaystyle=\frac{1}{2}(1-\epsilon-
B_{3})\omega^{3},$ $\displaystyle\omega^{3}_{1}$
$\displaystyle=(A_{3}-C)\omega^{1}+\frac{1}{2}\left(1+\epsilon+B_{3}\right)\omega^{2}+B_{2}\omega^{3},$
$\displaystyle\omega^{3}_{2}$ $\displaystyle=\frac{1}{2}\left(1+\epsilon-
B_{3}\right)\omega^{1}+(A_{3}+C)\omega^{2}+B_{1}\omega^{3}.$

where $B_{1}$ and $B_{2}$ are in terms of higher order derivatives of $C$, and
$B_{3}$ is the obstruction to integrability of $\omega^{3}$.

######  Proposition 10.

If $B_{3}=0$ then each leaf of the foliation defined by $\omega^{3}$ is flat
as a submanifold of $M$ and has mean curvature

$A_{3}=C\cos(2\xi)-\frac{1}{2}(\epsilon+1)\sin(2\xi).$

###### Proof.

By equation (58) and Theorem 1, we deduce that

$\iota^{*}(d\omega^{1}_{2})=\frac{1}{2}(1-\epsilon)\iota^{*}d\omega^{3}=0$

and therefore each leaf of $\omega^{3}$ is flat. Now by (57) and since
$\Gamma^{3}_{11}=A_{3}-C$ and $\Gamma^{3}_{22}=A_{3}+C$, we have that the mean
curvature is $A_{3}$. ∎

######  Corollary 2.

For a bi-contact structure that satisfies Theorem 1 with $C_{3}=B_{3}=0$
everywhere, then the integral surfaces of $\omega^{3}$ are never minimal in
$(M,g)$.

###### Proof.

Apply the formula for $A_{3}=0$ and then deduce the implications of this from
Proposition 5. ∎

###  4.2. Bi-contact Equivalence $C_{3}\neq 0$ Connection

Let $\theta^{Bi_{2}}$ be the Levi-Civita connection 1-form for the $C_{3}\neq
0$ case. Then $\theta^{Bi_{2}}$ has entries given by

$\displaystyle\omega^{1}_{2}$
$\displaystyle=A_{2}\omega^{1}-A_{1}\omega^{2}+\frac{1}{2}(\epsilon-1)\omega^{3},$
$\displaystyle\omega^{3}_{1}$
$\displaystyle=(A_{3}-C)\omega^{1}+\frac{1}{2}\left(1+\epsilon\right)\omega^{2}+\sin(\zeta)\omega^{3},$
$\displaystyle\omega^{3}_{2}$
$\displaystyle=\frac{1}{2}\left(1+\epsilon\right)\omega^{1}+(A_{3}+C)\omega^{2}+\cos(\zeta)\omega^{3}.$

######  Proposition 11.

Each leaf of the foliation defined by $\omega^{3}$ has mean curvature
$H=A_{3}$ and Gauss curvature $K=A_{3}^{2}-C^{2}-\frac{1}{2}(1+\epsilon)$.

###  4.3. Minimal Contact Hypersurfaces in a Symplectic Manifold

Let us consider the situation of a bicontact structure such that
$dC\wedge\omega^{1}\wedge\omega^{2}=C_{3}\Omega\neq 0$ and
$dC_{3}\wedge\omega^{3}=0$ on $M$. From Proposition 6 we know that we may
prolong to a symplectic manifold $\widetilde{M}$ where we have two independent
symplectic structures given by $\theta^{i}=d\omega^{1}$, $i=1,2$. On
$\widetilde{M}$ we have a natural invariant metric given by

$g=(\omega^{1})^{2}+(\omega^{2})^{2}+(\omega^{3})^{2}+(\omega^{4})^{2}.$

As in the previous section, we may inquire as to the nature of some natural
hyper surfaces. First we write out the connection 1-form $\theta$ so that
$d\omega=-\theta\wedge\omega$ where
$\omega=[\omega^{1}\quad\omega^{2}\quad\omega^{3}\quad\omega^{4}]^{T}$. If we
denote

$\theta=\begin{bmatrix}0&\omega^{1}_{2}&\omega^{1}_{3}&\omega^{1}_{4}\\\
\omega^{2}_{1}&0&\omega^{2}_{3}&\omega^{2}_{4}\\\
\omega^{3}_{1}&\omega^{3}_{2}&0&\omega^{3}_{4}\\\
\omega^{4}_{1}&\omega^{4}_{2}&\omega^{4}_{3}&0\end{bmatrix}$

where $\omega^{i}_{j}=-\omega^{j}_{i}$ for all $1\leq i,j\leq 4$, then the
structure equations $d\omega=-\theta\wedge\omega$ are compatible with our
structure equations (13) if

(59) $\displaystyle\omega^{1}_{2}$
$\displaystyle=\frac{1}{2}(1-\epsilon)\omega^{3}-\frac{1}{2}E\omega^{4},$
$\displaystyle\omega^{3}_{1}$
$\displaystyle=-C\omega^{1}+\frac{1}{2}\left(1+\epsilon\right)\omega^{2},$
$\displaystyle\omega^{3}_{2}$
$\displaystyle=\frac{1}{2}\left(1+\epsilon\right)\omega^{1}+C\omega^{2},$
$\displaystyle\omega^{4}_{1}$
$\displaystyle=\frac{1}{2}E\omega^{2}+\omega^{1},$
$\displaystyle\omega^{4}_{2}$
$\displaystyle=-\frac{1}{2}E\omega^{1}+\omega^{2},$
$\displaystyle\omega^{4}_{3}$ $\displaystyle=0.$

Integral surfaces of $\omega^{3}$ are contact submanifolds of $\widetilde{M}$
with contact form $\omega^{4}$. Moreover, the shape operator $S_{\Sigma}$ for
such submanifolds is given by

$S_{\Sigma}=\begin{bmatrix}-C&\frac{1}{2}\left(1+\epsilon\right)&0\\\
\frac{1}{2}\left(1+\epsilon\right)&C&0\\\ 0&0&0\end{bmatrix}.$

As such, we have that each $\Sigma$ an integral manifold of $\omega^{3}$ is
minimal as a hypersurface in $(\widetilde{M},g)$.

Given the relative simplicity of the connection, we may calculate the
curvature two form. Let $\Theta$ be the $\mathfrak{so}(4)$-valued 2-form with
nonzero entries given by

(60) $\displaystyle\Theta^{1}_{2}$
$\displaystyle=(C^{2}-\frac{3}{4}E^{2}-\frac{1}{2}(1-\epsilon))\omega^{1}\wedge\omega^{2}-\frac{1}{2}E_{1}\omega^{1}\wedge\omega^{4}-\frac{1}{2}E_{2}\omega^{2}\wedge\omega^{4},$
$\displaystyle\Theta^{1}_{3}$
$\displaystyle=-\left(C^{2}+C_{3}+\frac{1}{2}(1+\epsilon)\right)\omega^{1}\wedge\omega^{3}+\left(C-\frac{1}{4}E(1+\epsilon)\right)\omega^{1}\wedge\omega^{4}$
$\displaystyle+C(1-\epsilon)\omega^{2}\wedge\omega^{3}-\frac{1}{2}\left(1+\epsilon+CE\right)\omega^{2}\wedge\omega^{4},$
$\displaystyle\Theta^{1}_{4}$
$\displaystyle=-\frac{1}{2}E_{1}\omega^{1}\wedge\omega^{2}+\left(C-\frac{1}{4}E(1+\epsilon)\right)\omega^{1}\wedge\omega^{3}+\left(\frac{1}{4}E^{2}-1\right)\omega^{1}\wedge\omega^{4}$
$\displaystyle-\frac{1}{2}\left(1+\epsilon+CE\right)\omega^{2}\wedge\omega^{3},$
$\displaystyle\Theta^{2}_{3}$
$\displaystyle=C(1+\epsilon)\omega^{1}\wedge\omega^{3}-\frac{1}{2}(1+\epsilon+CE)\omega^{1}\wedge\omega^{4}$
$\displaystyle-\left(\frac{1}{2}(1+\epsilon)+C^{2}-C_{3}\right)\omega^{2}\wedge\omega^{3}+\left(-C+\frac{1}{4}E(1+\epsilon)\right)\omega^{2}\wedge\omega^{4},$
$\displaystyle\Theta^{2}_{4}$
$\displaystyle=-\frac{1}{2}E_{2}\omega^{1}\wedge\omega^{2}-\frac{1}{2}(1+\epsilon+CE)\omega^{1}\wedge\omega^{3}+\left(-C+\frac{1}{4}E(1+\epsilon)\right)\omega^{2}\wedge\omega^{3}$
$\displaystyle+\left(1-\frac{1}{4}E^{2}\right)\omega^{2}\wedge\omega^{4},$

and $\Theta^{3}_{4}=0$.

From the curvature 2-forms, we find that the scalar curvature is
$S=-(E^{2}/2+2C^{2}+7+\epsilon)<0$. Moreover, we may compute
$\text{Pf}(\Theta)$ and in so doing we find that

$\text{Pf}(\Theta)=2((1+\epsilon)+2C^{2})\Omega$

where $\Omega=\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\wedge\omega^{4}$.

##  5\. Conclusion and Further Work

We have demonstrated that bi-contact structures under bi-contactomorphisms
admit non-trivial local invariants. The invariant of primary concern is the
function $C$ and its derivatives. We found a local normal form in a special
case and demonstrated a Cartan style function count for initial data needed to
determine a bi-contactomorphism in case $C$ is constant. Finally, we presented
some invariant metric calculations and explored some local geometry of
foliations determined by $\omega^{3}$.

There are many ways to generalize this study of the local geometry of bi-
contact structures. One may increase the dimension and the number of
independent contact structures and study similar types of diffeomorphisms as
bi-contactomorphisms. The first author has already made some progress in this
case. Also, in upcoming work, the first author will also demonstrate a
relationship between Beltrami fields, Anosov flows, and bi-contact structures.
Notably, Theorem 4 yields a normal form for special types of projectively
Anosov flows.

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