\section{Introduction}

Helicoidal surfaces in $3$-dimensional space forms arise as a natural
generalization of rotational surfaces in such spaces. These surfaces
are invariant by a subgroup of the group of isometries of the ambient
space, called helicoidal group, whose elements can be seen
as a composition of a translation with a rotation for a given axis.

In the Euclidean space $\mathbb{R}^3$, do Carmo and Dajczer \cite{docarmo} describe the
space of all helicoidal surfaces that have constant mean
curvature or constant Gaussian curvature. This space behaves as
a circular cylinder, where a given generator corresponds to the
rotational surfaces and each parallel corresponds to a periodic family of helicoidal surfaces. Helicoidal surfaces with prescribed mean or Gaussian curvature are obtained by Baikoussis and Koufogiorgos \cite{BK}. More precisely, they obtain a closed form of such a surface by integrating the second-order ordinary differential equation satisfied by the generating curve of the surface. Helicoidal
surfaces in $\mathbb{R}^3$ are also considered by Perdomo \cite{P1} in the context
of minimal surfaces, and by Palmer and Perdomo \cite{PP2} where the
mean curvature is related with the distance to the $z$-axis. In the
context of constant mean curvature, helicoidal surfaces are considered by
Solomon and Edelen in \cite{edelen1}.

In the $3$-dimensional hyperbolic space $\mathbb{H}^3$, Mart\'inez, the second
author and Tenenblat \cite{MST} give a complete classification of the
helicoidal flat surfaces in terms of meromorphic data, which extends the
results obtained by Kokubu, Umehara and Yamada \cite{KUY} for rotational
flat surfaces. Moreover, the classification is also given by means of linear
harmonic functions, characterizing the flat fronts in $\mathbb{H}^3$ that
correspond to linear harmonic functions. Namely, it is well known that for
flat surfaces in $\mathbb{H}^3$, on a neighbourhood of a non-umbilical
point, there is a curvature line parametrization such that the first and
second fundamental forms are given by
\begin{equation}
\begin{array}{rcl}
I &=& \cosh^2 \phi(u,v) (du)^2 + \sinh^2 \phi(u,v) (dv)^2,  \\
II &=& \sinh \phi(u,v) \cosh \phi(u,v) \left((du)^2 + (dv)^2 \right),
\end{array} \label{firstff}
\end{equation}
where $\phi$ is a harmonic function, i.e., $\phi_{uu}+\phi_{vv}=0$. In
this context, the main result states that a surface in $\mathbb{H}^3$, parametrized
by curvature lines, with fundamental forms as in \eqref{firstff} and
$\phi(u,v)$ linear, i.e, $\phi(u,v) = a u + b v + c$, is flat if and only if, the
surface is a helicoidal surface or a {\em peach front}, where the second
one is associated to the case $(a,b,c) = (0,\pm1,0)$.
Helicoidal minimal surfaces were studied by Ripoll \cite{ripoll} and
helicoidal constant mean curvature surfaces in $\mathbb{H}^3$ are
considered by Edelen \cite{edelen2}, as well as the cases where
such invariant surfaces belong to $\mathbb{R}^3$ and $\mathbb{S}^3$.

Similarly to the hyperbolic space, for a given flat surface in the
$3$-dimensional sphere $\mathbb{S}^3$, there exists a parametrization by
asymptotic lines, where the first and the second fundamental forms
are given by
\begin{equation}
\begin{array}{rcl}
I &=& du^2 + 2 \cos \omega du dv + dv^2, \\
II &=& 2  \sin \omega du dv \label{primeiraffs}
\end{array}
\end{equation}
for a smooth function $\omega$, called the {\em angle function}, that
satisfy the homogeneous wave equation $\omega_{uv} = 0$.
Therefore, one can ask which surfaces are related to linear solutions
of such equation.

The aim of this paper is to give a complete classification of helicoidal flat
surfaces in $\mathbb{S}^3$, established in Theorems 1 and 2, by means of asymptotic lines coordinates, with first
and second fundamental forms given by \eqref{primeiraffs}, where the
angle function is linear. In order to do this, one uses the Bianchi-Spivak construction for flat surfaces in $\mathbb{S}^3$. This construction and the Kitagawa representation \cite{kitagawa1}, are important tools used
in the recent developments of flat surface theory. Examples of applications
of such representations can be seen in \cite{galvezmira1} and \cite{aledogalvezmira}. Our classification also makes use of a
representation for constant angle surfaces in $\mathbb{S}^3$, who comes
from a characterization of constant angle surfaces in the Berger
spheres obtained by Montaldo and Onnis \cite{MO}.

This paper is organized as follows. In Section 2 we give a brief description
of helicoidal surfaces in $\mathbb{S}^3$, as well as a ordinary differential equation
that characterizes those one that has zero intrinsic curvature.

In Section 3, the Bianchi-Spivak construction is introduced. It will be used to prove Theorem 1, which states that a flat surface in $\mathbb{S}^3$, with asymptotic
parameters and linear angle function, is invariant under helicoidal motions.

In Section 4, Theorem 2 establishes  the converse of Theorem 1, that is, a helicoidal flat surface admits a local parametrization,  given
by asymptotic parameters where the angle function is linear. Such local parametrization is obtained by using a characterization
of constant angle surfaces in Berger spheres, which is a consequence of the fact that a helicoidal flat surface is a
constant angle surface in $\mathbb{S}^3$, i.e., it has a unit normal that makes a constant angle with the Hopf vector field.

In section 5 we present an application for conformally flat hypersurfaces
in $\mathbb{R}^4$. The classification result obtained is used to give a geometric characterization for special conformally flat surfaces in $4-$dimensional
space forms. It is known that conformally flat hypersurfaces in
$4$-dimensional space forms are associated with solutions of a system
of equations, known as Lam\'e's system (see \cite{Jeromin1} and
\cite{santos} for details). In \cite{santos}, Tenenblat and the second author
obtained invariant solutions under symmetry groups of Lam\'e's system.
A class of those solutions is related to flat surfaces in $\mathbb{S}^3$,
parametrized by asymptotic lines with linear angle function. Thus
a geometric description of the correspondent conformally flat
hypersurfaces is given in terms of helicoidal flat surfaces in $\mathbb{S}^3$.

\section{Helicoidal flat surfaces}

Given any $\beta\in\mathbb{R}$, let $\{\varphi_\beta(t)\}$ be the one-parameter
subgroup of isometries of $\mathbb{S}^3$ given by
\[
\varphi_\beta(t) = \left(
\begin{array}{cccc}
1  & 0 &  0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \cos \beta t & -\sin \beta t \\
0 & 0 & \sin \beta t & \cos \beta t \\
\end{array}
\right).
\]
When $\beta\neq0$, this group fixes the set $l=\{(z,0)\in\mathbb{S}^3\}$, which
is a great circle and it is called the {\em axis of rotation}. In this case, the
orbits are circles centered on $l$, i.e., $\{\varphi_\beta(t)\}$ consists of
rotations around $l$. Given another number $\alpha\in\mathbb{R}$, consider now
the translations
$\{\psi_\alpha(t)\}$ along $l$,
\[
\psi_\alpha(t)=\left(
\begin{array}{cccc}
\cos \alpha t & -\sin \alpha t &  0 & 0 \\
\sin \alpha t & \cos \alpha t & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right).
\]

\begin{defi}\label{def:helicoidal}
{\em A {\em helicoidal} surface in $\mathbb{S}^3$ is a surface invariant under the
action of the helicoidal 1-parameter group of isometries
\begin{equation}\label{eq:movhel}
\phi_{\alpha,\beta}(t)=\psi_\alpha(t)\circ\varphi_\beta(t)=
\left(
\begin{array}{cccc}
\cos \alpha t & -\sin \alpha t &  0 & 0 \\
\sin \alpha t & \cos \alpha t & 0 & 0 \\
0 & 0 & \cos \beta t & -\sin \beta t \\
0 & 0 & \sin \beta t & \cos \beta t \\
\end{array}
\right),
\end{equation}}
{\em given by a composition of a translation $\psi_\alpha(t)$ and a rotation $\varphi_\beta(t)$ in $\mathbb{S}^3$. }
\end{defi}

\begin{rem}
{\em When $\alpha=\beta$, these isometries are usually called
{\em Clifford translations}. In this case, the orbits are all great circles,
and they are equidistant from each other. In fact, the orbits of the action
of $G$ coincide with the fibers of the Hopf fibration $h:\mathbb{S}^3\to\mathbb{S}^2$.
We note that, when $\alpha=-\beta$, these isometries are also, up to
a rotation in $\mathbb{S}^3$, Clifford translations. For this reason we will consider
in this paper only the cases $\alpha\neq\pm\beta$.}
\end{rem}

With these basic properties in mind, a helicoidal surface can be locally
parametrized by
\begin{equation}\label{eq:param-helicoidal}
X(t,s) = \phi_{\alpha,\beta}(t)\cdot\gamma(s),
\end{equation}
where $\gamma:I\subset\mathbb{R}\to\mathbb{S}^2_+$ is a curve parametrized by the arc
length, called the {\em profile curve} of the parametrization $X$. Here,
$\mathbb{S}^2_+$ is the half totally geodesic sphere of $\mathbb{S}^3$ given by
\[
\mathbb{S}^2_+ = \left\{(x_1, x_2, x_3, 0)\in\mathbb{S}^3: x_3>0\right\}.
\]
Then we have
\[
\begin{array}{rcl}
X_t &=& \phi_{\alpha,\beta}(t)\cdot(-\alpha x_2,\alpha x_1, 0,\beta x_3), \\
X_s &=& \phi_{\alpha,\beta}(t)\cdot\gamma'(s).
\end{array}
\]
Moreover, a unit normal vector field associated to the parametrization
$X$ is given by $N=\tilde N/ \|\tilde N\|$, where $\tilde N$ is explicitly given
by
\begin{equation}\label{normal-field}
\tilde{N} = \phi_{\alpha,\beta}(t)\cdot \big(\beta x_3(x_2'x_3-x_2 x_3',\beta x_3(x_1x_3'-x_1'x_3),
\beta x_3 (x_1'x_2-x_1 x_2'),-\alpha x_3' \big).
\end{equation}
Let us now consider a parametrization by the arc length of  $\gamma$
given by
\begin{equation} \label{eq:param-gamma}
\gamma(s) = \big(\cos\varphi(s)\cos\theta(s),\cos\varphi(s)\sin\theta(s),
\sin\varphi(s),0\big).
\end{equation}

We will finish this section discussing the flatness of helicoidal surfaces
in $\mathbb{S}^3$. Recall that a simple way to obtain flat surfaces in $\mathbb{S}^3$
is by means of the Hopf fibration $h:\mathbb{S}^3\to\mathbb{S}^2$. More precisely,
if $c$ is a regular curve in $\mathbb{S}^2$, then $h^{-1}(c)$ is a flat surface
in $\mathbb{S}^3$ (cf. \cite{spivak}). Such surfaces are called {\em Hopf cylinders}. The next result provides a necessary
and sufficient condition for a helicoidal surface, parametrized as in
\eqref{eq:param-helicoidal}, to be flat.

\begin{prop}\label{prop:HSF}
A helicoidal surface locally parametrized as in \eqref{eq:param-helicoidal},
where $\gamma$ is given by \eqref{eq:param-gamma},
is a flat surface if and only if the following equation
\begin{equation}
\beta^2\varphi''\sin^3\varphi\cos\varphi - \beta^2(\varphi')^2 \sin^4\varphi
+\alpha^2(\varphi')^4 \cos^4 \varphi = 0 \label{ode-helicoidal}
\end{equation} \label{prop-ode-helicoidal}
is satisfied.
\end{prop}
\begin{proof}
Since $\phi_{\alpha,\beta}(t)\in O(4)$ and $\gamma$ is parametrized by the arc length, the
coefficients of the first fundamental form are
given by
\[
\begin{array}{rcl}
E &=& \alpha^2\cos^2\varphi + \beta^2\sin^2\varphi, \\
F &=& \alpha\theta'\cos^2\varphi,\\
G &=& (\varphi')^2 + (\theta')^2 \cos^2 \varphi =1.
\end{array}
\]
Moreover, the Gauss curvature $K$ is given by
\[
\begin{array}{rcl}
4(EG - F^2)^2 K &=& E \left[ E_s G_s - 2 F_t G_s + (G_t)^2 \right] +
G \left[E_t G_t - 2 E_t F_s + (E_s)^2 \right]  \\
&&+ F (E_t G_s - E_s G_t - 2 E_s F_s + 4 F_t F_s - 2 F_t G_t )  \\
&& - 2 (EG-F^2)(E_{ss} - 2 F_{st} + G_{tt} ).
\end{array}
\]
Thus, it follows from the expression of $K$ and from the coefficients of
the first fundamental form that the surface is flat if, and only if,
\begin{equation}\label{eq:gauss}
E_s (EG - F^2)_s - 2 (EG-F^2)E_{ss}=0.
\end{equation}
When $\alpha=\pm\beta$, the equation \eqref{eq:gauss} is trivially
satisfied, regardless of the chosen curve $\gamma$. For the case
$\alpha\neq\pm\beta$, since
\[
EG-F^2 =\beta^2\sin^2\varphi+\alpha^2(\varphi')^2\cos^2\varphi,
\]
a straightforward computation shows that the equation \eqref{eq:gauss}
is equivalent to
\[
(\beta^2-\alpha^2)\big(\beta^2\varphi''\sin^3\varphi\cos\varphi
-\beta^2(\varphi')^2\sin^4\varphi+\alpha^2(\varphi')^4\cos^4\varphi\big) = 0,
\]
and this concludes the proof.
\end{proof}

\section{The Bianchi-Spivak construction}

A nice way to understand the fundamental equations of a flat surface $M$ in
$\mathbb{S}^3$ is by parameters whose coordinate curves are asymptotic curves
on the surface. As $M$ is flat, its intrinsic curvature vanishes identically.
Thus, by the Gauss equation, the extrinsic curvature of $M$ is constant
and equal to $-1$. In this case, as the extrinsic curvature is negative, it is
well known that there exist Tschebycheff coordinates around every point.
This means that we can choose local coordinates $(u,v)$ such that the
coordinates curves are asymptotic curves of $M$ and these curves are
parametrized by the arc length. In this case, the first and second fundamental
forms are given by
\begin{eqnarray}\label{eq:forms}
\begin{aligned}
I  &=  du^2 + 2 \cos \omega dudv + dv^2, \\
II&  =   2 \sin \omega du dv,
\end{aligned}
\end{eqnarray}
for a certain smooth function $\omega$, usually called the {\em angle function}.
This function $\omega$ has two basic properties. The first one is that as $I$
is regular, we must have $0<\omega<\pi$. Secondly, it follows from the Gauss
equation that $\omega_{uv}=0$. In other words, $\omega$ satisfies the homogeneous wave equation, and thus it can be locally decomposed as
$\omega(u,v) = \omega_1(u) + \omega_2(v)$, where $\omega_1$ and
$\omega_2$ are smooth real functions (cf. \cite{galvez1} and \cite{spivak}
for further details).

\vspace{.2cm}

Given a flat isometric immersion $f:M\to\mathbb{S}^3$ and a local smooth unit
normal vector field $N$ along $f$, let us consider coordinates $(u,v)$
such that the first and the second fundamental forms of $M$ are given
as in \eqref{eq:forms}. The aim of this work is to characterize the flat
surfaces when the angle function  $\omega$ is linear, i.e., when
$\omega=\omega_1+\omega_2$ is given by
\begin{eqnarray}\label{eq:linear}
\omega_1(u) + \omega_2 (v) = \lambda_1 u + \lambda_2 v + \lambda_3
\end{eqnarray}
where $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{R}$. I order to do this, let us
first construct flat surfaces in $\mathbb{S}^3$ whose first and second fundamental
forms are given by \eqref{eq:forms} and with linear angle function. This
construction is due to Bianchi \cite{bianchi} and Spivak \cite{spivak}.

\vspace{.2cm}

We will use here the division algebra of the quaternions, a very useful approach
to describe explicitly flat surfaces in $\mathbb{S}^3$. More precisely, we identify
the sphere $\mathbb{S}^3$ with the set of the unit quaternions
$\{q\in\mathrm{H}: q\overline q=1\}$ and $\mathbb{S}^2$ with the unit sphere in the
subspace of $\mathrm{H}$ spanned by $1$, $i$ and $j$.

\begin{prop}[Bianchi-Spivak representation]\label{teo:BS}
Let $c_a,c_b:I\subset\mathbb{R}\to\mathbb{S}^3$ be two curves parametrized by the
arc length, with curvatures $\kappa_a$ and $\kappa_b$, and whose torsions
are given by $\tau_a=1$ and $\tau_b=-1$. Suppose that $0 \in I$, $c_a(0)=c_b(0)=(1,0,0,0)$ e $c_a'(0) \wedge c_b '(0) \neq 0$.
Then the map
\[
X(u,v) = c_a (u) \cdot c_b (v)
\]
is a local parametrization of a flat surface in $\mathbb{S}^3$, whose first and
second fundamental forms are given as in \eqref{eq:forms}, where the
angle function satisfies $\omega_1'(u) = -\kappa_a (u)$ and
$\omega_2'(v) =\kappa_b (v)$.
\end{prop}

Since the goal here is to find a parametrization such that $\omega$ can
be written as in \eqref{eq:linear}, it follows from Theorem \ref{teo:BS}
that the curves of the representation must have constant curvatures.
Therefore, we will use the Frenet-Serret formulas in order to obtain curves
with torsion $\pm 1$ and with constant curvatures.

\vspace{.2cm}

Given a real number $r>1$, let us consider the curve
$\gamma_r:\mathbb{R}\to\mathbb{S}^3$ given by
\begin{equation}\label{eq:base}
\gamma_r(u) = \frac{1}{\sqrt{1+r^2}} \left(r\cos\frac{u}{r},r\sin\frac{u}{r},
\cos ru,\sin ru\right).
\end{equation}
A straightforward computation shows that $\gamma_r(u)$ is
parametrized by the arc length, has constant curvature
$\kappa=\frac{r^2-1}{r}$ and its torsion $\tau$ satisfies $\tau^2=1$.
Observe that $\gamma_r(u)$ is periodic if and only if $r^2\in\mathbb Q$.
When $r$ is a positive integer, $\gamma_r(u)$ is a closed
curve of period $2\pi r$. A curve $\gamma$ as in \eqref{eq:base} will
be called a {\em base curve}.

\vspace{.2cm}

Now we just have to apply rigid motions to a base curve in order to satisfy
the remaining requirements of the Bianchi-Spivak construction. It is easy
to verify that the curves
\begin{eqnarray}\label{eq:curves-condition}
\begin{array}{rcl}
c_a(u) &=& \dfrac{1}{\sqrt{1+a^2}}(a,0,-1,0) \cdot \gamma_a (u), \\
c_b(v) &=& \dfrac{1}{\sqrt{1+b^2}}T ( \gamma_b (v)) \cdot (b,0,0,-1),
\end{array}
\end{eqnarray}
are base curves, and satisfy $c_a(0)=c_b(0)=(1,0,0,0)$ and
$c_a'(0)\wedge c_b'(0)\neq0$, where
\begin{equation}
\begin{array}{rcl}
T &=& \left( \begin{array}{cccc}
1&0&0 & 0\\
0&1&0& 0 \\
0&0&0&1\\
0&0&1&0
\end{array} \right). \\
\end{array}
\end{equation}

Therefore we can establish our first main result:

\begin{thm}\label{teo:main1}
The map $X:U\subset\mathbb{R}^2\to\mathbb{S}^3$ given by
\[
X(u,v) = c_a(u) \cdot c_b(v),
\]
where $c_a$ and $c_b$ are the curves given in \eqref{eq:curves-condition},
is a parametrization of a flat surface in $\mathbb{S}^3$, whose first and second
fundamental forms are given by
\[
\begin{array}{rcl}
I &=& du^2 +2  \cos \left( \left( \frac{1-a^2}{a} \right) u + \left( \frac{b^2-1}{b} \right) v + c \right) du dv + dv^2, \\
II &=& 2 \sin \left(\left( \frac{1-a^2}{a} \right) u + \left( \frac{b^2-1}{b} \right) v + c \right) du dv,
\end{array}
\]
where $c$ is a constant. Moreover, up to rigid motions, $X$ is invariant under helicoidal motions.
\end{thm}
\begin{proof}
The statement about the fundamental forms follows directly from the
Bianchi-Spivak construction. For the second statement, note that
the parametrization $X(u,v)$ can be written as
\[
X(u,v) = g_a \cdot  Y(u,v) \cdot g_b,
\]
where
\[
\begin{array}{rcl}
g_a &=& \dfrac{1}{\sqrt{1+a^2}}(a,0,-1,0), \\
g_b &=& \dfrac{1}{\sqrt{1+b^2}}(b,0,0,-1),
\end{array}
\]
and
\[
Y(u,v) = \gamma_a(u) \cdot T(\gamma_b(v)).
\]
To conclude the proof, it suffices to show that $Y(u,v)$ is invariant by
helicoidal motions. To do this, we have to find $\alpha$ and $\beta$
such that
\[
\phi_{\alpha,\beta}(t)\cdot Y(u,v)=Y\big(u(t),v(t)\big),
\]
where $u(t)$ and $v(t)$ are smooth functions. Observe that $Y(u,v)$
can be written as
\begin{eqnarray}\label{eq:expY}
Y(u,v) = \dfrac{1}{\sqrt{(1+a^2)(1+b^2)}} (y_1, y_2, y_3, y_4),
\end{eqnarray}
where
\begin{equation*}
\begin{array}{rcl}
y_1(u,v) &=& ab \cos \left(\dfrac{u}{a} + \dfrac{v}{b} \right) - \sin (au+ bv), \\
y_2(u,v) &=& ab \sin \left(\dfrac{u}{a} + \dfrac{v}{b} \right) + \cos (au+ bv), \\
y_3(u,v) &=& b \cos \left(au- \dfrac{v}{b} \right) - a\sin\left(\dfrac{u}{a}-bv\right), \\
y_4(u,v) &=& b\sin\left(au-\dfrac{v}{b} \right)+a\cos\left(\dfrac{u}{a}-bv\right). \\
\end{array} \label{parametrizacao-y}
\end{equation*}
A straightforward computation shows that if $\phi_{\alpha,\beta}(t)$ is given by
\eqref{eq:movhel}, we have
\[
u(t) =u+z(t) \quad\textrm{and}\quad v(t)=v+w(t),
\]
where
\begin{eqnarray}\label{eq:z(t)w(t)}
z(t)=\frac{a(b^2-1)}{a^2b^2-1}\beta t \quad\text{and}\quad
w(t)=\frac{b(1-a^2)}{a^2b^2-1}\beta t,
\end{eqnarray}
with
\begin{eqnarray}\label{eq:alpha-beta}
\alpha=\dfrac{b^2-a^2}{a^2b^2-1}\beta,
\end{eqnarray}
showing that $Y(u,v)$ is invariant by helicoidal motions. Observe that
when $a=\pm b$ we have $\alpha=0$, i.e., $X$ is a rotational surface
in $\mathbb{S}^3$.
\end{proof}

\begin{rem}
{\em It is important to note that the constant $a$ and $b$ in
\eqref{eq:curves-condition} were considered in $(1,+\infty)$ in order to
obtain non-zero constant curvatures with its well defined torsions,
and then to apply the Bianchi-Spivak construction. This is not a strong
restriction since the curvature function $\kappa(t)=\frac{t^2-1}{t}$ assumes
all values in $\mathbb{R} \setminus\{0\}$ when $t \in (1, + \infty)$. However, by
taking $a=1$ and $b>1$ in \eqref{eq:curves-condition}, a long but
straightforward computation gives an unit normal vector field
\[
N(u,v) = \dfrac{1}{\sqrt{2(1+b^2)}} (n_1, n_2, n_3, n_4),
\]
where
\[
\begin{array}{rcl}
n_1(u,v) &=& -b \sin \left(u + \dfrac{v}{b} \right) + \cos (u+ bv), \\
n_2(u,v) &=& b \cos \left(u + \dfrac{v}{b} \right) + \sin (u+ bv), \\
n_3(u,v) &=& b \sin \left(u- \dfrac{v}{b} \right) - \cos\left(a-bv\right), \\
n_4(u,v) &=& -b\cos \left(u-\dfrac{v}{b} \right) - \sin\left(u-bv\right). \\
\end{array}
\]
Therefore, one shows that this parametrization is also by asymptotic
lines where the angle function is given by
$\omega(u,v) =\frac{1-b^2}{b}v -\frac{\pi}{2}$. Moreover, this is a
parametrization of a Hopf cylinder, since the unit normal vector field $N$
makes a constant angle with the Hopf vector field (see section 4).}
\end{rem}

We will use the parametrization $Y(u,v)$ given in \eqref{eq:expY}, compose with the stereographic projection in $\mathbb{R}^3$,  to visualize some examples with the corresponding constants $a$ and $b$.

\begin{figure}[!htb]
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura1.jpg}
\end{minipage}
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura2.jpg}
\end{minipage}
\caption{$a=2$ and $b=3$.}
\end{figure}

\begin{figure}[!htb]
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura3.jpg}
\end{minipage}
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura4.jpg}
\end{minipage}
\caption{$a=\sqrt2$ and $b=3$.}
\end{figure}

\begin{figure}[!htb]
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura5.jpg}
\end{minipage}
\begin{minipage}[]{0.45\linewidth}
\includegraphics[width=\linewidth]{figura6.jpg}
\end{minipage}
\caption{$a=\sqrt3$ and $b=\sqrt2$.}
\end{figure}

\section{Constant angle surfaces}

In this section we will complete our classification of helicoidal flat
surfaces in $\mathbb{S}^3$, by establishing our second main theorem, that
can be seen as a converse of Theorem \ref{teo:main1}. It is well
known that the Hopf map $h:\mathbb{S}^3\to\mathbb{S}^2$ is a Riemannian
submersion and the standard orthogonal basis of $\mathbb{S}^3$
\[
E_1(z,w)=i(z,w), \ \ E_2(z,w)=i(-\overline w,\overline z), \ \
E_3(z,w)=(-\overline w,\overline z)
\]
has the property that $E_1$ is vertical and $E_2$, $E_3$ are horizontal.
The vector field $E_1$, usually called the Hopf vector field, is an unit
Killing vector field.

\vspace{.2cm}

Constant angle surface in $\mathbb{S}^3$ are those surfaces whose its
unit normal vector field makes a constant angle with the Hopf vector
field $E_1$. The next result states that flatness of a helicoidal surface
in $\mathbb{S}^3$ turns out to be equivalent to constant angle surface.

\begin{prop}\label{prop:CAS}
A helicoidal surface in $\mathbb{S}^3$, locally parametrized by
\eqref{eq:param-helicoidal} and with the profile curve $\gamma$ parametrized
by \eqref{eq:param-gamma}, is a flat surface if and only if it is a
constant angle surface.
\end{prop}
\begin{proof}
Let us consider the Hopf vector field
\[
E_1(x_1, x_2, x_3, x_4) = (-x_2, x_1, -x_4, x_3),
\]
and let us denote by $\nu$ the angle between $E_1$ and the normal vector
field $N$ along the surface given in \eqref{normal-field}. Along the
parametrization \eqref{eq:param-helicoidal}, we can write the vector field
$E_1$ as
\[
E_1(X(t,s)) = \phi_{\alpha,\beta}(t)(-x_2, x_1,0,x_3).
\]
Then, since $\phi_{\alpha,\beta}(t)\in O(4)$, we have
\[
\langle N,E_1\rangle(t,s)=\langle N,E_1\rangle(s)=
(\beta-\alpha) \frac{x_3 x_3'}{\sqrt{\beta^2x_3^2 + \alpha^2 (x_3')^2}}.
\]
By considering the parametrization \eqref{eq:param-gamma} for the
profile curve $\gamma$, the angle $\nu=\nu(s)$ between $N$ and $E_1$
is given by
\begin{eqnarray}\label{eq:cosnu}
\cos \nu (s) = (\beta-\alpha) \frac{\varphi' \sin\varphi \cos\varphi}
{\sqrt{\beta^2\sin^2 \varphi + \alpha^2 (\varphi')^2 \cos^2 \varphi}}.
\end{eqnarray}
By taking the derivative in \eqref{eq:cosnu}, we have
\[
\frac{d}{ds}(\cos\nu(s))=\frac{(\beta-\alpha)\big(\beta^2\varphi''\sin^3
\varphi\cos\varphi - \beta^2(\varphi')^2 \sin^4 \varphi +
\alpha^2 (\varphi')^2 \cos^4 \varphi\big)}
{\big(\beta^2\sin^2\varphi+\alpha^2(\varphi')^2 \cos^2 \varphi \big)^{\frac{3}{2}}},
\]
and the conclusion follows from the Proposition \ref{prop:HSF}.
\end{proof}

Given a number $\epsilon>0$, let us recall that the Berger sphere
$\mathbb{S}^3_\epsilon$ is defined as the sphere $\mathbb{S}^3$ endowed with the
metric
\begin{eqnarray}\label{eq:berger}
\langle X,Y\rangle_\epsilon=\langle X,Y\rangle+(\epsilon^2-1)\langle X,E_1\rangle\langle Y,E_1\rangle,
\end{eqnarray}
where $\langle,\rangle$ denotes de canonical metric of $\mathbb{S}^3$. We define
constant angle surface in $\mathbb{S}^3_\epsilon$ in the same way that
in the case of $\mathbb{S}^3$. Constant angle surfaces in the Berger spheres
were characterized by Montaldo and Onnis \cite{MO}. More precisely,
if $M$ is a constant angle surface in the Berger sphere, with constant
angle $\nu$, then there exists a local parametrization $F(u,v)$ given by
\begin{eqnarray}\label{eq:paramMO}
F(u,v)=A(v)b(u),
\end{eqnarray}
where
\begin{eqnarray}\label{eq:curveb}
b(u)=\big(\sqrt{c_1}\cos(\alpha_1u),\sqrt{c_1}\sin(\alpha_1u),
\sqrt{c_2}\cos(\alpha_2u),\sqrt{c_2}\sin(\alpha_2u)\big)
\end{eqnarray}
is a geodesic curve in the torus $\mathbb{S}^1(\sqrt{c_1})\times\mathbb{S}^1(\sqrt{c_2})$,
with
\[
c_{1,2}=\frac{1}{2}\mp\frac{\epsilon\cos\nu}{2\sqrt{B}},\
\alpha_1=\frac{2B}{\epsilon}c_2, \ \alpha_2=\frac{2B}{\epsilon}c_1, \
B=1+(\epsilon^2-1)\cos^2\nu,
\]
and
\begin{eqnarray}\label{eq:xi_i}
A(v)=A(\xi,\xi_1,\xi_2,\xi_3)(v)
\end{eqnarray}
is a $1$-parameter family of $4\times4$ orthogonal matrices given by
\[
A(v)=A(\xi)\cdot\tilde A(v),
\]
where
\[
A(\xi)=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \sin\xi & \cos\xi \\
0 & 0 & -\cos\xi & \sin\xi
\end{array}
\right)
\]
and
\[
\tilde A(v)=\left(
\begin{array}{rrrr}
\cos\xi_1\cos\xi_2 & -\cos\xi_1\sin\xi_2 & \sin\xi_1\cos\xi_2 & -\sin\xi_1\sin\xi_3 \\
\cos\xi_1\sin\xi_2 & -\cos\xi_1\cos\xi_2 & \sin\xi_1\sin\xi_3 & \sin\xi_1\cos\xi_3 \\
-\sin\xi_1\cos\xi_3 & \sin\xi_1\sin\xi_3 & \cos\xi_1\cos\xi_2 & \cos\xi_1\sin\xi_2 \\
\sin\xi_1\sin\xi_3 & -\sin\xi_1\cos\xi_3 & -\cos\xi_1\sin\xi_2 & \cos\xi_1\cos\xi_2
\end{array}
\right),
\]
$\xi$ is a constant and the functions $\xi_i(v)$, $1\leq i\leq 3$, satisfy
\begin{eqnarray}\label{eq:xis}
\cos^2(\xi_1(v))\xi_2'(v)-\sin^2(\xi_1(v))\xi_3'(v)=0.
\end{eqnarray}

In the next result we obtain another relation between the function $\xi_i$,
given in \eqref{eq:xi_i}, and the angle function $\nu$.

\begin{prop}
{\em The functions $\xi_i(v)$, given in \eqref{eq:xi_i}, satisfy the following relation:
\begin{eqnarray}\label{eq:relxi_inu}
(\xi_1'(v))^2+(\xi_2'(v))^2\cos^2(\xi_1(v))+(\xi_3'(v))\sin^2(\xi_1(v))=
\sin^2\nu,
\end{eqnarray}
where $\nu$ is the angle function of the surface $M$.}
\end{prop}
\begin{proof}
With respect to the parametrization $F(u,v)$, given in \eqref{eq:paramMO},
we have
\[
F_v=A'(v)\cdot b(u)=A(\xi)\cdot\tilde A'(v)\cdot b(u).
\]
We have $\langle F_v,F_v\rangle=\sin^2\nu$ (cf. \cite{MO}). On the other hand,
if we denote by $c_1$, $c_2$, $c_3$, $c_4$ the columns of $\tilde A$,
we have
\[
\langle F_v,F_v\rangle\vert_{u=0}=g_{11}\langle c_1',c_1'\rangle+g_{33}\langle c_3',c_3'\rangle.
\]
As $\langle c_1',c_1'\rangle=\langle c_3',c_3'\rangle$, $\langle c_1',c_3'\rangle=0$ and
$g_{11}+g_{33}=1$, a straightforward computation gives
\begin{eqnarray*}
\sin^2\nu &=& \langle F_v,F_v\rangle = (g_{11}+g_{33})\langle c_1',c_1'\rangle \\
&=& (\xi_1'(v))^2+(\xi_2'(v))^2\cos^2(\xi_1(v))+(\xi_3'(v))\sin^2(\xi_1(v)),
\end{eqnarray*}
and we conclude the proof.
\end{proof}

\begin{thm}
Let $M$ be a helicoidal flat surface in $\mathbb{S}^3$, locally parametrized by
\eqref{eq:param-helicoidal}, and whose profile curve $\gamma$ is given
by \eqref{eq:param-gamma}. Then $M$ admits a new local
parametrization such that the fundamental forms are given as in
\eqref{eq:forms} and $\omega$ is a linear function.
\end{thm}
\begin{proof}
Consider the unit normal vector field $N$ associated to the local
parametrization $X$ of $M$ given in \eqref{eq:param-helicoidal}.
From Proposition \ref{prop:CAS}, the angle between $N$ and the
Hopf vector field $E_1$ is constant. Hence, it follows from
\cite{MO} (Theorem 3.1) that $M$ can be locally parametrized as
in \eqref{eq:paramMO}.
By taking $\epsilon=1$ in \eqref{eq:berger}, we can reparametrize the curve
$b$ given in \eqref{eq:curveb} in such a way that the new curve is a base
curve $\gamma_a$.
In fact, by taking $\epsilon=1$, we obtain $B=1$, and so
$\alpha_1=2c_2$ and $\alpha_2=2c_1$. This implies that
$\|b'(u)\|=2\sqrt{c_1c_2}$, because $c_1+c_2=1$. Thus, by writing
$s=2\sqrt{c_1c_2}$, the new parametrization of $b$ is given by
\[
b(s)=\frac{1}{\sqrt{1+a^2}} \left(a\cos\frac{s}{a},a\sin\frac{s}{a},
\cos(as),\sin(as)\right),
\]
where $a=\sqrt{c_1/c_2}$. On the other hand, we have
\[
A(v)\cdot b(s)=A(\xi)X(v,s),
\]
where $X(v,s)$ can be written as
\[
X(v,s)=\frac{1}{\sqrt{1+a^2}}(x_1,x_2,x_3,x_4),
\]
with
\begin{eqnarray}\label{eq:coef_xi}
\begin{array}{rcl}
x_1 &=& a\cos\xi_1\cos\left(\dfrac{s}{a}+\xi_2\right)+\sin\xi_1\cos(as+\xi_3), \\
x_2 &=& a\cos\xi_1\sin\left(\dfrac{s}{a}+\xi_2\right)+\sin\xi_1\sin(as+\xi_3), \\
x_3 &=& -a\sin\xi_1\cos\left(\dfrac{s}{a}-\xi_3\right)+\cos\xi_1\cos(as-\xi_2), \\
x_4 &=& -a\sin\xi_1\sin\left(\dfrac{s}{a}-\xi_3\right)+\cos\xi_1\sin(as-\xi_2). \\
\end{array}
\end{eqnarray}
On the other hand, the product $\phi_{\alpha,\beta}(t)\cdot X(v,s)$ can be written as
\[
\phi_{\alpha,\beta}(t)\cdot X(v,s)=\frac{1}{\sqrt{1+a^2}}(z_1,z_2,z_3,z_4),
\]
where
\begin{eqnarray}\label{eq:coef_zi}
\begin{array}{rcl}
z_1 &=& a\cos\xi_1\cos\left(\dfrac{s}{a}+\xi_2+\alpha t\right)
+\sin\xi_1\cos\left(as+\xi_3+\alpha t\right), \\
z_2 &=& a\cos\xi_1\sin\left(\dfrac{s}{a}+\xi_2+\alpha t\right)
+\sin\xi_1\sin\left(as+\xi_3+\alpha t\right), \\
z_3 &=& -a\sin\xi_1\cos\left(\dfrac{s}{a}-\xi_3+\beta t\right)
+\cos\xi_1\cos\left(as-\xi_2+\beta t\right), \\
z_4 &=& -a\sin\xi_1\sin\left(\dfrac{s}{a}-\xi_3+\beta t\right)
+\cos\xi_1\sin\left(as-\xi_2+\beta t\right).
\end{array}
\end{eqnarray}
As the surface is helicoidal, we have
\[
\phi_{\alpha,\beta}(t)\cdot X(v,s)=X(v(t),s(t)),
\]
for some smooth functions $v(t)$ and $s(t)$, which satisfy the
following equations:
\begin{eqnarray}
\xi_2(v(t)) + \dfrac{s(t)}{a} = \xi_2(v)+ \dfrac{s}{a}+\alpha t, \label{eq:xi2-alpha} \\
\xi_3(v(t)) + a s(t) = \xi_3 (v) + a s + \alpha t, \label{eq:xi3-alpha} \\
\dfrac{s(t)}{a} - \xi_3(v(t)) = \dfrac{s}{a} - \xi_3(v) + \beta t, \label{eq:xi3-beta} \\
a s(t) - \xi_2(v(t)) = as - \xi_2(v) + \beta t. \label{eq:xi2-beta}
\end{eqnarray}
It follows directly from \eqref{eq:xi2-alpha} and \eqref{eq:xi2-beta} that
\begin{equation}
s(t) = s + \dfrac{a(\alpha+\beta)}{a^2+1} t. \label{eq:s(t)}
\end{equation}
Note that the same conclusion is obtained by using \eqref{eq:xi3-alpha}
and \eqref{eq:xi3-beta}. By substituting the expression of $s(t)$ given in
\eqref{eq:s(t)} on the equations \eqref{eq:xi2-alpha} -- \eqref{eq:xi2-beta},
one has
\begin{eqnarray}
\xi_2(v(t))= \xi_2(v) + \left( \dfrac{a^2 \alpha - \beta}{a^2+1} \right) t, \label{eq:xi2(v(t))}  \\
\xi_3(v(t))= \xi_3(v) + \left( \dfrac{\alpha - a^2 \beta}{a^2+1} \right) t.
\label{eq:xi3(v(t))}
\end{eqnarray}
From now on we assume that $v'(t)\neq0$ since, otherwise, we would
have
\[
\frac{s(t)}{a}=\frac{s}{a}+\alpha t=\frac{s}{a}+\beta t
\quad\text{and}\quad
as(t)=as+\alpha t=as+\beta t.
\]
But the equalities above imply that $a^2=1$, which contradicts the
definition of base curve in \eqref{eq:base}. Thus, it follows from
\eqref{eq:xi2(v(t))} and \eqref{eq:xi3(v(t))} that
\begin{eqnarray}\label{eq:xi_2exi_3}
\xi_2'=\frac{a^2 \alpha-\beta}{a^2+1}\cdot\frac{1}{v'} \quad\text{and}\quad
\xi_3'=\frac{\alpha-a^2 \beta}{a^2+1}\cdot\frac{1}{v'}.
\end{eqnarray}
Therefore, from \eqref{eq:xis} and \eqref{eq:xi_2exi_3} we obtain
\begin{eqnarray}\label{eq:relxi_1}
\cos^2(\xi_1(v))(a^2\alpha-\beta)=\sin^2(\xi_1(v))(\alpha-a^2\beta).
\end{eqnarray}
As $a>1$, one has $a^2\alpha-\beta\neq0$ or $\alpha-a^2\beta\neq0$, and
we conclude from \eqref{eq:relxi_1} that $\xi_1(v)$ is constant. In this
case, there is a constant $b>1$ such that $\cos^2\xi_1=\dfrac{b^2}{1+b^2}$
and $\sin^2 \xi_1 = \dfrac{1}{1+b^2}$. Therefore, it follows from \eqref{eq:xis}
that
\begin{eqnarray}\label{eq:xi2-xi3}
\xi_2(v)= \dfrac{1}{b^2} \xi_3(v)+d,
\end{eqnarray}
for some constant $d$. On the other hand, if $\cos\xi_1\neq0$, it follows
from \eqref{eq:xis} that
\begin{eqnarray}\label{eq:xi_2xi_3}
(\xi_2'(v))^2=\tan^4\xi_1\cdot(\xi_3'(v))^2.
\end{eqnarray}
By substituting \eqref{eq:xi_2xi_3} in \eqref{eq:relxi_inu} we obtain
\[
\tan^2\xi_1\cdot(\xi_3'(v))^2=\sin^2\nu,
\]
ant this implies that we can choose $\xi_3(v)=bv$, and from \eqref{eq:xi3(v(t))}
we obtain
\begin{eqnarray}\label{eq:v(t)}
v(t)=v+\frac{\alpha-a^2\beta}{b(a^2+1)}t.
\end{eqnarray}
Moreover, from \eqref{eq:xi2-xi3}, the equation
$\xi_2(v(t)) = \dfrac{1}{b^2} \xi_3(v(t))+d$ implies that
\begin{eqnarray}\label{eq:b-alpha-beta}
\dfrac{1}{b^2} = \dfrac{a^2 \alpha - \beta}{\alpha-a^2 \beta},
\end{eqnarray}
and from \eqref{eq:b-alpha-beta} we obtain the same
relation \eqref{eq:alpha-beta} between $\alpha$ and $\beta$. This
relation, when substituted  in \eqref{eq:s(t)} and \eqref{eq:v(t)},
gives
\[
s(t)=s+\frac{a(b^2-1)}{a^2b^2-1}\beta t \quad\text{and}\quad
v(t)=v+\frac{b(1-a^2)}{a^2b^2-1}\beta t,
\]
that coincide with the expressions in \eqref{eq:z(t)w(t)}. Finally, from
the relation \eqref{eq:xi2-xi3} we obtain $\xi_2(v)=\frac{v}{b}-\frac{\pi}{2}$.
By taking $\xi=\frac{\pi}{2}$ and $\xi_1(v)=\arcsin\left(\frac{1}{\sqrt{1+b^2}}\right)$,
the new parametrization $F(u,v)$ thus obtained coincides with $Y(u,v)$
given in \eqref{eq:expY}, up to isometries of $\mathbb{S}^3$ and linear
reparametrization. The conclusion follows from Theorem \ref{teo:main1}.
\end{proof}

\section{Conformally flat hypersurfaces}

In this section, it will presented an application of the classification result
for helicoidal flat surfaces in $\mathbb{S}^3$ in a geometric description
for conformally flat hypersurfaces in four-dimensional space forms.

The problem of classifying conformally flat hypersurfaces in space forms
has been investigated for a long time, with special attention on $4$-dimensional
space forms. In fact, any surface in $\mathbb{R}^3$ is conformally flat, since it can be parametrized by isothermal coordinates. On the other hand, Cartan \cite{Cartan}
gave a complete classification of conformally flat hypersurfaces into a
$(n+1)$-dimensional space form, with $n+1\geq5$. Such hypersurfaces
are quasi-umbilic, i.e., one of the principal curvatures has multiplicity at least
$n-1$. In the same paper, Cartan showed that the quasi-umbilic surfaces are conformally flat, but the converse does not hold. Since then, there has been
an effort to obtain a classification of hypersurfaces with three distinct principal curvatures.

Lafontaine \cite{Lafontaine} considered hypersurfaces of type
$M^3 = M^2 \times I\subset\mathbb{R}^4$ and obtained the following classes of
conformally flat hypersurfaces: (a) $M^3$ is a cylinder over a surface, where
$M^2 \subset \mathbb{R}^3$ has constant curvature; (b) $M^3$ is a cone over a
surface in the sphere, where $M^2 \subset\mathbb{S}^3$ has with constant curvature;
(c) $M^3$ is obtained by rotating a constant curvature surface of the hyperbolic space and $M^2 \subset \mathbb{H}^3 \subset \mathbb{R}^4$, where $\mathbb{H}^3$
is the half space model (see \cite{Suyama2} for more details).

Hertrich-Jeromin \cite{Jeromin1} established a correspondence between
conformally flat hypersufaces in space forms, with three distinct principal
curvatures, and solutions $(l_1, l_2, l_3) : U \subset \mathbb{R}^3 \rightarrow \mathbb{R}$
for the Lam\'e's system \cite{reflame}
\begin{equation}
\begin{array}{rcl}
	l_{i,x_jx_k} - \dfrac{l_{i,x_j} l_{j,x_k}}{l_j} - \dfrac{l_{i,x_k} l_{k,x_j}}{l_k} &=& 0,  \\
	\left( \dfrac{l_{i,x_j}}{l_j} \right)_{,x_j} + \left( \dfrac{l_{j,x_i}}{l_i} \right)_{,x_i} + \dfrac{l_{i,x_k} l_{j,x_k}}{l_k^2} &=& 0,
\end{array} \label{lame}
 \end{equation}
where $i, j, k$ are distinct indices that satisfies the condition
\begin{equation} \label{guich}
l_1^2 - l_2^2 + l_3^2 = 0
\end{equation}
known as Guichard condition. In this case, the correspondent coformally
flat hypersurface in $M^4_K$ is parametrized by curvature lines, with
induced metric given by
\[
g = e^{2u} \left\{ l_1^2 (dx_1)^2 + l_1^2 (dx_1)^2 + l_3^2 (dx_3)^2  \right\}.
\]

In \cite{santos} the second author and Tenenblat obtained solutions of Lam\'e's system (\ref{lame}) that are invariant under symmetry groups. Among the solutions, there are those that are invariant under the action of the 2-dimensional subgroup of translations and dilations and depends only on two variables:
\begin{enumerate}
\item[(a)] $l_1 = \lambda_1, \; l_2 = \lambda_1 \cosh (b\xi + \xi_0), \; l_3 = \lambda_1 \sinh (b\xi + \xi_0)$, where $\xi = \alpha_2 x_2 + \alpha_3 x_3$, $\alpha_2^2 + \alpha_3^2 \neq 0$ and $b,\, \xi_0\in \mathbb{R}$ ;
\item[(b)] $ l_2 = \lambda_2, \; l_1 = \lambda_2 \cos \varphi(\xi) , \; l_3 = \lambda_2 \sin \varphi(\xi) $, where $\xi = \alpha_1 x_1 + \alpha_3 x_3$, $\alpha_1^2 + \alpha_3^2 \neq 0$ and $\varphi$ is one  of the following functions:
   \begin{enumerate}
      \item[(b.1)] $\varphi(\xi) = b \xi + \xi_0$, if $\alpha_1^2 \neq \alpha_3^2$, where $\xi_0, \,b\in \mathbb{R}$;
      \item[(b.2)] $\varphi$ is any function of $\xi$, if $\alpha_1^2 = \alpha_3^2$;
   \end{enumerate}
\item[(c)]  $l_3 = \lambda_3, \; l_2 = \lambda_3 \cosh (b\xi + \xi_0), \; l_1 = \lambda_3 \sinh (b\xi + \xi_0)$, where $\xi = \alpha_1 x_1 + \alpha_2 x_2$, $\alpha_1^2 + \alpha_2^2 \neq 0$ and   $b,\,\xi_0\in\mathbb{R} $.
\end{enumerate}
 It is known (see \cite{Suyama2}) that the solutions that do not depend on
one of the variables are associated to the products given by Lafontaine.
For the solutions given in (b), further geometric solutions can be obtained
with the classification result for helicoidal falt surfaces in $\mathbb{S}^3$.
These solutions are associated to  conformally flat hypersurfaces that are
conformal to the products $M^2 \times I \subset \mathbb{R}^4$ given by
\[
M^2 \times I = \left\{ tp:0<t<\infty, p\in M^2 \subset \mathbb{S}^3 \right\},
\]
where $M^2$ is a flat surface in $S^3$, parametrized by lines of curvature,
whose first and second fundamental forms are given by
\begin{equation}
\begin{array}{rcl} \label{firstffsphere}
I &=& \sin^2 (\xi + \xi_0) dx_1^2 + \cos^2 (\xi + \xi_0)  dx_3^2,  \\
II &=& \sin (\xi + \xi_0) \cos (\xi + \xi_0) (dx_1^2 - dx_3^2),
\end{array}
\end{equation}
which are, up to a linear change of variables, the fundamental forms that
are considered in this paper. Therefore, as an application of the
characterization of helicoidal flat surfaces in terms of first and second
fundamental forms, one has the following theorem:

\begin{thm} \label{characterization.helicoidal}
Let $ l_2 = \lambda_2, \; l_1 = \lambda_2 \cos \xi + \xi_0 , \;
l_3 = \lambda_2 \sin \xi + \xi_0 $ be solutions of the Lam\'e's sytem, where
$\xi = \alpha_1 x_1 + \alpha_3 x_3$ and $\alpha_1, \, \alpha_3, \, \lambda_2, \, \xi_0$ are real constants with $\alpha_1\cdot\alpha_3 \neq 0$. Then the associated conformally flat hypersurfaces are conformal to the product, $M^2 \times I$,
where $M^2 \subset \mathbb{S}^3$ is locally congruent to helicoidal flat surface.
\end{thm}

\bibliographystyle{acm}


\section{Introduction}

Let $M=\mathbb{H}^3/ \Gamma$, $\Gamma \subset \mathrm{PSL}_2(\mathbb{C})$ be an orientable hyperbolic $3$--manifold, and let $f\co F \rightarrow M$ be a proper immersion of a connected, orientable surface of genus at least $2$ such that $f_{*}\co\pi_1(F) \rightarrow \Gamma$ is injective.
$F$ (or more precisely $(f,F)$) is said to be {\em  totally geodesic\/} if $f_{*}(\pi_{1}(F)) \subset \Gamma$ is conjugate into $\mathrm{PSL}_2(\mathbb{R})$.  Thurston and Bonahon have described the geometry of surface groups in hyperbolic $3$--manifolds as falling into three classes: doubly degenerate groups, quasi-Fuchsian groups and groups with accidental parabolics.  The class of totally geodesic surface groups is a ``positive codimension'' subclass of the quasi-Fuchsian groups, so one may expect that hyperbolic $3$--manifolds containing totally geodesic surface groups are special.

Indeed, the presence of a totally geodesic surface in a hyperbolic $3$--manifold has important topological implications.  Long showed that immersed totally geodesic surfaces lift to embedded nonseparating surfaces in finite covers \cite{Long}, proving the virtual Haken and virtually positive $\beta_1$ conjectures for hyperbolic manifolds containing totally geodesic surfaces.  Given this, it is natural to wonder about the extent to which topology constrains the existence of totally geodesic surfaces in hyperbolic $3$--manifolds.  Menasco--Reid have made the following conjecture \cite{MeR}:
\begin{conj}[Menasco--Reid] No hyperbolic knot complement in $\mathrm{S}^3$ contains a closed embedded totally geodesic surface.
\end{conj}
They proved this conjecture for alternating knots.  The Menasco--Reid conjecture has been shown true for many other classes of knots, including almost alternating knots \cite{Aetal}, Montesinos knots \cite{Oer}, toroidally alternating knots \cite{Ad}, $3$--bridge and double torus knots \cite{IO} and knots of braid index 3 \cite{LP} and 4 \cite{Ma}.  For a knot in one of the above families, any closed essential surface in its complement has a topological feature which obstructs it from being even quasi-Fuchsian.  In general, however, one cannot hope to find such obstructions.  Adams--Reid have given examples of closed embedded quasi-Fuchsian surfaces in knot complements which volume calculations prove to be not totally geodesic \cite{AR}.

On the other hand, C\,Leininger has given evidence for a counterexample by constructing a sequence of hyperbolic knot complements in $\mathrm{S}^3$ containing closed embedded surfaces whose principal curvatures approach 0 \cite{Le}.  In this paper, we take an alternate approach to giving evidence for a counterexample.
\begin{thm} \label{cusped}
 There exist infinitely many hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
\end{thm}
This answers a question of Reid---recorded as Question 6.2 in \cite{Le}---giving counterexamples to the natural generalization of the Menasco--Reid conjecture to knot complements in rational homology spheres.  Thus the conjecture, if true, must reflect a deeper topological feature of knot complements in $\mathrm{S}^3$ than simply their rational homology.

Prior to proving \fullref{cusped}, in \fullref{sec2} we prove the following theorem.
\begin{thm} \label{closed}
There exist infinitely many hyperbolic rational homology spheres containing closed embedded totally geodesic surfaces.
\end{thm}
This seems of interest in its own right, and the proof introduces many of the techniques we use in the proof of \fullref{cusped}.  Briefly, we find a two-cusped hyperbolic manifold containing an embedded totally geodesic surface which remains totally geodesic under certain orbifold surgeries on its boundary slopes and use the Alexander polynomial to show that branched covers of these surgeries have no rational homology.  In \fullref{sec3} we prove \fullref{cusped}.  In the final section, we give some idea of further directions and questions suggested by our approach.

\subsubsection*{Acknowledgements}

The author thanks Cameron Gordon, Richard Kent, Chris Lein\-inger, Jessica Purcell and Alan Reid for helpful conversations.  The author also thanks the Centre Interfacultaire Bernoulli at EPF Lausanne for their hospitality during part of this work.

\section[Theorem 2]{\fullref{closed}}\label{sec2}

Given a compact hyperbolic manifold $M$ with totally geodesic boundary of genus $g$, gluing it to its mirror image $\wwbar{M}$ along the boundary yields a closed manifold $DM$---the ``double'' of $M$---in which the former $\partial M$ becomes an embedded totally geodesic surface.  One limitation of this construction is that this surface contributes half of its first homology to the first homology of $DM$, so that $\beta_1(DM) \geq g$.  This is well known, but we include an argument to motivate our approach. Consider the relevant portion of the rational homology Mayer--Vietoris sequence for $DM$:
\[ \cdots \rightarrow \mathrm{H}_1(\partial M,\mathbb{Q}) \stackrel{(i_*,-j_*)}{\longrightarrow} \mathrm{H}_1(M,\mathbb{Q}) \oplus \mathrm{H}_1(\wwbar{M},\mathbb{Q}) \rightarrow \mathrm{H}_1(DM,\mathbb{Q}) \rightarrow 0 \]
The labeled maps $i_*$ and $j_*$ are the maps induced by inclusion of the surface into $M$ and $\wwbar{M}$, respectively.  Recall that by the ``half lives, half dies'' lemma (see eg Hatcher \cite[Lemma 3.5]{Ha}), the dimension of the kernel of $i_*$ is equal to $g$.  Hence $\beta_1(M) \geq g$.  The gluing isometry $\partial M \rightarrow \partial \wwbar{M}$ (the identity) extends over $M$, thus $\mathrm{Ker}\ i_*=\mathrm{Ker}\ j_*$, and so $\mathrm{dim}\ \mathrm{Im}(i_*,-j_*)=g$.  Hence
\[ \mathrm{H}_1(DM, \mathbb{Q}) \cong \frac{\mathrm{H}_1(M, \mathbb{Q}) \oplus \mathrm{H}_1(\wwbar{M}, \mathbb{Q})}{\mathrm{Im}((i_*,-j_*))} \]
has dimension at least $g$.

Considering the above picture gives hope that by cutting $DM$ along $\partial M$ and regluing via some isometry $\phi\co \partial M \rightarrow \partial M$ to produce a ``twisted double'' $D_{\phi}M$, one may reduce the homological contribution of $\partial M$.  For then $j=i \circ \phi$, and if $\phi_*$ moves the kernel of the inclusion off of itself, then the argument above shows that the homology of $D_{\phi}M$ will be reduced.  Below we apply this idea to a family of examples constructed by Zimmerman and Paoluzzi \cite{ZP} which build on the ``Tripos'' example of Thurston \cite{Th}.

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\caption{The tangle $T$ and its double and twisted double.}
\label{links}
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The complement in the ball of the tangle $T$ in \fullref{links} is one of the minimal volume hyperbolic manifolds with totally geodesic boundary, obtained as an identification space of a regular ideal octahedron \cite{Mi}.  We will denote it by $O_{\infty}$.  For $n \geq 3$, the orbifold $O_n$ with totally geodesic boundary consisting of the ball with cone locus $T$ of cone angle $2\pi/n$ has been explicitly described by Zimmerman and Paoluzzi \cite{ZP} as an identification space of a truncated tetrahedron.  For each $k<n$ with $(k,n)=1$, Zimmerman and Paoluzzi describe a hyperbolic manifold $M_{n,k}$ which is an $n$--fold branched cover of $O_n$.  Topologically, $M_{n,k}$ is the $n$--fold branched cover of the ball, branched over $T$, obtained as the kernel of $\langle x,y \rangle = \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z} = \langle t \rangle$ via $x \mapsto t$, $y \mapsto t^k$, where $x$ and $y$ are homology classes representing meridians of the two components of $T$.

We recall a well-known fact about isometries of spheres with four cone points:

\begin{fact} Let $S$ be a hyperbolic sphere with four cone points of equal cone angle $\alpha$, $0 \leq \alpha \leq 2\pi/3$, labeled $a$, $b$, $c$, $d$.  Each of the following permutations of the cone points may be realized by an orientation-preserving isometry:
\begin{align*}
 &(ab)(cd)& &(ac)(bd)& &(ad)(bc)&
\end{align*}
\end{fact}

Using this fact and abusing notation, let $\phi$ be the isometry $(ab)(cd)$ of $\partial O_n$, with labels as in \fullref{links}.  Doubling the tangle ball produces the link $L_0$ in \fullref{links}, and cutting along the separating $4$--punctured sphere and regluing via $\phi$ produces the link $L$, a \textit{mutant\/} of $L_0$ in the classical terminology.  Note that $L$ and all of the orbifolds $D_{\phi} O_n$ contain the mutation sphere as a totally geodesic surface, by the fact above.  $\phi$ lifts to an isometry ${}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu$ of $\partial M_{n,k}$, and the twisted double $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,k}$ is the corresponding branched cover over $L$.

The homology of $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,k}$ can be described using the Alexander polynomial of $L$.  The two variable Alexander polynomial of $L$ is
\[ \Delta_L(x,y) = \frac{1}{x^3}(x-1)(xy-1)(y-1)^2(x-y). \]
For the regular $\mathbb{Z}$--covering of $\mathrm{S}^3-L$ given by $x \mapsto t^k$, $y \mapsto t$, the Alexander polynomial is
\[ \Delta_L^k(t)= (t-1)\Delta(t^k,t)= \frac{1}{t^{3k-1}}(t-1)^5\nu_{k-1}(t) \nu_k(t) \nu_{k+1}(t) \]
where $\nu_k(t)=t^{k-1} + t^{k-2} + \cdots + t + 1$.  By a theorem originally due to Sumners \cite{Su} in the case of links, the first Betti number of $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,k}$ is the number of roots shared by $\Delta_L^k(t)$ and $\nu_n(t)$.  Since this number is $0$ for many $n$ and $k$, we have a more precise version of \fullref{closed}.

\begin{thm1} For $n>3$ prime and $k \neq 0,1,n-1$, the manifold $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,k}$ is a hyperbolic rational homology sphere containing an embedded totally geodesic surface.
\end{thm1}

The techniques used above are obviously more generally applicable.  Given any hyperbolic two-string tangle in a ball with totally geodesic boundary, one may double it to get a $2$--component hyperbolic link in $\mathbb{S}^3$ and then mutate along the separating $4$--punctured sphere by an isometry.  By the hyperbolic Dehn surgery theorem and the fact above, for large enough $n$, $(n,0)$ orbifold surgery on each component will yield a hyperbolic orbifold with a separating totally geodesic orbisurface.  Then $n$--fold manifold branched covers can be constructed as above.  One general observation about such covers follows from the following well-known fact, originally due to Conway:

\begin{fact} The one variable Alexander polynomial of a link is not altered by mutation; ie,
\[ \Delta_{L_0}(t,t) = \Delta_L(t,t) \]
when L is obtained from $L_0$ by mutation along a $4$--punctured sphere.
\end{fact}

In our situation, this implies the following:

\begin{cor1} A $2$--component link in $\mathrm{S}^3$ which is the twisted double of a tangle has no integral homology spheres among its abelian branched covers.
\end{cor1}

\begin{proof} A link $L_0$ which is the double of a tangle has Alexander polynomial $0$.  Therefore by the fact above,
\[ \Delta_L^1(t)=(t-1)\Delta_L(t,t)=(t-1)\Delta_{L_0}(t,t)=0, \]
and so $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,1}$ has positive first Betti number by Sumners' theorem.  The canonical abelian $n^2$--fold branched cover of $L$ covers $D_{{}\mskip2.5mu\tilde{\mskip-2.5mu \vphantom{t}\smash{\phi}\mskip-1mu}\mskip1mu}M_{n,1}$ and so also has positive first Betti number.  Since the other $n$--fold branched covers of $L$ have $n$--torsion, no branched covers of $L$ have trivial first homology.
\end{proof}

\section[Theorem 1]{\fullref{cusped}}\label{sec3}

In this section we construct hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.  The following ``commutative diagram'' introduces the objects involved in the construction and the relationships between them.
\[ \disablesubscriptcorrection\xysavmatrix{ N_n \ar[rr]^{\mbox{\tiny{Dehn}}}_{\mbox{\tiny{filling}}} \ar[dd] && M_n \ar[rr]^{\mbox{\tiny{Dehn}}}_{\mbox{\tiny{filling}}} \ar[dd] && S_n \\ && && \\ N \ar[rr]^{\mbox{\tiny{orbifold}}}_{\mbox{\tiny{filling}}} && O_n && \\} \]
\fullref{cusped} may now be more precisely stated as follows.

\begin{thm1}  For each $n \geq 3$ odd, $O_n$ is a one-cusped hyperbolic orbifold containing a totally geodesic sphere with four cone points of order $n$, $M_n$ is a branched covering of $O_n$ which is a one-cusped hyperbolic manifold, and $S_n$ is a rational homology sphere.  \end{thm1}

Before beginning the proof, we give a brief sketch of the strategy.  We give an explicit polyhedral construction of a three-cusped hyperbolic manifold $N$ containing an embedded totally geodesic $4$--punctured sphere which intersects two of the cusps.  For $n \geq 3$, we give the polyhedral decomposition of the orbifold $O_n$ resulting from $n$--fold orbifold surgery on the boundary slopes of this $4$--punctured sphere.  From this it is evident that $O_n$ is hyperbolic and the sphere remains totally geodesic.  For odd $n \geq 3$, we prove that $O_n$ has a certain one-cusped $n$--fold manifold cover $M_n$ with a surgery $S_n$ which is a rational homology sphere.  This is accomplished by adapting an argument of Sakuma \cite{Sa} to relate the homology of the $n$--fold cover $N_n \rightarrow N$ corresponding to $M_n \rightarrow O_n$, to the homology of $S_n$.  $M_n$ is thus a hyperbolic knot complement in a rational homology sphere, containing the closed embedded totally geodesic surface which is a branched covering of the totally geodesic sphere with four cone points in $O_n$.

\begin{remark}  It follows from the construction that the ambient rational homology sphere $S_n$ covers an orbifold produced by $n$--fold orbifold surgery on each cusp of $N$.  Thus by the hyperbolic Dehn surgery theorem, $S_n$ is hyperbolic for $n >> 0$. \end{remark}

The proof occupies the remainder of the section.  We first discuss the orbifolds $O_n$.  For each $n$, the orbifold $O_n$ decomposes into the two polyhedra in \fullref{orbipoly}.  Realized as a hyperbolic polyhedron, $P_a^{(n)}$ is composed of two truncated tetrahedra, each of which has two opposite edges of dihedral angle $\pi/2$ and all other dihedral angles $\pi/2n$, glued along a face.  This decomposition is indicated in \fullref{orbipoly} by the lighter dashed and dotted lines.  The polyhedron $P_b^{(n)}$ has all edges with dihedral angle $\pi/2$ except for those labeled otherwise and realized as a hyperbolic polyhedron it has all combinatorial symmetries and all circled vertices at infinity.  By Andreev's theorem, polyhedra with the desired properties exist in hyperbolic space.  Certain face pairings (described below) of $P_a^{(n)}$ yield a compact hyperbolic orbifold with totally geodesic boundary a sphere with four cone points of cone angle $2\pi/n$.  Faces of $P_b^{(n)}$ may be glued to give a one-cusped hyperbolic orbifold with a torus cusp and totally geodesic boundary isometric to the boundary of the gluing of $P_a^{(n)}$.  $O_n$ is formed by gluing these orbifolds along their boundaries.

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\caption{Cells for $O_n$}
\label{orbipoly}
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The geometric limit of the $O_n$ as $n \rightarrow \infty$ is $N$, a $3$--cusped manifold which decomposes into the two polyhedra in \fullref{idealpoly}.  As above, realized as a convex polyhedron in hyperbolic space $Q_a$ has all circled vertices at infinity.  The edge of $Q_a$ connecting face $A$ to face $C$ is finite length, as is the corresponding edge on the opposite vertex of $A$; all others are ideal or half-ideal and all have dihedral angle $\pi/2$.  $Q_a$ has a reflective involution of order $2$ corresponding to the involution of $P_a^{(n)}$ interchanging the two truncated tetrahedra.  The fixed set of this involution on the back face is shown as a dotted line, and notationally we regard $Q_a$ as having an edge there with dihedral angle $\pi$, splitting the back face into two faces $X_5$ and $X_6$.  $Q_b$ is the regular all-right hyperbolic ideal cuboctahedron.

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Another remark on notation: the face opposite a face labeled with only a letter should be interpreted as being labeled with that letter ``prime''.  For instance, the leftmost triangular face of $Q_a$ has label $C'$.  Also, each ``back'' triangular face of $Q_b$ takes the label of the face with which it shares a vertex.  For example, the lower left back triangular face is $Y_1'$.

We first consider face pairings of $Q_a$ producing a manifold $N_a$ with two annulus cusps and totally geodesic boundary.  Let $r$, $s$ and $t$ be isometries realizing face pairings $X_1 \mapsto X_3$, $X_6 \mapsto X_4$ and $X_2 \mapsto X_5$, respectively.  Poincar\'{e}'s polyhedron theorem yields a presentation
\[ \langle\ r,s,t\ \,|\,\ rst=1\ \rangle \]
for the group generated by $r$, $s$ and $t$.  Note that this group is free on two generators, say $s$ and $t$, where by the relation $r=t^{-1}s^{-1}$.  Choose as the ``boundary subgroup'' (among all possible conjugates) the subgroup fixing the hyperbolic plane through the face $A$.  A fundamental polyhedron for this group and its face-pairing isometries are in \fullref{faces}.  Note that the boundary is a $4$--punctured sphere, and two of the three generators listed are the parabolics $t^{-1}s^{-1}ts^{-1}$ and $sts^{-1}t$, which generate the two annulus cusp subgroups of $\langle\,s,t\,\rangle$.

We now consider $Q_b$ and the $3$--cusped quotient manifold $N_b$.  For $i \in \{1,2,3,4\}$, let $f_i$ be the isometry pairing the face $Y_i \rightarrow Y_{i+1}'$ so that $v_i \mapsto v_{i+1}$.  Let $g_1$ be the \textit{hyperbolic\/} isometry (that is, without twisting) sending $E \rightarrow E'$ and $g_2$ the hyperbolic isometry sending $F \rightarrow F'$.  The polyhedron theorem gives presentation
\begin{align*}
\langle f_1, f_2, f_3, f_4, g_1, g_2 \,|\, & f_1g_2f_2^{-1}g_1^{-1}=1, \\
   & f_2^{-1}g_2^{-1}f_3g_1^{-1}=1, \\
   & f_3g_2^{-1}f_4^{-1}g_1=1, \\
   & f_4^{-1}g_2f_1g_1=1 \rangle \end{align*}
for the group generated by the face pairings.  The first three generators and relations may be eliminated from this presentation using Nielsen--Schreier transformations, yielding a presentation
\[\langle\ f_4, g_1, g_2\ \,|\,\ f_4^{-1}[g_2,g_1]f_4[g_2,g_1^{-1}] = 1\ \rangle \]
(our commutator convention is $[x,y]=xyx^{-1}y^{-1}$), where the first three relations yield
\[ f_1=g_1g_2^{-1}g_1^{-1}f_4g_2g_1^{-1}g_2^{-1},\quad f_2=g_2^{-1}g_1^{-1}f_4g_2g_1^{-1}\quad \text{and}\quad f_3=g_1^{-1}f_4g_2. \]
The second presentation makes clear that the homology of $N_b$ is free of rank $3$, since each generator has exponent sum $0$ in the relation.  Faces $D$ and $D'$ make up the totally geodesic boundary of $N_b$.  \fullref{faces} shows a fundamental polyhedron for the boundary subgroup fixing $D$, together with the face pairings generating the boundary subgroup.

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\caption{Closed cusp of $N_b$}
\label{closedcusp}
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\end{figure}

$N_b$ has two annulus cusps, each with two boundary components on the totally geodesic boundary, and one torus cusp.  A fundamental domain for the torus cusp in the horosphere centered at $v_1$ is shown in \fullref{closedcusp}, together with face pairing isometries generating the rank--$2$ parabolic subgroup fixing $v_1$.  The generators shown are
\[ bob=(f_4g_1^{-1})^2f_4g_2g_1^{-1}g_2^{-1} \quad\text{and}\quad rita=(f_4g_1^{-1})^3f_4g_2g_1^{-1}g_2^{-1}. \]
Note that $(bob)^{-4}(rita)^3$ is trivial in homology.  This and $rita \cdot (bob)^{-1}=f_4 g_1^{-1}$ together generate the cusp subgroup fixing $v_1$.  For later convenience, we now switch to the conjugate of this subgroup by $f_4^{-1}$, fixing $v_4$ and refer to the conjugated elements $m=f_4^{-1}(f_4g_1^{-1})f_4=g_1^{-1}f_4$ and $l=f_4^{-1}((bob)^{-4}(rita)^3)f_4$ as a ``meridian-longitude'' generating set for the closed cusp of $N_b$.

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\label{faces}
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\end{figure}

The totally geodesic $4$--punctured spheres on the boundaries of $N_a$ and $N_b$ are each the double of a regular ideal rectangle, and we construct $N$ by gluing $N_a$ to $N_b$ along them.  Let us therefore assume that the polyhedra in \fullref{idealpoly} are realized in hyperbolic space in such a way that face $A$ of $Q_a$ and face $D$ of $Q_b$ are in the same hyperbolic plane, with $Q_a$ and $Q_b$ in opposite half-spaces.  Further arrange so that the polyhedra are aligned in the way suggested by folding the page containing \fullref{faces} along the dotted line down the center of the figure.  With this arrangement, Maskit's combination theorem gives a presentation for the amalgamated group: \begin{align*}
 \langle\ f_4, g_1, g_2, s, t  \,|\, &
 f_4^{-1}[g_2,g_1]f_4[g_2,g_1^{-1}] = 1, \\
 & t^{-2}s^{-2}=f_4^{-1}g_2^{-1}g_1^{-1}f_4g_2g_1^{-1}, \\
 & sts^{-1}t=g_2g_1g_2^{-1}f_4^{-1}g_1g_2g_1^{-1}f_4, \\
 & t^{-1}s^{-1}ts^{-1}=f_4^{-1}g_1^{-1}f_4g_2\ \rangle \end{align*}
The first relation comes from $N_b$ and the others come from setting the boundary face pairings equal to each other.  Observe that the last relation can be solved for $g_2$. Using Nielsen--Schreier transformations to eliminate $g_2$ and the last relation results in the presentation:
\begin{align*}
  \langle\ f_4, g_1, s, t\ \,|\,
 &f_4^{-2}g_1[f_4t^{-1}s^{-1}ts^{-1},g_1]f_4[f_4t^{-1}s^{-1}ts^{-1},g_1^{-1}]g_1^{-2}f_4g_1 = 1  \\
 &t^{-2}s^{-2}=f_4^{-1}st^{-1}stf_4^{-1}g_1^{-1}f_4^2t^{-1}s^{-1}ts^{-1}g_1^{-1},\\
 &sts^{-1}t=f_4^{-1}g_1f_4t^{-1}s^{-1}ts^{-1}g_1st^{-1}stf_4^{-2}g_1f_4t^{-1}s^{-1}ts^{-1}g_1^{-1}f_4\ \rangle\end{align*}
Replace $g_1$ with the meridian generator $m=\smash{g_1^{-1}}f_4$ of the closed cusp of $N_b$ and add generators $m_1=f_4^{-1}mf_4$ and $m_2=st^{-1}stm_1t^{-1}s^{-1}ts^{-1}$, each conjugate to $m$, yielding:
\begin{align*}
 \langle\ f_4,m,m_1,m_2,s,t\ \,|\, &
 m_1=f_4^{-1}mf_4,\ m_2=st^{-1}stm_1t^{-1}s^{-1}ts^{-1}  \\
 &m_1^{-1}t^{-1}s^{-1}ts^{-1}f_4m^{-1}m_2mf_4^{-1}st^{-1}stm_1m^{-1}=1  \\
 &s^2t^2f_4^{-1}m_2mf_4^{-1}=1  \\
 &t^{-1}st^{-1}s^{-1}m^{-1}f_4t^{-1}s^{-1}ts^{-1}f_4m^{-1}m_2^{-1}m=1\ \rangle
\end{align*}
Note that after abelianizing, each of the last two relations expresses $f_4^2=m^2s^2t^2$, since $m_1$ and $m_2$ are conjugate to $m$ and therefore identical in homology.  In light of this, we replace $f_4$ by $u=t^{-1}s^{-1}f_4m^{-1}$, which has order $2$ in homology.  This yields the presentation:\begin{align}
 \nonumber \langle\ m,m_1,m_2,s,t,u\ \,|\, \\
 &m_1^{-1}m^{-1}u^{-1}t^{-1}s^{-1}mstum=1 \\
 &m_2^{-1}st^{-1}stm_1t^{-1}s^{-1}ts^{-1}=1 \\
 &m_1^{-1}t^{-1}s^{-1}t^{2}um_2u^{-1}t^{-2}stm_1m^{-1}=1 \\
 &s^2t^2m^{-1}u^{-1}t^{-1}s^{-1}m_2u^{-1}t^{-1}s^{-1}=1 \\
 &t^{-1}st^{-1}s^{-1}m^{-1}stumt^{-1}s^{-1}t^2um_2^{-1}m=1\ \rangle
\end{align}
Let $R_i$ denote the relation labeled $(i)$ in the presentation above.  In the abelianization, $R_1$ sets $m_1=m$, $R_2$ sets $m_2=m_1$, $R_3$ disappears, and the last two relations set $u^2=1$.  Therefore
\[ \mathrm{H}_1(N) \cong \mathbb{Z}^3 \oplus \mathbb{Z}/2\mathbb{Z} = \langle m \rangle \oplus \langle s \rangle \oplus \langle t \rangle \oplus \langle u \rangle. \]
(In this paper we will generally blur the distinction between elements of $\pi_1$ and their homology classes.)

The boundary slopes of the totally geodesic $4$--punctured sphere coming from $\partial N_a$ and $\partial N_b$ are represented in $\pi_1(N)$ by $t^{-1}s^{-1}ts^{-1}$ and $sts^{-1}t$.  Let $O_n$ be the finite volume hyperbolic orbifold produced by performing face identifications on $P^{(n)}_a$ and $P^{(n)}_b$ corresponding to those on $Q_a$ and $Q_b$.  $O_n$ is geometrically produced by $n$--fold orbifold filling on each of the above boundary slopes of $N$.  Appealing to the polyhedral decomposition, we see that the separating $4$--punctured sphere remains totally geodesic, becoming a sphere with four cone points of order $n$.  Our knots in rational homology spheres are certain manifold covers of the $O_n$.  In order to understand the homology of these manifold covers, we compute the homology of the corresponding abelian covers of $N$.

Let $p\co\widetilde{N} \rightarrow N$ be the maximal free abelian cover; that is, $\widetilde{N}$ is the cover corresponding to the kernel of the map $\pi_1(N) \rightarrow \mathrm{H}_1(N) \rightarrow \mathbb{Z}^3 =\langle x,y,z \rangle$ given by:
\begin{align*}
& m \mapsto x & & s \mapsto y & & t \mapsto z & & u \mapsto 1 &
\end{align*}
Let $X$ be a standard presentation $2$--complex for $\pi_1(N)$ and $\widetilde{X}$ the $2$--complex covering $X$ corresponding to $\widetilde{N} \rightarrow N$.  Then the first homology and Alexander module of $\widetilde{X}$ are naturally isomorphic to those of $\widetilde{N}$, since $N$ is homotopy equivalent to a cell complex obtained from $X$ by adding cells of dimension three and above.  The covering group $\mathbb{Z}^3$ acts freely on the chain complex of $\widetilde{X}$, so that it is a free $\mathbb{Z}[x,x^{-1},y,y^{-1},z,z^{-1}]$--module.  Below we give a presentation matrix for the Alex\-ander module of $\widetilde{X}$:
\[ \left( \begin{array}{ccccc}
 \frac{1-yz+xyz}{x^2yz} & 0 & -1 & -\frac{y^2z^2}{x} & \frac{-1+yz+z^2}{xz^2} \\
 -\frac{1}{x} & \frac{y^2}{x} & \frac{x-1}{x} & 0 & 0 \\
 0 & -\frac{1}{x} & \frac{z}{xy} & \frac{yz}{x} & -\frac{1}{x} \\
 \frac{x-1}{x^2yz} & -\frac{(x-1)(y+z)}{xz} & \frac{x-1}{xyz} & \frac{y(x-z)}{x} & \frac{1-2x+xz}{xz^2} \\
 \frac{x-1}{x^2z} & -\frac{y(x-1)(y-1)}{xz} & \frac{(x-1)(-1+y-z)}{xyz} & \frac{y(-x+xy+xyz-yz)}{x} & \frac{x+y-2xy}{xz^2} \\
 \frac{x-1}{x^2} & 0 & -\frac{z(x-1)}{xy} & -\frac{yz(x+yz)}{x} & \frac{y+xz}{xz}
\end{array} \right) \]
The rows of the matrix above correspond to lifts of the generators for $\pi_1(N)$ sharing a basepoint, ordered as $\{\tilde{m},\wwtilde{m_1},\wwtilde{m_2},\tilde{s},\tilde{t},\tilde{u}\}$ reading from the top down.  These generate $\mathrm{C}_1(\widetilde{X})$ as a $\mathbb{Z}[x,x^{-1},y,y^{-1},z,z^{-1}]$--module.  The columns are the Fox free derivatives of the relations in terms of the generators, giving a basis for the image of $\partial \mathrm{C}_2(\widetilde{X})$.  For a generator $g$ above, let $p_g$ be the determinant of the square matrix obtained by deleting the row corresponding to $\tilde{g}$.  These polynomials are:
\[ \begin{array}{lcl}
p_m & = & -(x^{-4}z^{-2})(x-1)^2(y-1)(z-1)(y+z+4yz+y^2z+yz^2) \\
p_{m_1} & = & (x^{-4}z^{-2})(x-1)^2(y-1)(z-1)(y+z+4yz+y^2z+yz^2) \\
p_{m_2} & = & -(x^{-4}z^{-2})(x-1)^2(y-1)(z-1)(y+z+4yz+y^2z+yz^2) \\
 p_s & = & (x^{-4}z^{-2})(x-1)(y-1)^2(z-1)(y+z+4yz+y^2z+yz^2) \\
 p_t & = & -(x^{-4}z^{-2})(x-1)(y-1)(z-1)^2(y+z+4yz+y^2z+yz^2) \\
 p_u & = & 0
\end{array} \]
The Alexander polynomial of $\mathrm{H}_1(\widetilde{N})$ is the greatest common factor:
\[ \Delta(x,y,z)=(x-1)(y-1)(z-1)(y+z+4yz+y^2z+yz^2) \]
up to multiplication by an invertible element of $\mathbb{Z}[x,x^{-1},y,y^{-1},z,z^{-1}]$.

Let $N_{\infty}$ be the infinite cyclic cover of $N$ factoring through $\widetilde{N}$ given by: \begin{align*}
& m \mapsto x^2 & & s \mapsto x & & t \mapsto x & & u \mapsto 1 &
\end{align*}
Then the chain complex of $N_{\infty}$ is a $\Lambda$--module, where $\Lambda=\mathbb{Z}[x,x^{-1}]$ and specializing the above picture yields an Alexander polynomial
\[ \begin{array}{lcl}
  \Delta_{\infty}(x) & = & (x^2-1)(x-1)^2(2x+4x^2+2x^3) \\
   & = & 2x(x-1)^3(x+1)^3. \end{array} \]
Let $N_n$ be the $n$--fold cyclic cover of $N$ factoring through $N_{\infty}$.  For $n$ odd, $N_n$ has three cusps, since $m$, $sts^{-1}t$, and $t^{-1}s^{-1}ts^{-1}$ map to $x^{\pm2}$, which generates $\mathbb{Z}/n\mathbb{Z}$.  Let $S_n$ be the closed manifold obtained by filling $N_n$ along the slopes covering $m$, $sts^{-1}t$, and $t^{-1}s^{-1}ts^{-1}$.  Theorem \ref{cusped} follows quickly from the following lemma.

\begin{lem} For odd $n \geq 3$, $S_n$ is a rational homology sphere.  \end{lem}

\begin{proof} The proof is adapted from an analogous proof of Sakuma concerning link complements in $S^3$.

The chain complex of $N_n$ is isomorphic to $C_*(N_{\infty}) \otimes (\Lambda/(x^n-1))$.  Note that $x^n - 1 = (x-1)\nu_n$, where $\nu_n(x) = x^{n-1}+x^{n-2} + \hdots + x + 1$.  Sakuma observes that the short exact sequence of coefficient modules
\[ 0 \rightarrow \mathbb{Z} \stackrel{\nu_n}{\longrightarrow} \Lambda/(x^n-1) \rightarrow \Lambda/(\nu_n) \rightarrow 0 \]
where the map on the left is multiplication by $\nu_n$, gives rise to a short exact sequence in homology
\[ 0 \rightarrow \mathrm{H}_1(N) \stackrel{tr}{\longrightarrow} \mathrm{H}_1(N_n) \rightarrow \mathrm{H}_1(N_{\infty})/\nu_n \mathrm{H}_1(N_{\infty}) \rightarrow 0 \]
where $tr$ is the transfer map, $tr(h)=h+x.h+\cdots+x^{n-1}.h$ for a homology class $h$. Define $\mathrm{H}_n=\mathrm{H}_1(N_{\infty})/\nu_n \mathrm{H}_1(N_{\infty})$.  Since the Alexander polynomial of $N_{\infty}$ does not share roots with $\nu_n$, $\mathrm{H}_n$ is a torsion $\mathbb{Z}$--module.

The lemma follows from a comparison between $\mathrm{H}_1(S_n)$ and $\mathrm{H}_n$.  The Mayer--Vietoris sequence implies that $\mathrm{H}_1(S_n)$ is obtained as the quotient of $\mathrm{H}_1(N_n)$ by the subgroup generated by transfers of the meridians.  If $N$ were a link complement in $\mathrm{S}^3$, it would immediately follow that $\mathrm{H}_n = \mathrm{H}_1(S_n)$, since the homology of a link complement is generated by meridians.  In our case we have
\[ \mathrm{H}_n = \mathrm{H}_1(N_n) / \langle\ tr(m),tr(s),tr(t),tr(u)\ \rangle, \]
whereas
\[ \mathrm{H}_1(S_n) = \mathrm{H}_1(N_n) / \langle\ tr(m),tr(2s),tr(2t)\ \rangle. \]
However one observes that $\mathrm{H}_1(S_n) \rightarrow \mathrm{H}_n$ is an extension of degree at most 8 (since $u$ has order 2 in $\mathrm{H}_1(N)$), and so $\mathrm{H}_1(S_n)$ is also a torsion group.
\end{proof}

Let $M_n$ be the manifold obtained by filling two of the three cusps of $N_n$ along the slopes covering $sts^{-1}t$ and $t^{-1}s^{-1}ts^{-1}$.  We have geometrically described $M_n$ as a branched cover of $O_n$, produced by $n$--fold orbifold filling along $sts^{-1}t$ and $t^{-1}s^{-1}ts^{-1}$.  There is a closed totally geodesic surface in $M_n$ covering the totally geodesic sphere with four cone points in $O_n$.  A closed manifold $S_n$ is produced by filling the remaining cusp of $M_n$ along the meridian covering $m$.  Since $S_n$ is a rational homology sphere, $M_n$ is a knot complement in a rational homology sphere, and we have proven \fullref{cusped}.

\section{Further directions}

Performing ordinary Dehn filling along the three meridians of $N$ specified in the previous section yields a manifold $S$, which is easily seen to be the connected sum of two spherical manifolds.  The half arising from the truncated tetrahedra is the quotient of $\mathrm{S}^3$, regarded as the set of unit quaternions, by the subgroup $\langle i,j,k \rangle$.  The half arising from the cuboctahedron is the lens space $\mathrm{L}(4,1)$.  The manifolds $S_n$ may be regarded as $n$--fold branched covers over the three-component link $L$ in $S$ consisting of the cores of the filling tori.

Since the meridians $t^{-1}s^{-1}ts^{-1}$ and $sts^{-1}t$ represent squares of primitive elements in the homology of $N$, any cover of $S$ branched over $L$ will have nontrivial homology of order $2$ coming from the transfers of $s$ and $t$.  However, it is possible that techniques similar to those above may be used to create knot complements in integral homology spheres.  If the manifold $N$ above---in addition to its geometric properties---had trivial nonperipheral \textit{integral\/} homology, then $S$ would be an integral homology sphere.  Porti \cite{Po} has supplied a formula in terms of the Alexander polynomial for the order of the homology of a cover of an integral homology sphere branched over a link, generalizing work of Mayberry--Murasugi in the case of $S^3$ \cite{MM}.  Using this formula, the order of the homology of branched covers of $S$ could be easily checked.

In fact, the Menasco--Reid conjecture itself may be approached using a variation of these techniques.  A genus $n-1$ handlebody may be obtained as the $n$--fold branched cover of a ball over the trivial $2$--string tangle, so knot complements in the genus $n-1$ handlebody may be obtained as $n$--fold branched covers over the trivial tangle of a knot complement in the ball.  In analogy with \fullref{sec3}, allowing the complement of $T$ to play the role of $N_a$ we ask the following:

\begin{ques} Does there exist a hyperbolic $3$--manifold with one rank $2$ and two rank $1$ cusps, which is the complement of a tangle in the ball, with totally geodesic boundary isometric to the totally geodesic boundary of the complement of the tangle $T$? \end{ques}

Such a manifold would furnish an analog of the manifold $N_b$ in \fullref{sec3}.  If the glued manifold $N$ was a $2$--component link complement in $\mathrm{S}^3$, with an unknotted component intersecting the totally geodesic Conway sphere, and this sphere remained totally geodesic under the right orbifold surgery along its boundary slopes, branched covers would give a counterexample to the Menasco--Reid conjecture.  In any case, Thurston's hyperbolic Dehn surgery theorem implies that as $n \rightarrow \infty$, the resulting surfaces would have principal curvature approaching 0, furnishing new examples of the phenomenon discovered by Leininger in \cite{Le} (although unlike Leininger's examples this would not give bounded genus).

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