text:
\begin{document}

\title{Real elements in the mapping class group of  $T^2$}

\author{Nerm\.{\i}n Salepc\.{\i}}
\address{Universit\'e de Pau et des Pays de l'Adour
Avenue de l'Universit\'e - BP 1155
Batiment IPRA
64013 PAU CEDEX}
\email{nermin.salepci@univ-pau.fr}

\begin{abstract}
We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by  features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with $SL(2,\Z)$ as well as that the quotient group $PSL(2,\Z)$ is the symmetry group of the {\em Farey tessellation} of the Poincar\'e disk.
\end{abstract}

\maketitle
\section{Introduction}
In this work, we are interested in finding the isotopy classes of orientation preserving diffeomorphisms of  the torus which have a representative that decomposes into a product of two orientation reversing involutions. The question arose when we were studying properties of the monodromy maps of real Lefschetz fibrations.

Let us recall that a Lefschetz fibration is  a projection of an oriented smooth 4-manifold onto an oriented surface such that
apart from finitely many fibers, which have a single node, the fibers are actually smooth oriented surfaces.
Intuitively, a real structure can be regarded as a topological generalization of the complex conjugation. We define a {\em real structure} on an oriented 4-manifold as an orientation preserving involution whose fixed point set, if it is not empty, has dimension 2.  Likewise, a {\em real structure} on an oriented surface is an orientation reversing involution. A smooth manifold together with a real structure is called a {\em real manifold} and the fixed point set of a real structure is called the {\em real part}.
A {\em real Lefschetz fibration}  is, thus, defined as  a Lefschetz fibration of a real 4-manifold over a real surface where the fiber structure is compatible with the real structures.

The fundamental fact about real Lefschetz fibrations is that their monodromy maps along the loops on which the real structure induces an orientation reversing action decompose into a product of two orientation reversing involutions. These involutions are the real structures of the two real fibers over the two real points of the loop. The diffeomorphisms (as well as their classes) with such a feature  are called {\em real}.  Let us remark that if, in particular, the base space is a surface with a single boundary component, then  the monodromy map along the boundary component satisfies this property.

We can rephrase the decomposition property as follows: the monodromy maps as above of real Lefschetz fibrations are conjugated to their inverses by a real structure.  At this point, a weaker property, that of being conjugated to its inverse by an orientation reversing diffeomorphism  appears naturally. Diffeomorphisms (as well as their classes) with the latter property are called {\em weakly real}.
We study weakly real and real diffeomorphisms simultaneously.

As is well known, the restriction of a Lefschetz fibration to  loops  in the complement of the critical set is a usual fibration over a circle with fiber compact connected oriented smooth $2$-manifold $F$.  We call such fibrations  {\em $F$-fibrations} and extend the above definitions to $F$-fibrations to proceed with them.

The aim of this article is to answer the following questions in the case of $F=T^2$ (elliptic):

\begin{enumerate}

\item  Which classes are real/ weakly real?

\item  Obviously real classes are weakly real. What about the converse:  are weakly real classes real?

\end{enumerate}

Employing the identification of the mapping class group of the torus with  $SL(2,\Z)$ we state the main results as follows:

\begin{t.}
All elliptic and parabolic matrices in  $SL(2,\Z)$ are real.

A hyperbolic matrix $A \in SL(2,\Z)$ is real if and only if its cutting period-cycle  $[a_1 a_2 \ldots a_{2n}]_A$ is odd-bipalindromic.

Moreover, a matrix in $SL(2,\Z)$ is real if and only if it is weakly real.
\end{t.}

(The three statements of the above theorem are indeed presented separately as Theorem~\ref{elpar}, Theorem~\ref{hiper} and Theorem~\ref{wrr}, respectively.)

Elliptic, parabolic and hyperbolic matrices in $SL(2,\Z)$ differ by the nature of their fixed points.  In Section~\ref{eliparabol},
we discuss the conjugacy classes of elliptic and parabolic matrices.  Conjugacy classes of hyperbolic matrices are explained in
Section~\ref{hiperbol}, where  we also  give the definition of the cutting period-cycle. The last two sections  are devoted to the real factorizations of the matrices. We define odd-bipalindromic cutting period-cycle in the last section.

Finally, we would like to note that this work is self-contained and the tools used are explained in detail.

\textbf{Acknowledgements.}
This work is a part of my PhD thesis where we introduced real Lefschetz fibrations and gave a classification of real elliptic Lefschetz fibrations over $S^2$ with only real critical values.  I would like to express my gratitude to my supervisors  Sergey Finashin and Viatcheslav Kharlamov  for  their guidance and endless support throughout  my research.

I would also like to thank  Caroline Series for her interest to my questions and for sending me her articles and  Allen Hatcher for pointing out the reference I needed as well as Slava Matveyev for fruitful discussions.

\section{Real  $F$-fibrations and their monodromies}
Let  $Y$ be a compact connected oriented smooth 3-manifold,
and  $\pi:Y\to S^1$ be a fibration whose fiber is an
oriented smooth $2$-manifold $F$. We call
$\pi$ an {\em $F$-fibration}. In particular, when $F=T^2$, we call $\pi$ an \emph{elliptic $F$-fibration}.

\begin{d.} \label{realS1}
An $F$-fibration $\pi:Y\to S^1$ is called \emph{weakly real} if
there is an orientation preserving diffeomorphism $H:Y\to Y$ which
sends fibers into fibers reversing their orientations. If
$H^2=id$, then $H$ will be called a \emph{real structure} on the
$F$-fibration $Y\to S^1$. An $F$-fibration equipped with a real
structure will be called \emph{real}.

Note that $H$ induces an orientation reversing diffeomorphism
$h:S^1\to S^1$ such that the following diagram commutes:

$$\xymatrix{Y \ar[r]^{H} \ar[d]_{\pi} & Y \ar[d]^{\pi} \\
          S^1 \ar[r]^{h}  & S^1.}$$
\end{d.}

Here, it is not difficult to see that the set of orientation reversing
involutions form a single conjugacy class in the diffeomorphism
group of $S^1$ (the crucial observation is that any such
involution has precisely two fixed points).
Therefore, any real $F$-fibration is equivariantly isomorphic
to an $F$-fibration whose involution $h$ is standard. Let it be the complex conjugation $c_{S^1}: S^1\to S^1$,
$z\mapsto\bar z$, $z\in S^1\subset\C$.

In the case of a weakly real $F$-fibration, $h$ may be not
an involution; however, it also has precisely two fixed points and
can be changed into an involution by an isotopy. This isotopy can be lifted to an isotopy of
$H$. Thus, by modification of $H$ we can always make $h$ an
involution.
 So it is not restrictive for us to suppose always that  $h=c_{S^1}$ for both
real and weakly real $F$-fibrations.

The restrictions of $H$ to the invariant fibers
$F_\pm=\pi^{-1}(\pm1)$ will be denoted $h_\pm:F_\pm\to F_\pm$.
 In the case of real $F$-fibrations, we will prefer using  the notation
$c_Y$ for the involution $H$, and $c_\pm$ for the involutions
$h_\pm$.

In what follows, we choose the point $b$ in the upper semi-circle,
$S^+$.  The restriction $Y_+=\pi^{-1}(S^+)\to S^+$ of $\pi$
admits a trivialization $\phi_+:Y_+\to F\times S^+$ which is
identical on the fiber $F=F_b$. In the case of real fibrations, this allows us to consider the
pull-back of $c_\pm$ via $\phi$, namely, the two involutions
$x\mapsto \phi_+(c_\pm(\phi_+^{-1}(x\times\pm1)))$ on the same
fiber $F$. We will stick to the notation $c_\pm$ for these
involutions.

It is well known that any
$F$-fibration $\pi:Y\to S^1$ is isomorphic to the projection
$M_f\to S^1$ of a mapping torus $M_f=F\times
I\diagup_{{(f(x),0)\sim (x,1)}}$ of some orientation preserving diffeomorphism $f:F\to
F$. More precisely, if we fix a particular fiber $F=F_b=\pi^{-1}(b)$, $b\in S^1$,
then an isomorphism $\phi:M_f\to Y$ can be chosen so that
$F\times0$ and $F\times1$ are identified with the fiber $F_b$, so
that $x\times0\mapsto x$ and $x\times1\mapsto f(x)$.

An $F$-fibration $\pi$ determines an orientation preserving diffeomorphism $f$
up to isotopy and thus, provides a well-defined element in the
mapping class group $[f]\in\M(F)$ called the {\em monodromy of
$\pi$} (relative to the fiber $F=F_b$).
A map $f$ representing the
class $[f]$ will be also often called {\em monodromy}, or more
precisely, a {\em monodromy map}.

In some cases, we fix a marking $\rho:\Sg\to F_b$ (an identification of $F_{b}$ with an abstract genus-$g$ surface $\Sigma_{g}$). Then the
diffeomorphism $\rho^{-1}\circ f\circ\rho:\Sg\to\Sg$ (the
pull-back of $f$) as well as its isotopy class $[\rho^{-1}\circ
f\circ\rho]\in\M(\Sg)$ will be called the {\em monodromy of $\pi$
relative to the marking $\rho$}.

\begin{t.}\label{fcc}
Let $\pi: Y \to S^1$ be a weakly real $F$-fibration with a distinguished fiber $F=F_b$, $b\in S^+$. Then the two
product diffeomorphisms of the fiber $F$, $(h_+)^{-1} \circ h_-$,
and $h_+\circ (h_-)^{-1}$ are isotopic and equal the monodromy
of $\pi$ relative to the fiber $F$.
In particular, if $\pi$ is a real $F$-fibration, then the monodromy can
be factorized as $c_+ \circ c_-$.
\end{t.}

\noindent {\it Proof:}
Consider a trivialization $Y_-\to F\times S^-$ of the
restriction $Y_-=\pi^{-1}(S^-)\to S^-$ of $\pi$ over the lower
semi-circle, $S^-$, which is the composition of $\phi_+\circ H
:Y_-\to F\times S^+$, with the map $F\times S^+\to F\times S^-$,
$(x,z)\mapsto (x, c_{S^1}(z))$.

If $S^1$ is split into several arcs and a fibration over $S^1$ is
glued from trivial fibrations over these arc, then the monodromy
is clearly the product of the gluing maps of the fibers over the
common points of the arcs, ordered in the counter-clockwise
direction beginning from a marked point $b\in S^1$.
In our case, the arcs are $S^+$, $S^-$, their common points
follow in the order $-1$, $+1$, and the corresponding gluing maps,
are $h_-$ and $h_+^{-1}$. This gives monodromy $(h_+)^{-1}\circ
h_-$. If we consider another trivialization $Y_-\to F\times
S^-$ replacing in its definition $H$ by $H^{-1}$, then the
gluing maps will be $h_-^{-1}$ and $h_+$, and the monodromy is
factorized as $h_+\circ (h_-)^{-1} $.
\hfill  $\Box$\\

\begin{c.}\label{2cfc}
(1) If $f:F \to F$ is a monodromy map of  a weakly real $F$-fibration, then the diffeomorphisms $h^{-1}\circ f \circ h$ as well as $h\circ f\circ h^{-1}$, where $h$ stands either for $h_+$, or for $h_-$, are all isotopic to the inverse $f^{-1}$.

(2) If $f$ is a monodromy map  of a real $F$-fibration, then $f^{-1}=c_{+}\circ f\circ c_{+}=c_{-}\circ f\circ c_{-}$. \hfill  $\Box$\\
\end{c.}

\begin{r.}
It is obvious that  $f=c_{+} \circ c_{-}$  for some real structures $c_{-}, c_{+}$ if and only if  $f^{-1}=c_{\pm} \circ f \circ c_{\pm}$. It follows from the known cases of the Nielsen realization problem that  $[f]=[c_{+} \circ c_{-}] $  if and only if $[f^{-1}]=[c_{\pm} \circ f \circ c_{\pm}] $, (cf. \cite{He}, \cite{K}).  In another words,  if
$[f^{-1}]=[c \circ f \circ c] $,  then there is a diffeomorphism $g$ isotopic to $f$ such that  $g^{-1}=c \circ g \circ c $.  Similarly, if  $[f^{-1}]=[h\circ f \circ h^{-1} ] $ for some orientation reversing diffeomorphism $h$, then there are diffeomorphisms $g, k$  such that $[g]=[f]$ and $[k]=[h]$ with  $g^{-1}=k\circ g \circ k^{-1}$.   (All  the class equalities above are considered  in the extended mapping class group $\Mm(T^2)$ of the torus.)   \end{r.}

\begin{d.}\label{zayifguclureel}
A diffeomorphism $f: F\to F$  as well as its isotopy class $[f]\in \M(F)$ will be called {\em real}  if there is a real structure $c:F \to F$ such that $f^{-1}=c\circ f\circ c$.

We call $f:F\to F$ as well as its isotopy class $[f]\in \M(F)$  {\em weakly real}  if $f^{-1}=h\circ f \circ h^{-1}$ for some orientation reversing diffeomorphism $h$ of $F$.
\end{d.}

\begin{p.}\label{s1fcc}
An $F$-fibration is real (respectively weakly real) if and only if
its monodromy $f$ is real (respectively weakly real).
\end{p.}

\noindent {\it Proof:}
We give the proof for real $F$-fibrations; the proof for weakly real ones is
analogous.

 Necessity of the condition follows from Corollary~\ref{2cfc}.
As for the converse, consider an $F$-fibration $\pi:Y\to
S^1$ with the monodromy class $[f]\in \M(F)$,
and $f$ its representative such that   $f^{-1}=c\circ f\circ c$
for some real structure  $c$  on $F$. Presenting $Y$ as
$F\times [0,1]\diagup_{(f(x), 0)\sim(x,1)}$, we obtain a well-defined
involution $c_Y:Y\to Y$ induced from the involution
$(x,t)\mapsto(c(x),1-t)$ on $F\times [0,1]$.
 It preserves the fibration structure and acts as $c$ and $f\circ
 c$ on the real fibers $F\times\frac12$ and $F\times0=F\times1$
respectively.
 \hfill  $\Box$\\

\section{Homology monodromy factorization of elliptic $F$-fibrations}

It is well known that $\M(T^2)=SL(2,\Z)$, due to the fact that every
diffeomorphism $f:T^2 \rightarrow T^2$ is isotopic to a {\em linear
diffeomorphism}. The latter diffeomorphisms by definition are
induced on $T^2=\R^2/\Z^2$ by a linear map $\R^2\to\R^2$ defined by
a matrix $A\in SL(2,\Z)$. Note that we can naturally identify
$T^2=H_1(T^2,\R)/H_1(T^2, \Z)$ and interpret matrix $A$ as the induced
automorphism $f_*$ in $H_1(T^2, \Z)$. The latter automorphism is
 called the \emph{homology monodromy}. Since
isotopic diffeomorphisms have the same homology monodromy in
$H_1(T^2,\Z)$, we obtain well-defined homomorphisms $\M(T^2)\to
Aut^+(H_1(T^2,\Z))\to SL(2,\Z)$ which are in fact isomorphisms (here
$Aut^+$ stand for the orientation preserving automorphisms).
Let $a$ denote the simple closed curve on $T^2$ represented by the
equivalence class of the horizontal interval
$I\times0\subset\R^2$, and $b$ is similarly represented by the
vertical interval $0\times I$. We have $a\circ b=1$; hence, the
homology classes represented by these curves are integral
generators of $H_1(T^2,\Z)$. The mapping class group $\M(T^2)$ of $T^2$ is
generated by the Dehn twists $t_a$ and $t_b$, which can be
characterized by their homology monodromy homomorphism matrices
${t_a}_*=\tiny{\left(\begin{array}{cc}
           1 & 1\\
           0 & 1
         \end{array} \right)}$ and
${t_b}_*=\tiny{\left(\begin{array}{cc}
           1 & 0\\
           -1& 1
         \end{array} \right)}$.

In a like manner, it can be shown that  the extended mapping class group $\Mm(T^2)$  is isomorphic to the general linear group $GL(2,\Z)$.
As a consequence, for elliptic $F$-fibrations, the question of
characterization of real monodromy classes $[f]\in\M(T^2)$ can be
interpreted as the question on the decomposability of their
homology monodromy $f_* \in SL(2,\Z)$ into a product of two {\em
linear real structures} in the group $GL(2,\Z)$. The latter structures by definition  are
linear orientation reversing maps of order 2 defined by integral
$(2\times 2)$-matrices.
 Such decomposability is equivalent to the
property that $f_*$ is conjugate to its inverse by a linear real
structure. Hence, a necessary condition for a matrix $A$ to be real
is that both $A$ and $A^{-1}$ lie in the same conjugacy classes
in the group $GL(2,\Z)$.

Recall that there are three types of real structures  on $T^2$
distinguished by the number of their real components: $0$, $1$, or
$2$. Note that the automorphisms of $H_1(T^2,\Z)$ induced by
the real structures with 0 or 2 real components  are diagonalizable over $\Z$, namely, their
matrices are conjugate to
 $\tiny{\left(\begin{array}{cc}
           1 & 0\\
           0 & -1
         \end{array} \right)}$
in $GL(2,\Z)$. So we cannot determine if the number of components
$0$ or $2$ knowing only the matrix representing the homology
action of the real structure. The homology action of a real structure with 1 real component is presented by a
matrix conjugate to
 $\tiny{\left(\begin{array}{cc}
           0 & 1\\
           1 & 0
         \end{array} \right)}$.

\section{The modular action on the hyperbolic half-plane}

Let $\C^2$ be considered as the vector space of $2\times1$ matrices over $\C$. Then a matrix $A=\tiny{\left(\begin{array}{cc}
           a & b\\
          c  & d
         \end{array} \right)}$ in $GL(2,\Z)$  acts on $\C^2$ from the left as matrix multiplication. $$\small{\left(\begin{array}{cc}
           a & b\\
          c  & d
         \end{array} \right)\left(\begin{array}{c}
           z_1\\
           z_2
         \end{array} \right)=\left(\begin{array}{c}
           az_1 + bz_2\\
          cz_1  + dz_2
         \end{array} \right)}$$

This action can be extended to $\CP^1= \C^2 \setminus \{(0,0)\}\diagup_{\tiny{(z_1, z_2) \sim (\lambda z_1, \lambda z_2)}}$.
Let us identify $\CP^1\cong \{(z_1, z_2)\in \C^2, z_2\neq 0 \}\cup
\{\infty \} \cong  \C \cup \{\infty\}$  and rewrite the action of
$GL(2,\Z)$. We obtain a linear fractional transformation
$z\rightarrow  \frac{az+b}{cz+d}$ where $z=\frac{z_1}{z_2}$. In
particular, if $A \in SL(2 ,\Z)$, then the transformation preserves
the orientation of $\C$ and takes $\R \cup \{\infty\}$ to itself
preserving its orientation. Hence, it gives rise to a
diffeomorphism of the upper half plane $\mathbb{H}$ which can be
seen as a model for the hyperbolic plane where the geodesics are
the semi-circles centered at a real point or vertical half-lines
which can also be considered as arcs of infinite radius. By
identifying the upper half plane with the lower half plane by the  complex
conjugation, one extends the action of $SL(2,\Z)$ to an action of
$GL(2,\Z)$. The standard fundamental domain of the action is the
set $\{z| \left|Re(z)\right|\leq \frac{1}{2}, \left|z\right|\geq 1
\}$ which is shown in Fig.~\ref{fundomain}.

\begin{figure}[ht]
   \begin{center}
      \includegraphics[ width=6cm]{2temelal.jpg}
           \caption{\small{The upper half plane model of the hyperbolic plane and the standard fundamental domain of $SL(2,\Z)$}.}
       \label{fundomain}
      \end{center}
 \end{figure}

\section{The Farey Tessellation}

Let us identify the upper half plane model with the Poincar\'e disk model $\D$. We will consider the disk $\D$ together with its boundary $\R \cup \infty$ and define a tessellation on $\D$ as follows:

Set $\infty$ as $\frac{1}{0}$ and consider the two fractions $\frac{0}{1}$ and $\frac{1}{0}$, spot them on $\D$ as the south and the north poles respectively and connect them with a line which will be the vertical diameter. Consider their mediant $\frac{0+1}{1+0}= \frac{1}{1}$ and connect each of them with a geodesic to the mediant. Apply the same to the fractions $\{\frac{0}{1}, \frac{1}{1}\}$ and $\{\frac{1}{1}, \frac{1}{0}\}$. Iterating this process one obtains a tessellation of the right semi-disk. By taking the symmetry one extends the tessellation to $\D$, see Fig.~\ref{tesel}.

\begin{figure}[ht]
   \begin{center}
   \includegraphics[width=4.8cm, scale=0.3]{2tesel2.jpg}
   \caption{\small{Tessellation of $\D$.}}
       \label{tesel}
      \end{center}
 \end{figure}

In the literature this tessellation is called the \emph{Farey
tessellation}. Let us denote the disk
together with the Farey tessellation by $\D_F$. Note that the Farey
tessellation is a tessellation of $\D$ by ideal triangles ( i.e. triangles with vertices on the boundary $\D_F$).
In fact, the set of vertices of the triangles is exactly $\Q \cup \{\infty\}$.
Moreover, two fractions $\frac{m_1}{n_1},  \frac{m_2}{n_2}$ are
connected by a {\em line} if and only if $m_1n_2-m_2n_1=\pm1$. Hence, the action of
$GL(2,\Z)$ on $\D$ induces an action on $\D_F$ which is
transitive on the geodesics of $\D_F$. Only $\pm I$ acts as the identity; hence, the modular
group $PGL(2,\Z)=GL(2,\Z) / \pm I$ is the symmetry group of $\D_F$
where the subgroup $PSL(2,\Z) =SL(2,\Z) / \pm I$ gives the
orientation preserving symmetries. In what follows we denote by $\Gamma$ the triangle with vertices $\{0, 1,
\infty\}$. Note that $\Gamma$ splits in 3 copies of the fundamental region.

The fixed points of the modular action of a matrix $A\in
PSL(2,\Z)$, $A\neq I$, in $\D_F$ are solutions of
$z=\frac{az+b}{cz+d}$. This gives a quadratic equation
with the discriminant
$tr(A)^2-4$. If
the trace $\left| tr(A)\right|<2$, then the discriminant is
negative and the modular action is a rotation around an imaginary
point (an interior point of $\D_F$). Such matrices are called {\em
elliptic}. If $\left| tr(A)\right|=2$, then the discriminant
vanishes, and $A$ acts as a translation with one fixed rational
point, $\frac{d-a}2$ (on the boundary of $\D_F$). Such matrices
are called {\em parabolic}.
 The {\em hyperbolic matrices} have
$\left| tr(A)\right|>2$ and define a translation of $\D_F$ with
two fixed quadratically irrational real points on the boundary of
$\D_F$.

\section{Conjugacy classes of elliptic and parabolic matrices}\label{eliparabol}

\textbf{Elliptic matrices:} As mentioned above an elliptic matrix  $A\in PSL(2,\Z)$ acts on $\D_F$ as a rotation around a point in the interior of $\D_F$. The center of the rotation belongs to one of the triangles of the tessellation. Without loss of generality, let us assume that the fixed point belongs to the triangle $\Gamma$.
If the fixed point belongs to an edge of $\Gamma$, then $A$ rotates $\Gamma$ by angles $\pm \pi$. Let us denote such rotations by $E_{\pm \pi}$.
The other possibility is the rotations $E_{\pm\frac{2\pi}3}$ by  angles $\pm\frac{2\pi}3$ around the center of $\Gamma$, see
Fig.~\ref{elliptic}. It is not hard to see that the matrices  $E_{\pm \pi}$ (respectively $E_{\pm\frac{2\pi}3}$) are conjugate to each other via an orientation reversing matrix in $PGL(2,\Z)$.

Since $PGL(2,\Z)$ acts transitively on the triangles of the
tessellation, $E_\pi=\tiny{\left(\begin{array}{cc}
           0 & 1\\
          -1 & 0
         \end{array} \right)}$ and $E_{\frac{2\pi}{3}}=\tiny{\left(\begin{array}{cc}
           0 & 1\\
          -1 & 1
         \end{array} \right)}$ represent the conjugacy classes in $PGL(2,\Z) $ of elliptic matrices in $PSL(2,\Z)$.

\begin{figure}[ht]
   \begin{center}
      \includegraphics[width=4.8cm, scale=0.3]{2elip.jpg}\hspace{.5cm}
   \includegraphics[width=4.8cm, scale=0.3]{2elip2.jpg}
       \caption{\small{Modular actions of elliptic matrices, $E_{\pm \pi}$,$\,E_{\pm \frac{2\pi}{3}}$.}}
       \label{elliptic}
      \end{center}
 \end{figure}

Each matrix $A$ in $PSL(2,\Z)$ defines two matrices $\pm A$ in $SL(2,\Z)$. It is not hard to see that the matrices $\pm E_\pi$ are conjugate to each other via reflection with respect to the edge containing the fixed point while $\pm E_{\frac{2\pi}{3}}$ are not, simply by the fact that they have different traces. Hence, there are three conjugacy classes (represented by $E_\pi, \pm E_{\frac{2\pi}{3}}$) in $GL(2,\Z)$ of elliptic matrices in $SL(2,\Z)$.

\textbf{Parabolic matrices:}  The fixed point of the action of a parabolic matrix in $PSL(2,\Z)$ is rational; thus, it is a common vertex of an infinite set of the triangles of $\D_F$. Since $PGL(2,\Z)$ acts transitively on the rational points, it is
not restrictive to assume that the fixed point of the translation is 0.

     \begin{figure}[ht]
   \begin{center}
      \includegraphics[ width=4.8cm, scale=0.3]{2parabol.jpg}
       \caption{\small{Modular actions of parabolic matrices $P_n$.}}
       \label{parabolic}
      \end{center}
 \end{figure}

Hence, a parabolic element can shift the triangle $\Gamma$ by  an arbitrary number
$n$ of triangles to the right or to the left (see Fig.~\ref{parabolic})
fixing the point 0. The left shift is conjugated to the right shift by the
reflection with respect to the vertical line. Hence, the
equivalence classes in $PGL(2,\Z)$ are determined by the number
$n$ of shifts.
Such a shift can be represented by the matrix $P_n= \tiny{\left(\begin{array}{cc}
           1 & 0\\
           n & 1
         \end{array} \right)}$,  $n \in \N$.

The matrix $P_n\in PSL(2,\Z)$ corresponds to two matrices $\pm P_n$ in $SL(2,\Z)$. Having traces with opposite signs, they belong to two different conjugacy classes which are, thus, determined by the integer $\pm n$. Representatives of the conjugacy classes can be chosen as $\pm \tiny{\left(\begin{array}{cc}
           1 & 0\\
           n & 1
         \end{array} \right)}$, $n \in \N$.

\section{Conjugacy classes of hyperbolic matrices}\label{hiperbol}
A hyperbolic matrix $A \in PSL(2,\Z)$ acts on $\D_F$ as translation fixing two irrational points. The geodesic (a semicircle), $l_A$, connecting these fixed points, oriented in the direction of translation, remains invariant under the translation, so $A$ preserves also the set of the triangles of $\D_F$ which are cut by $l_A$, see Fig.\ref{hyperbolic}.
\begin{figure}[ht]
   \begin{center}
      \includegraphics[ width=4.8cm, scale=0.3]{2hiper.jpg}
       \caption{\small{Modular action of a hyperbolic matrix.}}
       \label{hyperbolic}
  \end{center}
\end{figure}

With respect to the orientation of $l_A$ (orientation is defined by the action of $A$),  such triangles are situated in two different ways: a set of triangles with a common vertex lying on the left of $l_A$ followed by a set of triangles with common vertex lying on the
right of $l_A$, see Fig.~\ref{period}.
\begin{figure}[ht]
   \begin{center}
      \includegraphics[width=4.8cm, scale=0.27]{2hiperneyapar.jpg}\hspace{.4cm}
       \caption{\small{Periodic pattern of the truncated triangles of the Farey tessellation.}}
       \label{period}
      \end{center}
\end{figure}

Let us label right and left triangles by $R$ and  $L$,
respectively. Then we encode the arrangement of left and right
triangles with respect to $l_A$ as an infinite word, $\ldots LL
\ldots L RR \ldots RLL \ldots L \ldots$, of  2 letters. This word
is called  the {\em cutting word} of $l_A$. Let us fix a point $p$
at the intersection of $l_A$ with an edge of a triangle. Relative to this point, we obtain a sequence, $(a_1,
a_2, a_3,\ldots)_p$,  from the cutting word  where $a_{2i-1}$
stands for the number of consecutive triangles of one type while
$a_{2i}$ is the number of consecutive triangles of
the other type. For example, if the cutting word with respect to
$p$ reduced to the word $\underbrace{LL\ldots
L}_{a_1}\underbrace{RR\ldots R}_{a_2}\underbrace{LL\ldots
L}_{a_3}\ldots =L^{a_1}R^{a_2}L^{a_3}\ldots$, then we obtain
$(a_1, a_2,\ldots)_p$. This sequence is called the {\em cutting
sequence} relative to the point $p$.

Left and right triangles form a periodic pattern, and the action of $A$ is a shift by a period. Since the choice of $p$ is arbitrary the cutting sequence relative to $p$ is periodic after possibly some finite terms. Moreover, its period is of even length. Note that the choice of the point $p$ is not canonical; hence, we can encode the period only as a cycle, $[a_1 a_2\ldots a_{2n-1} a_{2n}]_A$, which we call the {\em cutting period-cycle associated to the matrix $A$}.

Because of the fact that $PGL(2, \Z)$ is the full symmetry group
of $\D_F$, the cutting period-cycle of a
 hyperbolic matrix $A\in PSL(2,\Z)$ gives the complete invariant of the conjugacy class in
$PGL(2,\Z)$ of $A$. In other words, two matrices $A, B \in
PSL(2,\Z)$ are in the same conjugacy class in $PGL(2,\Z)$ if and
only if $[a_1 a_2\ldots a_{2n}]_A=[a_{\sigma(1)}
a_{\sigma(2)}\ldots a_{\sigma(2n)}]_B$ for a cyclic permutation
$\sigma$. Hence, we will denote the conjugacy classes in
$PGL(2,\Z)$ of hyperbolic matrices of $PSL(2,\Z)$ by the cycle
$[a_1a_2\ldots a_{2n}]$ (defined up to cyclic ordering).

It can be seen geometrically that with respect to the triangle $\Gamma$ a matrix representing a translation corresponding to the cutting period-cycle $[a_1 a_2 \ldots a_n]$ can be chosen as the following product of parabolic matrices.

$$\small{\left(\begin{array}{cc}
           1 & a_1\\
           0 & 1
         \end{array} \right)
         \left(\begin{array}{cc}
           1 & 0\\
         a_2 & 1
         \end{array} \right)\cdots\left(\begin{array}{cc}
           1 & a_{2n-1}\\
           0 & 1
         \end{array} \right) \left(\begin{array}{cc}
           1 & 0\\
         a_{2n} & 1
         \end{array} \right)}.$$

For the sake of simplicity, let us
denote $U=\tiny{\left(\begin{array}{cc}
           1 & 1\\
           0 & 1
         \end{array} \right)}$ and $V=\tiny{
         \left(\begin{array}{cc}
           1 & 0\\
          1 & 1
         \end{array} \right)}$. Then the above product is written as $U^{a_1}V^{a_2}\ldots V^{a_{2n}}$. Note that $U$ is conjugate to $V$ in $PGL(2,\Z)$ but not in $PSL(2,\Z)$.

Let us note that in certain cases, namely if $l_A$ intersects the vertical line of $\D_F$ (since the action of $PGL(2,\Z)$ is transitive on the geodesics of $\D_F$, up to conjugation this property is always satisfied),  the cutting sequence of $l_A$ with respect to the point of intersection of $l_A$ with the vertical line is related to the continued fraction expansion of the fixed point $\xi$ which is the ``end point'' of $l_A$ with respect to the orientation. The corresponding theorem is due to C.~Series \cite{S2, S1}.

\begin{t.}\cite{S2, S1} Let $x > 1$, and let $l$ be any geodesic ray joining some point $p$
on the vertical line of $\D_F$ to $x$, oriented from $p$ to
$x$. Suppose that the cutting word of $l$ with respect to $p$ is
$L^{a_1}R^{a_2}L^{a_3}\ldots$. Then
$x=a_1+\frac{1}{a_2+\frac{1}{a_3+\cdots}}$.

Note that if $0< x< 1$, then the sequence starts with $R$ and
$x=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\cdots}}}$. \\

If $x<0$ everything applies with $x$ replaced by $-x$ and with $R$
and $L$ interchanged.
\end{t.}

A matrix $A\in PSL(2,\Z)$ corresponds to $\pm A$ in $SL(2,\Z)$. Since $\pm A$ have
traces with opposite signs, the cutting period-cycle $[a_1 a_2 \ldots
a_{2n}]$, together with the sign of the trace determine the conjugacy classes of
$\pm A$ in $GL(2,\Z)$.  Representatives of them
can be chosen as $\pm U^{a_1} V^{a_2}\ldots
V^{a_{2n}}$.

\section{Real factorization of elliptic and parabolic matrices}\label{monrelf}

The modular action of the linear real structures
$\tiny{\left(\begin{array}{cc}
          1 & 0\\
          0 & -1
         \end{array} \right)},\tiny{\left(\begin{array}{cc}
           0 & 1\\
           1 & 0
         \end{array} \right)}$
on the hyperbolic plane $\D_{F}$ is $z\mapsto -\bar z$ and
$z\mapsto\frac1{\bar z}$, respectively.
Geometrically, these are the reflections with respect to the vertical and respectively the
horizontal lines, see Fig.~\ref{yansimalar}. In particular, the first
reflection takes the basic triangle $\Gamma$  to the triangle with vertices $\{0, -1, \infty \}$,
and the second one takes $\Gamma$ to itself.

\begin{figure}[ht]
   \begin{center}
         \includegraphics[width=4.8cm, scale=0.3]{2yansimalar.jpg}
       \caption{\small{Modular actions of the linear real structures.}}
       \label{yansimalar}
      \end{center}
 \end{figure}

\begin{t.}\label{elpar}
All elliptic and parabolic matrices in $SL(2,\Z)$ are products of two linear real structures.
\end{t.}

\noindent {\it Proof:}
The explicit real decompositions for the representatives of  the conjugacy classes of
the elliptic matrices are given below:

$$\begin{array}{rcl}
E_{\frac{2\pi}{3}}&=&\small{\left(\begin{array}{cc}
           0 & 1\\
           -1 & 1
         \end{array} \right)=\left(\begin{array}{cc}
           1 & 0\\
           1 & -1
         \end{array} \right)\left(\begin{array}{cc}
           0 & 1\\
           1 & 0
         \end{array} \right)}\\
-E_{\frac{2\pi}{3}}&\cong &\small{\left(\begin{array}{cc}
           -1 & 1\\
           -1 & 0
         \end{array} \right)=\left(\begin{array}{cc}
           1 & -1\\
           0 & -1
         \end{array} \right)\left(\begin{array}{cc}
           0 & 1\\
           1 & 0
         \end{array} \right)}\\
 E_\pi&=&\small{\left(\begin{array}{cc}
           0 & 1\\
          -1 & 0
         \end{array} \right)=\left(\begin{array}{cc}
           1 & 0\\
           0 & -1
         \end{array} \right)\left(\begin{array}{cc}
           0 & 1\\
           1 & 0
        \end{array} \right)}.\end{array}$$

Fig.~\ref{eref} illustrates geometrically the above decompositions in terms of the
corresponding modular action of the matrices.

\begin{figure}[ht]
   \begin{center}
         \includegraphics[width=4cm, scale=0.3]{2e2.jpg}
      \includegraphics[width=4cm, scale=0.3]{2e3.jpg}
      \includegraphics[width=4cm, scale=0.3]{2e1.jpg}
       \caption{\small{Decompositions of the modular actions of the elliptic matrices.}}
       \label{eref}
      \end{center}
 \end{figure}

Real decompositions for  the representatives of the conjugacy classes of the parabolic matrices are as
follows:

         $$P_n=\small{\left(\begin{array}{cc}
           1 & 0\\
           n & 1
         \end{array} \right)=\left(\begin{array}{cc}
           1 & 0\\
           n & -1
         \end{array} \right)\left(\begin{array}{cc}
           1 & 0\\
           0 & -1
         \end{array} \right)}$$

           $$-P_n=\small{\left(\begin{array}{cc}
           -1 & 0\\
           -n & -1
         \end{array} \right)=\left(\begin{array}{cc}
           1 & 0\\
           n & -1
         \end{array} \right)\left(\begin{array}{cc}
           -1 & 0\\
            0 & 1
         \end{array} \right)}.$$
\hfill  $\Box$\\

Fig.~\ref{p} shows  the real decompositions of the
modular action of  matrices $\tiny{\left(\begin{array}{cc}
           1 & 0\\
           n & 1
         \end{array} \right)}$ $n=1,2$.

\begin{figure}[ht]
   \begin{center}
      \includegraphics[width=4cm, scale=0.3]{2p2.jpg}\hspace{.5cm}
      \includegraphics[width=4cm, scale=0.3]{2p1.jpg}
       \caption{\small{Decompositions of the modular actions of the parabolic matrices $P_1,\, P_2$.}}
       \label{p}
      \end{center}
 \end{figure}

\section{The criterion of factorizability for hyperbolic matrices}

\begin{l.} \label{2sym} If the cutting period-cycle of a hyperbolic matrix
$A$ is $[a_1 a_2 \ldots a_{2n}]$, then the cutting period-cycle of
$A^{-1}$ is $[a_{2n} a_{2n-1} \ldots a_{1}]$.
\end{l.}

\noindent {\it Proof:}
Note that $l_A= l_{A^{-1}}$ with the opposite orientation. So the cutting word of $A^{-1}$ can be obtained from the cutting word of $A$ by taking the mirror image of the word and interchanging $L$ with $R$. Interchanging $L$ and $R$ does not effect the cutting period-cycle; hence, the cutting period-cycle of $A^{-1}$ is the reverse $[a_{2n} a_{2n-1}\ldots a_1]$ of the cutting period-cycle $[a_{1} a_{2}\ldots a_{2n}]$ of $A$.
\hfill  $\Box$\\

\begin{d.}
A finite sequence $(a_1a_2\ldots a_k)$ is called \emph{palindromic} if it is equal to the reversed sequence $(a_k a_{k-1}\ldots a_1)$.
We call $k$ the \emph{length of the sequence}.

A finite sequence is called \emph{bipalindromic} if it can be subdivided into two palindromic sequences.
\end{d.}

\begin{d.}
A cutting period-cycle is called \emph{bipalindromic} if there is a cyclic permutation such that the permuted period is bipalindromic.

In particular, if the cutting period-cycle is subdivided into two palindromic sequences of odd length (respectively even length) we call it \emph{odd-bipalindromic} (respectively \emph{even-palindromic}).
\end{d.}

To illustrate, the period $[1213]$ is odd-bipalindromic, while the period $[1122]$ is even-bipalindromic.

\begin{l.} If $A^{-1}=Q^{-1}AQ$ for some $Q\in PGL(2,\Z)$, then the cutting period-cycle $[a_1 a_2\ldots a_{2n}]_{A}$ is bipalindromic.
\end{l.}

\noindent {\it Proof:}  If  $A$ and $A^{-1}$ are in the same conjugacy class in $PGL(2,\Z)$, then they have the same cutting period-cycles up to cyclic permutation. By Lemma~\ref{2sym}  we know that the cutting period-cycle of $A^{-1}$ is $[a_{2n} a_{2n-1}\ldots a_{1}]$ while the cutting period-cycle of $A$ is $[a_1 a_2\ldots a_{2n}]$. Hence, $[a_{\sigma(1)}  a_{\sigma(2)} \ldots a_{\sigma{(2n)}}]= [a_{2n} a_{2n-1} \ldots a_{1}]$ for some cyclic permutation $\sigma$.
Without loss of generality, let us assume that $\sigma$ is a shift by $k=2n-l$, so we have $(a_{l+1}  a_{l+2} \ldots a_{2n} a_{1} a_{2} \ldots a_{l})= (a_{2n} a_{2n-1} \ldots a_{l+1}a_{l}a_{l-1} \ldots a_{1})$ as finite sequences. Thus,  $(a_{l+1}  a_{l+1} \ldots a_{2n})= (a_{2n} a_{2n-1} \ldots a_{l+1})$ and  $(a_{1} a_{2} \ldots a_{l})=(a_{l} a_{l-1} \ldots a_{1})$ which implies that the cutting period-cycle is bipalindromic.
\hfill  $\Box$\\

Note that if the cutting period-cycle is odd-bipalindromic, then the symmetry of the palindromic pieces lifts to a symmetry of the left and the right triangles corresponding to the cutting period-cycle. However, this is not true for even-bipalindromic periods. In the case of $[1213]$, we have $121\sim LR^2L=LRRL$ and $3\sim R^3=RRR$, whereas of $[1122]$, we have $11\sim LR$ and $22\sim L^2R^2=LLRR$.

\begin{t.} \label{hiper}
A hyperbolic matrix $A$ is a product of two linear real structures if and only if its cutting period-cycle  $[a_1 a_2 \ldots a_{2n}]_A$ is odd-bipalindromic.
\end{t.}

\begin{l.}\label{c(l)=l} Let $A \in PSL(2,\Z)$ such that $A^{-1}=Q^{-1}AQ$ for some $Q \in PGL(2,\Z)$, and let $l_A$ be the geodesic invariant under the action of $A$. Then $Q(l_A)=l_A$.
\end{l.}

\noindent {\it Proof:}
Clearly, if $A(l_A)=l_A$, then $A^{-1}(l_A)=l_A$. Hence, $$A^{-1}(l_A)=Q^{-1}AQ(l_A) \Leftrightarrow  Q(l_A)=A(Q(l_A)).$$
By the uniqueness of the invariant geodesic we obtain $Q(l_A)=l_A$.
\hfill  $\Box$\\

\begin{l.}\label{orpres}
Let $A, Q, l_A$ be as above. If the cutting period-cycle  $[a_1
a_2\ldots a_{2n}]_A$ is even-bipalindromic, then $Q$ is
orientation preserving.
\end{l.}

\noindent {\it Proof:}
By means of Lemma~\ref{c(l)=l}, we have $Q(l_A)=l_A$; hence, $Q$  preserves triangles meeting $l_A$.  The action of $Q$ on $\D_F$ is a linear fractional transformation; thus, it preserves the angles. An analysis on the angles at the meeting points of $l_A$ and the edges of the triangles will forbid the existence of the orientation reversing map when the cutting period-cycle is even-bipalindromic.

Let us assume that the cutting period-cycle has the form:
 $$[\underbrace{a_1 a_2\ldots a_k a_k \ldots a_2 a_1}_{P}\underbrace{a'_1 a'_2 \ldots a'_s a'_s \ldots a_2 a'_1}_{P'}]$$ where $s+k=n$ and $P$ and $P'$ are two palindromic pieces.
Substituting the pieces $P, P'$ to the cutting sequence, we obtain a sequence of  $P$ and $P'$ of the form $\ldots PP'PP'\ldots$.  Clearly, the action of the matrix $A$ corresponds to a shift by two: it  takes $P$ to $P$, $P'$ to $P'$.
Let us consider the edges which separate the triangles corresponding to $P$ from the triangles corresponding to $P'$. They are of two types:  with respect to the orientation of  $l_{A}$, we encounter the edges where we move from $P$ to $P'$, and inversely, the edges where we pass from $P'$ to $P$. We denote such edges by $e_{i}$ and $e'_{i}$ respectively, see Fig.~\ref{acilar}.  (We enumerate the edges with respect to an auxiliary point $p$ fixed on $l_{A}$.)

  \begin{figure}[ht]
   \begin{center}
   \includegraphics[width=4.8cm, scale=0.3]{2sinirlar.jpg}\hspace{.5cm}
\includegraphics[width=6.5cm, scale=0.3] {2acilar.jpg}
       \caption{Triangles $\tau_{i}$ and $\tau'_{i}$ and the interior angles $\alpha_{i}, \alpha'_{i}$, $\beta_{i}$, $\beta'_{i}$.}
       \label{acilar}
  \end{center}
\end{figure}

 Each triangle of $\D_\F$ which is cut by $l_A$ splits into two pieces, one of which is a triangle. Let $\tau_i$ (respectively $\tau'_i$) denote the triangle having one edge $e_i$ (respectively $e'_i$) and obtained as the union of  triangle-pieces of the triangles of $\D_F$ with a common vertex on one side of $l_A$ (respectively on the other side of $l_{A}$), see Fig.~\ref{acilar}. Let $\alpha_i$ (respectively $\alpha'_i$) be the interior angles of $\tau_i$ (respectively $\tau'_i$) between the edges $e_i$ (respectively $e'_i$) and $l_A$.
In addition, let $\beta_i$ (respectively $\beta'_i$) be the other interior angle of $\tau_i$ (respectively $\tau'_i$) on $l_A$.

Note that since $A$ shifts triangles  by the period $PP'$, $A$ takes $\alpha_i$ to $\alpha_{i+1}$ (respectively $\alpha'_i$ to $\alpha'_{i+1}$). Hence, all $\alpha_i$ (respectively all $\alpha'_i$) are equal. Let $\alpha=\alpha_i$ for all $i$ (respectively  $\alpha'=\alpha'_i$ for all $i$).

The crucial observation is that if the cutting period cycle is even-bipalindromic, then there is an elliptic matrix in the conjugacy classes of $E_{\pi}$ which fixes the point of intersection of $l_A$ with the middle edge of $P$ or $P'$, see Fig.~\ref{don}. Such matrix interchanges the edges $e_i$ to $e'_i$. Hence, $\alpha=\alpha'$.
(In the same way, we obtain $\beta=\beta_i=\beta'_i$ for all $i$.)

  \begin{figure}[ht]
   \begin{center}
   \includegraphics[width=7.5cm, scale=0.3]{2dondur.jpg}
       \caption{Elliptic rotations of even-bipalindromic cutting period-cycles.}
       \label{don}
  \end{center}
\end{figure}

Let us assume that $\alpha< \frac{\pi}{2}$ (otherwise, we can replace  $\alpha$ with $\beta$.)
We choose an orientation of $\D_F$ by specifying $(v_1, v_2)$ where $v_1$ is a tangent vector of $l_A$ and $v_2$ is the tangent vector of $e_i$ (or $e'_i$) such that the angle $\alpha$ between $v_1$ and $v_2$ is $\alpha < \frac{\pi}{2}$.  The matrix  $Q$ takes $(v_1, v_2)$ to itself since it preserves $l_A$ and the set of edges $e_{i}, e'_{i}$; hence, it preserves the angles between the two. However, an orientation reversing map can not preserve both the angle  $\alpha < \frac{\pi}{2}$ between the vectors $(v_1, v_2)$ and the vectors $v_{1},v_{2}$ at the same time. Thus, $Q$ is orientation preserving.\hfill  $\Box$\\

\noindent {\it Proof of Theorem \ref{hiper}:}
If $A$ is  a product of two linear real
structures. Then the cutting period-cycle is odd-bipalindromic by
Lemma~\ref{orpres}.

If the cutting period-cycle is odd-bipalindromic, then up
to cyclic ordering, it has two palindromic
pieces of odd length. Let us assume that the cutting period-cycle
is of the form $$\pm[a_1 a_2\ldots a_k a_{k+1} a_k\ldots a_2 a_1 a'_1
a'_2\ldots a'_s a'_{s+1} a'_s\ldots a'_2 a'_1]$$ where
$(2k+1)+(2s+1)=2n$. Then for some $R\in PGL(2,\Z)$, we have
$B=R^{-1}AR$ such that $B=\pm U^{a_1}V^{a_2}\ldots U^{a_2}
V^{a_1}U^{a'_1}V^{a'_2}\ldots U^{a'_2} V^{a'_1}.$ Matrices
$U^{a_i}$ and $V^{a_i}$ have the following real decompositions:
$$ {U^{a_i}=\small{\left(\begin{array}{cc}
           1 & -a_i\\
           0 & -1
         \end{array} \right)\left(\begin{array}{cc}
           1 & 0\\
           0& -1
         \end{array} \right)}} \, \textrm{and} \, {V^{a_i}=\small{\left(\begin{array}{cc}
           1 & 0\\
           0 & -1
         \end{array} \right)\left(\begin{array}{cc}
           1 & 0\\
          -a_i& -1
         \end{array} \right)}}.$$

\vspace{0.3cm}

Hence, the product $U^{a_1}V^{a_2}\ldots U^{a_2}V^{a_1}$ can be
rewritten as follows:

 $$\small{\left(\begin{array}{cc}
           1 & -a_1\\
           0& -1
         \end{array} \right)\cdots\left(\begin{array}{cc}
           1 & 0\\
           -a_k& -1
         \end{array} \right)\left(\begin{array}{cc}
           1 & -a_{k+1}\\
           0& -1
         \end{array} \right)\left(\begin{array}{cc}
           1 & 0\\
           -a_k& -1
         \end{array} \right)\cdots\left(\begin{array}{cc}
           1 & -a_1\\
           0 & -1
         \end{array} \right)}. $$

      \vspace{0.3cm}
This gives a linear real structure, since it is a conjugate of
$\tiny{\left(\begin{array}{cc}
           1 & -a_{k+1}\\
           0& -1
         \end{array} \right)}$. Similarly, the product $U^{a'_1}V^{a'_2}\ldots U^{a'_2}V^{a'_1}$ gives a linear real structure conjugate to $\tiny{\left(\begin{array}{cc}
           1 & -a_{s+1}\\
           0& -1
         \end{array} \right)}$.
         \hfill  $\Box$\\

\begin{t.} \label{wrr}
A matrix $A \in SL(2,\Z)$ is a product of two linear real structures if and only if  there is a matrix $Q$ with $\det Q=-1$ such that $A^{-1}=Q^{-1}AQ$.
\end{t.}

\noindent {\it Proof:}  Necessity of the condition is trivial.
As for the converse, we only need to consider the case of hyperbolic matrices.
Let $A$ be a hyperbolic matrix such that $ A^{-1}=Q^{-1} A Q$, for some $Q$ with $\det{Q}=-1$.
Then the cutting period-cycle $[a_1a_2\ldots a_{2n-1} a_{2n}]_{A}$ is odd-bipalindromic by Lemma~\ref{orpres}. So  $A$ is real by
Theorem~\ref{hiper}.
\hfill  $\Box$\\

Theorem~\ref{wrr} together with Proposition~\ref{s1fcc}  lead to the following corollary.

\begin{c.}
An elliptic $F$-fibration is real if and only if it is weakly real.
 \hfill  $\Box$\\
\end{c.}


TITLE: Is it bad math style to use the same index symbol in different indexed objects?
QUESTION [3 upvotes]: The title says it all (I'm referring to the case, when writing for example an article to be published or something similar - something where the writing itself should also be of quality).
Example: "Let $A_1,A_2,\ldots,A_n$ be subsets of a set $L$ and $f:L\rightarrow L$. Then define $B=\cup_{i=0}^n A_i$ and $C=\cup_{i=1}^{\infty} f^i (A_2)$, where $f^i$ is $f$ iterated with itself $i$ times."
So (in this example), would it be a problem to use $i$ twice as an index variable the ranges over different sets ? Or is it ok to think of $i$ as being define like a (lets do a computer-science analogy) "local variable", that is only used to create the object "in" which it is defined ($B$ for example) and after the object is created, the variable is "freed" again (so it can be used again to define $C$) ?
If it is NOT okay to do that, when is it allowed to use the same symbol again, after its first use ? After half a page ? After two pages ? In index-heavy proofs one would very soon have finished all letters commonly used as indices, of one would always have to use a different letter. And indexing sets with letters like $x$ or $f$ seems even worse style to me (ok, maybe there are areas in math, in which this would be common practice, but I don't know any).

REPLY [3 votes]: One time when you might want to avoid that is in expressions like
$$
\sum_{i\in A} f(i) \cdot \sum_{j\in B} g(j)
$$
since you might want to use the distributive law to get
$$
\sum_{i\in A}\sum_{j\in B} f(i)g(j).
$$
The expression $\sum_{j\in B} g(j)$ is what logicians call an "alphabetic variant" of $\sum_{i\in B} g(i)$.  This idea also shows up in thing like
$$
(\forall i\  \varphi(i))\  \&\  (\forall j\  \chi(j))
$$
and you want to put it in "prenex form" with all quantifiers at the beginning:
$$
\forall i\  \forall j\  (\varphi(i)\  \&\  \chi(j)).
$$


TITLE: How to find $X$ and $Y$ using $Z$ between two points in $3$ dimensions?
QUESTION [1 upvotes]: How to find $X$ and $Y$ using $Z$ between two points in $3$ dimensions?
$(1, 1, 1)$, $(X, Y, 1.5)$ and $(2, 2, 2)$ $\leftarrow$ this is the line
$X = 1.5$
$Y = 1.5$
I am looking for the formula
Using answer below:
$z_p=t_pz_1+(1-t_p)*z_2$, replace $t_p$ with $x$
$z_p=xz_1+(1-x)+z_2$
we know $z_p=1.5$, $z_1=1$, $z_2=2$, put it into the formula
$1.5=x*1+(1-x)+2$
$1.5=x+2-2x$
$2x-x=2-1.5$
$x=0.5$, $(x=t_p)$,
In case you need it using $t_p$
$t_p = (z_2-z_p)/(z_2-z_1)$;

REPLY [2 votes]: In general, the line through points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ will have parametric equation
$x=tx_1 + (1-t)x_2\\y=ty_1 + (1-t)y_2\\z=tz_1 + (1-t)z_2$
If you know any one of the three co-ordinates for a point $P$ on this line then you can solve for $t$ and so find the other two co-ordinates at $P$.

REPLY [1 votes]: Your unknown point is on the line segment between $A$ and $B$. The points on that segment have the form $tA + (1-t)B$. Since  you know the third coordinates you can find $t$, then use $t$ to find the other two coordinates.
In your example this is easy: $t = 1/2$. You have the midpoint of the segment.


TITLE: Why do photons have spin of 1 when they have no mass?
QUESTION [0 upvotes]: How can a photon have a spin of $1$ when it has no mass? Since the spin is intrinsic angular momentum, if $m = 0$, then the spin must be zero.

REPLY [7 votes]: A classical picture of spin angular momentum treats particles like tops, and says that their spin is given by some form of $I\,\omega$, where $I$ is derivable from the objects' geometry and mass, and $\omega$ is its angular velocity.
For quantum mechanics, this picture does not work, at least when describing spin angular momentum.  Even for the electron, it if you try and work out some boundaries on what $I$ can be from experiments about the size of the electron, you get implausible values for the angular velocity.
It is better to just think of spin as an intrinsic property of quantum particles that relates to the particle's symmetery under rotation.  (spin 0 particles do not alter under rotation, spin 1 particles rotate to themselves after being rotated through a full circle, spin 1/2 particles rotate to themselves after being rotated through 720 degrees, etc.).  Within the formalism of quantum mechanics, there are ways of relating these concepts clearly and showing why rotations should give rise to this sort of property.  But it is not best to think of the photon spinning like a top.


TITLE: Sketching a graph of $f(x)$
QUESTION [3 upvotes]: Given a function $f(x)$ is increasing in the intervals $(-\infty, -1 )$ and $(3, \infty)$, and decreasing in the interval $(-1,3)$
If I want to sketch a graph of $f(x)$,
I know that the maximum point $\implies x=-1$
Minimum Point $\implies x=3$
However, let's say given the graph
$-5<x<5$
$-5<y<5$
Is there anyway to make the graph nearly accurate from only the given information?
For example, at which $y$ coordinate point would the maximum and minimum point be at?  Or do I just take a random guess from the given information since I only want to sketch it?
Here is my sketch -

REPLY [1 votes]: Good question. I'll assume your function is of the form $f:\mathbf{R}\to \mathbf{R}$. When you're asked to sketch a function or curve like this, you should just seek to include the information specified. In this case, just draw the graph so that it is increasing on $(-\infty,-1)$ and $(3,\infty)$, while decreasing on $(-1,3)$. Also, draw it so that the maximum point occurs at $x=-1$, and the minimum point occurs at $x=3$.
If you feel uncomfortable not knowing the $y-$values of your graph at $x=-1,3$, try simply labeling the $y-$value as $y=f(-1)$ and $y=f(3)$, respectively.
EDIT: I just saw your sketch now. That looks fine.

REPLY [1 votes]: You have no information about the $y$ values, so your sketch is fine. There are lots of functions that meet your description.


TITLE: Single-slit Fraunhofer diffraction within finite baffle/barrier
QUESTION [0 upvotes]: Many standard expositions of Fraunhofer diffraction from a single slit (such as here: https://en.wikipedia.org/wiki/Diffraction#Single-slit_diffraction) assume that the slit is situated in a baffle or barrier of infinite extent.
What happens when the baffle/barrier is of finite width? For example, what if the apparatus consists of plane waves propagating along the axis of symmetry of a cylinder, and the 'slit' in this context is a hole at the end of the cylinder? Can anyone point to a derivation of the intensity pattern on a screen a long distance from the slit?

REPLY [0 votes]: Light passing the edge of a barrier will also produce a diffraction pattern beyond the barrier.  The question then becomes; Do the two patterns overlap?  What is their relative intensity?  Will they be “washed out” by scattered light from beyond the barrier?  What you would like is a well focused plane wave which impacts just the area around the slit.


TITLE: Approximation of semicontinuous function
QUESTION [0 upvotes]: I'm looking for a reference for the theorem saying that a real-valued lower (upper) semicontinuous function on any metric space can be reached as a pointwise limit by a non-decreasing (non-increasing) sequence of continuous functions.
To provide some context, I'm interested in the special case, where the metric space is $\overline{\mathbb{R}}\times\mathbb{R}^N$ with $\overline{\mathbb{R}}$ being the two-point compactification of a real line and a lower semicontinuous function $f:\overline{\mathbb{R}}\times\mathbb{R}^N\to[0,\infty)$ is an extension of a continuous function $g\in C(\mathbb{R}\times\mathbb{R}^N;[0,\infty))$ given by
$$f(x,y):=\begin{cases} g(x,y) &for &(x,y)\in \mathbb{R}\times\mathbb{R}^N  \\ 0 &for & (x,y)\in\{\pm\infty\}\times\mathbb{R}^N\end{cases}.$$
The function $g$ satisfies an additional homogeneity condition: $g(x,\alpha y)=\alpha g(x,y)$ for $\alpha\geq 0$ and all $x,y$ and this property carries over to $f$ obviously. I'm wondering if I can guarantee that the approximating sequence also fulfills this condition.
Thank you.

REPLY [1 votes]: As your function has a particularly simple form, it might be the easiest to just explicitly write down a nice approximating sequence.
$$f_n(x,y) = \begin{cases} g(x,y) & |x| \leq n \\ (n + 1 - |x|)g(x,y) & n \leq |x| \leq n+1 \\ 0 & |x| > |n+1|\end{cases} $$
Homogeniety of $f_n$ in $y$ is now inherited directly from $g$. Moreover, for each fixed $x$ and $y$ the sequence $(f_n(x,y))_{n \in \mathbb{N}}$ is eventually constant.


text:
\label{sec:experiments}
\begin{figure*}[ht]
     \centering
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/sparse_vector_training_curve.pdf}
         \put(5, 70){\textbf{a}}
         \end{overpic}
         \caption{\label{fig:sparse_vector_training_curve}}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/sparse_vector_fixed_d.pdf}
         \put(5, 70){\textbf{b}}
         \end{overpic}
         \caption{\label{fig:sparse_vector_fixed_d}}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/sparse_vector_fixed_n.pdf}
         \put(5, 70){\textbf{c}}
         \end{overpic}
         \caption{\label{fig:sparse_vector_fixed_n}}
     \end{subfigure}
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/sparse_vector_weights.pdf}
         \put(5, 70){\textbf{d}}
         \end{overpic}
         \caption{\label{fig:sparse_vector_weights}}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/low_rank_sysid_fixed_d.pdf}
         \put(5, 70){\textbf{e}}
         \end{overpic}
         \caption{\label{fig:low_rank_fixed_d}}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.3275\textwidth}
         \centering
         \begin{overpic}[width=\textwidth]{figures/low_rank_sysid_fixed_ratio.pdf}
         \put(5, 70){\textbf{f}}
         \end{overpic}
         \caption{\label{fig:low_rank_fixed_ratio}}
     \end{subfigure}
    \caption{(a) Risk curves. Solid indicates training error and dashed indicates holdout error. (b) Test error with $d$ and $s$ fixed as $n$ varies. (c) Test error with $n$ and $s/d$ fixed as $d$ varies. (d) Weights learned without explicit projection. (e) Test error with $d$ and $r$ fixed as $n$ varies. (f) Test error with fixed $r/d$ as $n=d$ varies.}
    \label{fig:main_fig}
\end{figure*}
We now illustrate our theoretical results in two concrete problem settings. We first study a scalar-valued output problem where the Bayes-optimal parameter vector is sparse. We then consider a vector-valued system identification problem where the Bayes-optimal parameter matrix is low-rank.

We compare three variants of the Reflectron with three different choices of potential. The first is the GLM-tron,
which is equivalent to the use of the Euclidean
potential $\psi_2(\bx) = \frac{1}{2} \norm{\bx}^2_2$.
The second is the $p$-norm algorithm~\citep{p_norm},
which uses the potential $\psi_p(\bx) = \frac{1}{2} \norm{\bx}_p^2$
for $p \in [1, \infty]$.
The third variant is the hypentropy
algorithm~\citep{ghai20hypentropy}, which generalizes the setup considered in Corollary~\ref{cor:ent} beyond the probability simplex and uses the potential $\psi_\beta(\bx) = \sum_{i=1}^{d} (\bx_i \arcsinh(\bx_i/\beta) - \sqrt{\bx_i^2 + \beta^2})$ for $\beta \in (0, \infty)$.

A complete description of the experimental setup is given in the appendix.

\subsection{Sparse vector GLMs}
\label{sec:exp:sparse_vector}
In this setting, the learner receives measurements
$y_i = \sigma(\ip{\bthet}{\bx_i}) + w_i$ with $\bx_i \sim \mathrm{Unif}([-1, 1]^d)$, $w_i \sim \mathrm{Unif}([-\sigma_w, \sigma_w]^d)$, and where $\sigma(\cdot)$ is the sigmoid activation. $\bthet$ is taken to be an $s$-sparse vector with $s \ll d$, and we compare the GLM-tron with \textit{explicit} $\ell_1$ projection to the Reflectron with matching \textit{explicit} $\ell_1$ projection and the \textit{implicit} regularization due to the either the hypentropy or $p$-norm potentials.

The learner has knowledge that $\bthet$ is sparse, as well as access to the upper bounds $\norm{\bthet}_p \leq W_p$. In the experiments, we set $W_p = 2\norm{\bthet}_p$. Let $\mathbb{B}_p(r)$ denote the closed $\ell_p$-ball in $\R^d$ of radius $r$ centered at the origin. For the GLM-tron, we set $\calC = \mathbb{B}_1(W_1)$ and the projection onto $\calC$ is Euclidean. For the $p$-norm algorithm, we set $\calC = \mathbb{B}_p(W_p)$ and apply a Bregman projection onto $\calC$. For hypentropy, we set $\calC = \mathbb{B}_1(W_1)$ and again use the Bregman projection onto $\calC$. We compare each algorithm in two experimental regimes.

In the first regime, the ambient dimension $d=10000$ and sparsity $s=100$ are fixed, and we study the performance of each algorithm as a function of the number of data points $n$.

For each pair of $(n, \mathrm{alg})$, we run the full-batch pseudogradient algorithm for
$5000$ iterations
over a grid of hyperparameters, and we tune the step size $\lambda$ and the $p$ value for the $p$-norm algorithm (resp.\ $\beta$ for hypentropy).
As suggested by Theorem~\ref{thm:stat_disc}, we use a holdout set of size $n_{\mathrm{hold}} = 500$ to select the parameters with lowest validation error over $5000$ iterations.

Each algorithm is run for $5$ trials and the configuration that achieves the lowest median test error (over the $5$ trials) is shown in the figures. The size of the test set is $n_{\mathrm{test}} = 1000$, and the error bars correspond to the min/median/max over the $5$ trials.

Figure~\ref{fig:sparse_vector_training_curve} shows the training error
and holdout error of the best configuration for each algorithm with
$n=1000$. Each algorithm overfits, and the holdout set is necessary to find the predictor with lowest generalization error.

Figure~\ref{fig:sparse_vector_fixed_d} shows the resulting test error of each algorithm. For each value of $n$, both the $p$-norm and hypentropy algorithms have lower test error when compared to the GLM-tron, in line with the generalization error predictions of Theorem~\ref{thm:stat_disc}.

In the second regime, the number of data points is fixed at $n=1000$ while the ambient dimension $d$ is varied for fixed $s/d = 0.01$. Figure~\ref{fig:sparse_vector_fixed_n} shows the test error for each algorithm, which increases with the ambient dimension $d$ in all cases. As in Figure~\ref{fig:sparse_vector_fixed_d}, for fixed $d$, both the $p$-norm and hypentropy algorithms have lower test error than the GLM-tron. Taken together, Figures~\ref{fig:main_fig}(a)-(c) validate the claims of Theorem~\ref{thm:stat_disc}.

To verify the predictions of Theorem~\ref{thm:imp_reg_disc}, we remove the explicit projection onto $\calC$ for both the GLM-tron and hypentropy and visualize the structure of the learned parameter vector in Figure~\ref{fig:sparse_vector_weights} ($d = 1000$, $s=10$, and $n=1000$). Figure~\ref{fig:sparse_vector_weights} shows that hypentropy recovers a much sparser solution than the GLM-tron despite the lack of an explicit projection onto the $\ell_1$-ball. In particular, $971$ coordinates have absolute value greater than $0.001$ for the parameters found by the GLM-tron, while there are only $56$ for hypentropy. Moreover, the qualitative structure of the parameter vector found by hypentropy is much closer to that of the true parameters, and quantitatively $\norm{\wh{\bthet}_{\mathrm{glm}} - \bthet}_1 = 24.609$ while $\norm{\wh{\bthet}_{\mathrm{hyp}} - \bthet}_1 = 0.421$. The parameters found by the $p$-norm algorithm have similar structure to those found by hypentropy and are omitted for visual clarity.
\subsection{Low rank system identification}
\label{sec:exp:low_rank}
We now consider a nonlinear system identification problem similar to~\citet{foster2020learning}, where the system dynamics are given by a vector-valued GLM and the parameters may be identified using a spectral variant of the Reflectron. In this setting, the learner observes $n$ trajectories $\{\bx^i_t\}_{t=0, i=1}^{T, n}$ from the discrete-time dynamical system $\bx_{t+1}^i = \rho \bx_t^i + \sigma(\bThet \bx_t^i) + w_t^i$. The system is initialized from $\bx_0^i \sim \mathrm{Unif}([-1,1]^d)$, the process noise is given by $w_t^i \sim \mathrm{Unif}([-\sigma_w, \sigma_w]^d)$, $\sigma(\cdot)$ is the element-wise sigmoid activation, and $\bThet$ is a $d \times d$ matrix with $r = \mathrm{rank}(\bThet) \ll d$. This model is motivated by applications in computational neuroscience, where the system state can be interpreted as a vector of firing rates in a recurrent neural network, and the learned parameters represent the synaptic weights~\citep{rutishauser15}. In the experiments, we fix $\rho = 0.9$, $T=5$, and $\sigma_w = 0.1$.

The generalization error for an estimate $\wh{\bThet}$ is
$\varepsilon(\wh{\bThet}) = \frac{1}{2T} \sum_{t=0}^{T-1} \Expsub{\bx_0}{ \norm{ \bx_{t+1} - \rho \bx_t - \sigma(\wh{\bThet} \bx_t)}^2  }$, which measures the ability of the learned connectivity to correctly predict a new random trajectory in a mean square sense. We search for $\wh{\bThet}$ by minimizing the empirical loss
$\wh{\varepsilon}(\wh{\bThet}) = \frac{1}{2nT} \sum_{i=1, t=0}^{n, T-1} \norm{ \bx_{t+1}^i - \rho \bx_t^i - \sigma(\wh{\bThet} \bx_t^i) }^2$.

Similar to Section~\ref{sec:exp:sparse_vector}, the learner has knowledge that $\bThet$ is low-rank, and we compare how the implicit bias of each method impacts its generalization performance. Let $\lambda(\mathbf{M})$ denote the vector of singular values of a matrix $\mathbf{M}$. For both the GLM-tron and hypentropy algorithms, we project onto
$\calC = \left\{ \wh{\bThet} \in \R^{d \times d} : \norm{\lambda(\wh{\bThet})}_1 \leq 2 \norm{\lambda(\bThet)}_1 \right\}$. For the $p$-norm algorithm, we project onto $\calC = \left\{ \wh{\bThet} \in \R^{d \times d} : \norm{\lambda(\wh{\bThet})}_p \leq 2 \norm{\lambda(\bThet)}_p \right\}$. Hyperparameters are tuned just as in the sparse vector setting.

In Figure~\ref{fig:low_rank_fixed_d}, the ambient dimension
and rank are fixed to be $d=1000$ and $r=20$, and we study the impact of the number of trajectories $n$ on the generalization error. Both the $p$-norm and hypentropy algorithms achieve lower test error than the GLM-tron algorithm for fixed $n$. In Figure~\ref{fig:low_rank_fixed_ratio}, the ambient dimension $d$ is varied with fixed $r/d = 0.02$, and the number of trajectories is held equal to the dimension $n=d$. A heuristic explanation of this scaling is provided in Appendix~\ref{app:heur}. As the dimension increases, the gap between the test error for the GLM-tron and hypentropy increases. This trend also holds for the $p$-norm algorithm for $n < 1200$, which begins to become brittle to the choice of hyperparameter for large $n$. As in the previous section, these results validate the predictions of Theorem~\ref{thm:stat_disc}, now with vector-valued outputs.

config: arxiv


TITLE: why $\arctan(nx)$ doesn't converge uniformly?
QUESTION [5 upvotes]: I'd love your help with understanding what led me to conclude that  $\arctan(nx)$ converges uniformly for $x \in (0, \infty)$
I wanted to check that $\lim_{n \to \infty} \sup |f_n(x)-f(x)|$ is 0 where $f(x)=\lim_{n \to \infty} f_n (x)$.
$f(x)=\frac {\pi}{2}$.
I interpret $\lim_{n \to \infty} \sup |f_n(x)-f(x)|$ as: for every $x_0$ in my interval, compute $\lim_{n\to \infty} |f_n(x_0)-f(x)|$=$L_{x_0}$ and then take the $\sup (L_{x_0})$ for all $x_0$ in my range. Am I right?
So, In my case I have $f(x)=\lim \arctan (nx)= \frac {\pi}{2}$
$r_n(x)=f_n(x)-f(x)=\frac {\pi}{2}-\arctan(nx)$
So for every $x \in (0, \infty)$ , small as can be, when I'll compute $\lim \frac {\pi}{2}-\arctan(nx) =0$ when $n \to \infty$ so $\sup (L_{x_0})=0$ since all the values in this set is $0$. What do I get wrong here? what is the correct defenition for$\lim_{n \to \infty} \sup |f_n(x)-f(x)|$? I feel like I got it all wrong.
Thanks guys!

REPLY [4 votes]: Uniform convergence means that the distance between $f_n(x)$ and $f(x)$ becomes small for all $x$, at the same time. To emphasisze here are the two definitions:

$f_n$ converges to $f$ pointwise if for every $x$ and every $\varepsilon >0$ there exists $n_0$ such that $|f(x)-f_n(x)|<\varepsilon$ for all $n \geq n_0$.
$f_n$ converges uniformly to $f$ if for every $\varepsilon >0$ there exists $n_0$ such that $|f(x)-f_n(x)|<\varepsilon$ for all $n \geq n_0$ and for all $x$.

In your case, the problem is that every function $f_n$ can be extended continuously by $f_n(0)=0$, so near $0$ the functions $f_n$ cannot approach $\pi/2$ simultaneously for small $x$. If your functions would have been defined on $(\delta,\infty)$ with $\delta>0$ then the convergence would have been uniform.
Another way to see that $f_n$ does not converge uniformly to $f$ is to consider $f_n :[0,\infty)$, extended by continuity. It is relatively easy to prove that if $f_n$ converges uniformly to $f$ and all $f_n$ are continuous then $f$ is also continuous. But the limit of your functions $f_n(x)=\arctan (nx)$ is
$$ f(x)=\begin{cases}\frac{\pi}{2},& x>0 \\ 0,& x=0\end{cases}$$
which is not continuous.

REPLY [0 votes]: Here is another reason to see that the convergence is not uniform.  The functions $\arctan(nx)$ converge pointwise to the function that is 0 at 1 and $\pi/2$ for any positive number.  Each function is continuous, but the limit  is not.  Hence, the convergence cannot be uniform.


text:
\begin{document}
\title[Generalized Chen's conjecture on biharmonic submanifolds is false]{ The generalized Chen's conjecture on biharmonic submanifolds is false }
\date {12/31/2010} \subjclass{58E20, 53C12, 53C42}
\keywords{Biharmonic maps, biharmonic submanifolds, biharmonic
foliations, conformally flat spaces, Chen's conjecture on biharmonic
submanifolds.}
 \maketitle

\section*{Abstract}
\begin{quote}
{\footnotesize The generalized Chen's conjecture on biharmonic
submanifolds asserts that any biharmonic submanifold of a
non-positively curved manifold is minimal (see e.g., \cite{CMO1},
\cite{MO}, \cite{BMO1}, \cite{BMO2}, \cite{BMO3}, \cite{Ba1},
\cite{Ba2}, \cite{Ou1}, \cite{Ou2}, \cite{IIU}). In this paper, we
prove that this conjecture is false by constructing  foliations of
proper biharmonic hyperplanes in a $5$-dimensional conformally flat
space with negative sectional curvature. Many examples of proper
biharmonic submanifolds of non-positively curved spaces are also
given.}
\end{quote}
\section{Biharmonic submanifolds and the generalized Chen's conjecture}

We work on the category of smooth objects so all manifolds, maps, and tensor fields discussed in
this paper are smooth unless there is an otherwise statement.\\

A {\bf biharmonic map} is a map $\varphi:(M, g)\longrightarrow (N,
h)$ between Riemannian manifolds which is a local solution of the
$4$th order PDEs
\begin{equation}\label{BTF}
\tau^{2}(\varphi):={\rm
Trace}_{g}(\nabla^{\varphi}\nabla^{\varphi}-\nabla^{\varphi}_{\nabla^{M}})\tau(\varphi)
- {\rm Trace}_{g} R^{N}({\rm d}\varphi, \tau(\varphi)){\rm d}\varphi
=0,
\end{equation}
 where $R^{N}$ denotes the curvature operator of $(N, h)$ defined by
$$R^{N}(X,Y)Z=
[\nabla^{N}_{X},\nabla^{N}_{Y}]Z-\nabla^{N}_{[X,Y]}Z,$$ and
 $\tau(\varphi)={\rm Trace}_{g}\nabla {\rm d} \varphi$ is the tension
field of $\varphi$ and $\tau(\varphi)=0$ means the map $\varphi$ is
harmonic.\\

Clearly, it follows from (\ref{BTF}) that any harmonic map is
biharmonic and we
call those non-harmonic biharmonic maps {\bf proper biharmonic maps}.\\

A submanifold $M$ of $(N,h)$ is called a {\bf biharmonic
submanifold} if the inclusion map ${\bf i}: (M, {\bf
i}^*h)\longrightarrow (N,h)$ is a biharmonic isometric immersion. It
is well known that an isometric immersion is minimal if and only if
it is harmonic. So a minimal submanifold is trivially biharmonic and
we call a non-minimal biharmonic submanifold a {\bf proper
biharmonic submanifold}.\\

Among the fundamental problems in the study of biharmonic maps are
the following: {\bf (1) existence problem:} given two model spaces
(e.g., some ``good" spaces such as spaces of constant sectional
curvature or more general symmetric or homogeneous spaces), does
there exist a proper biharmonic map from one space into another?
{\bf (2) classification problem:} classify all proper biharmonic
maps between two model spaces where the existence is known. A
typical and
challenging classification problem is the following \\

{\bf Chen's conjecture \cite{CH}:} any biharmonic submanifold in a
Euclidean
space is minimal.\\

The conjecture was proved to be true for biharmonic surfaces in
$\r^{3}$ (by Jiang \cite{Ji2} and Chen-Ishikawa \cite{CI}
independently) and for biharmonic hypersurfaces in $\r^4$
(\cite{HV}). Dimitri$\acute{\rm c}$ \cite{Di} showed that the
conjecture is also true for any biharmonic curve, any biharmonic
submanifold of finite type, any pseudo-umbilical biharmonic
submanifold $M^m\subset \r^n$ with $m\ne 4$, and  any biharmonic
hypersurface in $\r^n$ with at most two distinct principal
curvatures. However, the
conjecture is still open in general.\\

In the same direction of classifying proper biharmonic submanifolds
of non-positively curved manifolds, Caddeo, Montaldo and Oniciuc
\cite{CMO2} proved that any biharmonic submanifold in hyperbolic
$3$-space $H^{3}(-1)$ is minimal, and any pseudo-umbilical
biharmonic submanifold $M^m\subset H^n$ with $m\ne 4$ is minimal. It
is also shown in \cite{BMO1} that any biharmonic hypersurface of
$H^n$ with at most two distinct principal curvatures is minimal. All
these results suggest the following generalized Chen's conjecture on
biharmonic submanifolds which was proposed by Caddeo, Montaldo and
Oniciuc \cite{CMO1}:\\

{\bf The generalized Chen's conjecture}: any biharmonic submanifold
of $(N, h)$ with ${\rm Riem}^N\leq 0$ is minimal (see e.g.,
\cite{CMO1}, \cite{MO}, \cite{BMO1}, \cite{BMO2}, \cite{BMO3},
\cite{Ba1}, \cite{Ba2}, \cite{Ou1}, \cite{Ou2}, \cite{IIU}).\\

The gaol of this paper is to  prove that the generalized Chen's
conjecture for biharmonic submanifolds is false. We accomplished
this by using the idea of constructing foliations of proper
biharmonic hyperplanes in a conformally flat space given in
\cite{Ou1}. The idea is to determine a conformally flat metric on
$\r^{m+1}$ so that a foliation by the hyperplanes defined by the
graphs of  linear functions becomes a proper biharmonic foliation.
It turns out that when $m=4$ the system of biharmonic equations
reduces to a single equation which has infinitely many solutions
including counter examples to the generalized Chen's conjecture.

\section{Foliations of conformally flat spaces by biharmonic hyperplanes}

\indent  As conformally flat spaces are going to play a central role
in this paper we summarize, in this section, some basic definitions
and the relations between various curvatures of two Riemannian
manifolds which are conformally related. Two Riemannian metrics $g$
and $\bar{g}$ on $M$ are {\em conformally equivalent}, if $\bar{g}=
e^{2\sigma}g$ for some  function $\sigma$ on $M$. A map $\varphi:(M,
g)\longrightarrow (N, h)$ between Riemannian manifolds is {\em
conformal} if ${\varphi}^{*} h= e^{2\sigma}g$ for some  function
$\sigma$ on $M$. Say two Riemannian manifolds ($M,g$) and ($N,h$)
are conformally diffeomorphic, if there exists a conformal
diffeomorphism from one space into another. A Riemannian manifold
($M^{m},g$) is {\em a conformally flat space} if for any point of
$M$ there exists a neighborhood which is conformally diffeomorphic
to the Euclidean space $\mathbb{R}^{m}$. It is well known that any
two-dimensional Riemannian manifold is conformally flat due to the
existence of isothermal coordinates. For $m= 3$, ($M^{m},g$) is
conformally flat if and only if the Schouten tensor $H$ satisfies
$({\nabla}_{X} H)(Y,Z) = ({\nabla}_{Y}H)(X,Z)$ for any vector fields
$X,\;Y$ and $Z$ on $M$. For $m\geq 4$, $(M^{m},g)$ is conformally
flat if and only if the Weyl curvature vanishes identically. It is
easy to see that a space of constant sectional curvature is
conformally flat but there exist many conformally flat
spaces which are not of constant sectional curvature.\\

\indent Let $\nabla, {\rm R,\;\; Ric, \; \;K}$ (respectively ${\bar
\nabla},\; \bar {\rm R},\; {\bar {\rm R}}{\rm ic},  \bar {\rm K}$)
denote the Levi-Civita connection, Riemannian curvature, Ricci
curvature,
 and sectional curvature of the Riemannian metric
$g$ (respectively ${\bar g}=e^{2\sigma}g$). Then,  it is not
difficult to check (see also \cite{Ha} and \cite{Wa}) the following
relations between the connections and the curvatures of the two
Riemannain metrics that are conformally equivalent:

\begin{eqnarray}\label{Cnabla}
{\bar \nabla}_{X}Y=\nabla_{X}Y+(X\sigma)Y+(Y\sigma)X-g(X,Y)\,{\rm
grad}_g\, \sigma,\;\;\forall \;\; X,Y\in TM,
\end{eqnarray}

\begin{eqnarray}\notag
{\bar R}(W,Z, X,Y)&=&e^{2\sigma}\{R(W,Z,
X,Y)+g(\nabla_{X}\nabla\sigma,Z)g(Y,W)\\\label{RX}
&&-g(\nabla_{Y}\nabla\sigma,Z)g(X,W)+g(X,Z)g(\nabla_{Y}\nabla
\sigma,W)\\\notag &&-g(Y,Z)g(\nabla_{X}\nabla \sigma,W)
+[(Y\sigma)(Z\sigma)-g(Y,Z)|\nabla \sigma|^{2}]g(X,W)\\\notag &&
-[(X\sigma)(Z\sigma)-g(X,Z)|\nabla \sigma|^{2}]g(Y,W)\\\notag
&&+[(X\sigma)g(Y,Z)-(Y\sigma)g(X,Z)]g(\nabla \sigma,W)\}
\end{eqnarray}
for any $W, Z, X, Y\in TM$. \\

With respect to local coordinates $\{x_i\}$ and the natural frame
$\{\frac{\partial}{\partial x_i}=\partial_i\}$, Equation (\ref{RX})
is equivalent to
\begin{eqnarray}\notag\label{d}
e^{-2\sigma}{\bar
R}_{ij\,kl}&=&R_{ij\,kl}+g_{il}\sigma_{jk}-g_{ik}\sigma_{jl}+g_{jk}
\sigma_{il}-g_{jl}\sigma_{ik}\\\label{246}&&
+(g_{il}g_{jk}-g_{ik}g_{jl})|\nabla \sigma|^{2},
\end{eqnarray}
where we have used the  notation
$\sigma_{jl}=\nabla_{l}\sigma_j-\sigma_l\sigma_j=\nabla_{l}\nabla_j\sigma-\sigma_l\sigma_j$.\\
By contracting (\ref{d}) we have
\begin{eqnarray}\label{Ric}
{\bar R}_{jk}=R_{jk}-(n-2)\sigma_{jk}-g_{jk}[ \Delta\sigma
+(n-2)|\nabla \sigma|^{2}],
\end{eqnarray}
where $\Delta$ and $\nabla$ denote the Laplacian and the gradient
operator defined by the metric $g$.

\indent Let $P$ be a section spanned by an orthonormal basis $X, Y$
with respect to $g$ (hence ${\bar X}=e^{-\sigma}X, {\bar
Y}=e^{-\sigma}Y$ form an orthonormal basis with respect to ${\bar
g}$). Then, we have the relationship between the sectional
curvatures with respect to metrics ${\bar g}$ and $g$ given by
\begin{eqnarray}\label{QL}
e^{2\sigma}{\bar K}(P)=K(P)-(g(\nabla_{X}\nabla
\sigma,X)+g(\nabla_{Y}\nabla \sigma,Y))\\\notag
-(|\nabla\sigma|^{2}-(X\sigma )^{2}-(Y\sigma)^{2}).
\end{eqnarray}

 We also need the following theorem which will be used to prove our main theorem
about biharmonic hypersurfaces in a conformally flat space.

\begin{theorem}\label{EQ1}\cite{Ou1}
Let $\varphi:M^{m}\longrightarrow N^{m+1}$ be an isometric immersion
of codimension-one with mean curvature vector $\eta=H\xi$. Then
$\varphi$ is biharmonic if and only if:
\begin{equation}\label{eq1}
\begin{cases}
\Delta_g H-H |A|^{2}+H{\rm
Ric}^N(\xi,\xi)=0,\\
 2A\,({\rm grad}_g\,H) +\frac{m}{2} {\rm grad}_g\, H^2
-2\, H \,({\rm Ric}^N\,(\xi))^{\top}=0,
\end{cases}
\end{equation}
where  $\Delta_g$ and ${\rm grad}_g$ are the Laplacian and the
gradient operators of the hypersurface, and ${\rm Ric}^N :
T_qN\longrightarrow T_qN$ denotes the Ricci operator of the ambient
space defined by $\langle {\rm Ric}^N\, (Z), W\rangle={\rm Ric}^N
(Z, W)$ and  $A$ is the shape operator of the hypersurface with
respect to the unit normal vector $\xi$.
\end{theorem}

Now we are ready to prove one of the main theorems of this paper.

\begin{theorem}\label{f1}
 For positive integer $m\ge 2$, let $a_i, i=1, 2, \ldots, m$ and $c$ be constants. Then, the
isometric immersion $\varphi : \mathbb{R}^m\longrightarrow
(\mathbb{R}^{m+1},h=f^{-2}(z)(\sum_{i=1}^m{\rm d}{x_i}^{2}+{\rm
d}{z}^{2})$, $\varphi(x_1,\ldots, x_m)=(x_1,\ldots,
x_m,\sum_{i=1}^{m}a_ix_i+c)$ into the conformally flat space is
biharmonic if and only if one of the following three cases happens
\begin{itemize}
\item[(i)] $f'=0$,
in this case $\varphi$ is minimal (actually, totally geodesic), or
\item[(ii)]  $m=4$ and
$f$ is a solution of the equation
\begin{equation}\label{single}
\sum_{i=1}^4a_i^2f^2f'''+(4-\sum_{i=1}^4a_i^2)ff'f''-4(2+\sum_{i=1}^4a_i^2)(f')^3=0,
\end{equation}
or
\item[(iii)] $a_i=0$ for $i=1,\;\cdots,\;m \;{\rm and}\;
f(z)=\frac{1}{Az+B}$, where  $A$ and $B$ are constants. In this case
each hyperplane is a proper biharmonic hypersurface. This recovers a
result (Theorem 3.1) obtained earlier in \cite{Ou1}.
\end{itemize}
\end{theorem}

\begin{proof}
Using the notations $\partial_i=\frac{\partial}{\partial x_i},i=1,2,
\ldots, m,
 \partial_{m+1}=\frac{\partial}{\partial z},$ \;we can easily check that $\{\bar e_\alpha=f(z)\partial_\alpha,\;\; \alpha=1, 2, \ldots, m+1\}$
constitute an orthonormal frame  on the conformally flat space
$(\mathbb{R}^{m+1}, h)$. One can also check that
\begin{eqnarray}
\begin{cases}
\varphi(\partial _i)=\frac{1}{f}({\bar e} _i+a_i{\bar e}
_{m+1}),\;\;\; i=1, 2,
\ldots, m;\\
\eta=\frac{1}{f}(\sum_{j=1}^m a_j{\bar e} _j-{\bar e} _{m+1})
 \end{cases}
\end{eqnarray}
constitute a natural frame adapted to the hypersurface with $\eta$
being a normal vector.

Applying Gram-Schmidt orthonormalization process to the natural
frame
\begin{eqnarray}
\begin{cases} {\bar e} _i+a_i{\bar e} _{m+1},\;\;\; i=1, 2, \ldots,
m;\\\sum_{j=1}^m a_j{\bar e} _j-{\bar e} _{m+1},
 \end{cases}
\end{eqnarray}
or by a straightforward checking one can verify the following \\

{\bf Claim I.} Let
$k_i=\frac{1}{\sqrt{1+\sum_{l=1}^{i}a_l^2}},\;i=1,\cdots,\;m,\;k_0=1,\;a_0=0$.
Then, the vector fields
\begin{eqnarray}
\begin{cases} e_i=-a_ik_ik_{i-1}\sum_{l=1}^{i-1}a_l\bar
e_l+\frac{k_i}{k_{i-1}}\bar e_i+a_ik_ik_{i-1}\bar e_{m+1},\;\;
i=1,2,\ldots, m;\\e_{m+1}=\sum_{l=1}^{m}a_lk_{m}\bar e_l-k_{m}\bar
e_{m+1},
\end{cases}
\end{eqnarray}
 form an orthonormal frame adapted to the hypersurface
$z=\sum_{i=1}^{m}a_ix_i+c$ with $\xi=e_{m+1}$ being the unit normal
vector field.\\

\indent  Let $\bar h=\sum_{i=1}^m{\rm d}{x_i}^{2}+{\rm d}{z}^{2}$
denote the Euclidean metric on $\mathbb{R}^{m+1}$. Then,
$h=e^{-2\sigma}\bar h$ with  $\sigma=\ln f(z)$.  It follows that
\begin{equation}\label{Grad}
{\rm grad}_h\,\sigma =\bar e_{m+1}(\sigma)\bar e_{m+1}=f^{'}\bar
e_{m+1}.
\end{equation}
Using the fact that $\bar
\nabla_{\partial_\alpha}{\partial_\beta}=0, \;\;\forall\;\; \alpha,
\beta=1, 2,\ldots, m+1 $, and the relation
\begin{eqnarray*}
{\bar \nabla}_{X}Y=\nabla_{X}Y+(X\sigma)Y+(Y\sigma)X-h(X,Y){\rm
grad}_h\, \sigma,
\end{eqnarray*}
we can compute the Levi-Civita connection $\nabla$ of the
conformally flat metric $h$ with respect to the orthonormal frame
$\{\bar e_i\}$ to get
\begin{equation}\label{Conn}
(\nabla_{\bar e_\alpha}\bar e_{\beta})=\left(\begin{array}{ccccc}
f'\bar e_{m+1} &0&\ldots&0&-f'\bar e_1
\\ 0 & f'\bar e_{m+1}&\ldots&0&-f'\bar e_2\\
\ldots&\ldots&\ldots&\ldots&\ldots
\\ 0 & 0&\ldots&f'\bar e_{m+1}&-f'\bar e_m\\
0&0&\ldots&0&0\\
\end{array}\right)_{(m+1)\times (m+1)}.
\end{equation}
A further computation using (\ref{Conn}) yields
\begin{eqnarray}\notag
\nabla_{e_i}e_i&=&k_i^2k_{i-1}^2\nabla_{(-a_i\sum_{l=1}^{i-1}a_l\bar
e_l+\frac{1}{k_{i-1}^2}\bar e_i+a_i\bar
e_{m+1})}(-a_i\sum_{l=1}^{i-1}a_l\bar e_l+\frac{1}{k_{i-1}^2}\bar
e_i+a_i\bar e_{m+1})\\\label{Cii}
&=&k_i^2k_{i-1}^2f'[(a_i^2\sum_{l=1}^{i-1}a_l^2+\frac{1}{k_{i-1}^4})\bar
e_{m+1}+a_i^2\sum_{l=1}^{i-1}a_l\bar e_l-\frac{a_i}{k_{i-1}^2}\bar
e_i],
\end{eqnarray}
\begin{eqnarray}\notag
\nabla_{e_i}e_{m+1}&=&\nabla_{(-a_ik_ik_{i-1}\sum_{l=1}^{i-1}a_l\bar
e_l+\frac{k_i}{k_{i-1}}\bar e_i+a_ik_ik_{i-1}\bar
e_{m+1})}(\sum_{l=1}^{m}a_lk_{m}\bar e_l-k_{m}\bar
e_{m+1})\\\label{m+1} &=&k_{m}f'[a_ik_ik_{i-1}\bar
e_{m+1}+\frac{k_i}{k_{i-1}}\bar
e_i-a_ik_ik_{i-1}\sum_{l=1}^{i-1}a_l\bar e_l]\\\notag &=&
k_{m}f'e_i,
\end{eqnarray}
and
\begin{eqnarray}\notag
\nabla_{e_{m+1}}e_{m+1}&=&\nabla_{\sum_{l=1}^{m}a_lk_{m}\bar
e_l-k_{m}\bar e_{m+1}}(\sum_{l=1}^{m}a_lk_{m}\bar e_l-k_{m}\bar
e_{m+1})\\\label{2m} &=&(1-k_{m}^2)f'\bar
e_{m+1}+\sum_{l=1}^{m}a_lk_{m}^2f'\bar e_l.
\end{eqnarray}

On the other hand, one can use the relation $e_i=C_i^{\alpha}{\bar
e}_{\alpha}=fC_i^{\alpha}\partial_{\alpha},
\xi=C_{m+1}^{\alpha}{\bar
e}_{\alpha}=fC_{m+1}^{\alpha}\partial_{\alpha}$ and the Ricci
curvature
(\ref{Ric}) to verify the following\\

{\bf Claim II.} For a hypersurface in the conformally flat space
$(\mathbb{R}^{m+1}, h=e^{-2\sigma}(\sum_{i=1}^m{\rm d}{x_i}^{2}+{\rm
d}{z}^{2}))$, we have
\begin{eqnarray}\label{RiNf}
{\rm Ric}\,(\xi, \xi)&=&\Delta\sigma+(m-1)[{\rm Hess}\,(\sigma)(\xi,
\xi)-(\xi\sigma)^2+|{\rm grad}_h\,\sigma|^{2}]\\\label{RiTf} ({\rm
Ric}\,(\xi))^T&=&(m-1)[{\rm grad}_g(\xi\sigma)-\xi(\sigma){\rm
grad}_g\,\sigma+A({\rm grad}_g \,\sigma)],
\end{eqnarray}
where ${\rm grad}_g $ is the gradient defined by the induced metric
on the hypersurface.\\

Substituting
\begin{eqnarray*} \notag
\Delta\sigma&=&\sum_{\alpha=1}^{m+1}[ \bar e_\alpha \bar
e_\alpha(\sigma)-(\nabla_{\bar e_\alpha}\bar
e_\alpha)(\sigma)]=ff^{''}-m(f^{'})^2,\\\notag {\rm
Hess}\,(\sigma)(\xi,\xi)&=&e_{m+1}e_{m+1}(\sigma)-(\nabla_{e_{m+1}}e_{m+1})(\sigma)\\\notag
&=&k_{m}^2ff^{''}-(1-k_{m}^2)(f^{'})^2,\\\notag {\rm
grad}_h\,\sigma&=&f^{'}\bar e_{m+1},|{\rm
grad}_h\,\sigma|^2=(f^{'})^2,\;\;\;\xi(\sigma)=-k_{m}f^{'}.
\end{eqnarray*}
into (\ref{RiNf}) we obtain
\begin{eqnarray}\label{RiN}
{\rm Ric}\,(\xi, \xi)&=&\Delta\sigma+(m-1)[\rm Hess(\sigma)(\xi,
\xi)-(\xi\sigma)^2+|{\rm grad}_h\,\sigma|^{2}]\\\notag
&=&[1+(m-1)k_m^2]ff^{''}-m(f^{'})^2.
\end{eqnarray}

Noting that $\xi=e_{m+1}$ is the unit normal vector field we can
easily compute the components of the second fundamental form to get
\begin{eqnarray}\notag
h(e_i,e_i)=\langle\nabla_{e_i}e_i,e_{m+1}\rangle=-\langle
\nabla_{e_i}e_{m+1},e_i\rangle=-k_{m}f',\\\notag
h(e_i,e_j)=\langle\nabla_{e_i}e_j,e_{m+1}\rangle=-\langle\nabla_{e_i}e_{m+1},e_j\rangle=0,\;\;i\neq
j,
\end{eqnarray}
from which we conclude that each hyperplane
$z=\sum_{i=1}^{m}a_ix_i+c$ is a totally umbilical hypersurface in
the conformally flat space and all principal normal curvature are
equal to
\begin{equation}\label{H}
H=\xi(\sigma)=-k_{m}f'.
\end{equation}

It follows that
\begin{eqnarray}\label{RiT}
[Ric(\xi)]^T&=&(m-1)[{\rm grad}_g(\xi\sigma)-\xi(\sigma){\rm
grad}_g\,\sigma+A({\rm grad}_g \,\sigma)]\\\notag &=& (m-1){\rm
grad}_g\,H,
\end{eqnarray}
and the norm of the second fundamental form is given by
\begin{equation}\label{A2}
|A|^2=\sum_{i=1}^m\langle\nabla_{e_i}\xi,\nabla_{e_i}\xi\rangle^2=mk_{m}^2(f')^2.
\end{equation}
A further computation gives
\begin{eqnarray}\notag
e_i(H)&=&(-a_ik_ik_{i-1}\sum_{l=1}^{i-1}a_l\bar
e_l+\frac{k_i}{k_{i-1}}\bar e_i+a_ik_ik_{i-1}\bar
e_{m+1})(-k_mf')\\\notag &=&-a_ik_ik_{i-1}k_mff'',
\end{eqnarray}
\begin{eqnarray}\label{gradH}
{\rm grad}_gH&=&
\sum_{i=1}^me_i(H)e_i=-\sum_{i=1}^ma_ik_ik_{i-1}k_{m}ff''e_i,
\end{eqnarray}

\begin{eqnarray}
 e_ie_i(H)&=&-e_i(a_ik_ik_{i-1}k_{m}ff'')\\\notag
&=&-a_i^2k_i^2k_{i-1}^2k_m(f^2f'''+ff'f''),\\(\nabla_{e_i}e_i)(H)&=&-k_i^2k_{i-1}^2k_mf'(a_i^2\sum_{l=1}^{i-1}a_l\bar
e_l-\frac{a_i}{k_{i-1}^2}\bar e_i)(f')\\\notag
&=&a_i^2k_i^2k_{i-1}^2k_mff'f''
\end{eqnarray}
and

\begin{eqnarray}\label{LapH}
\triangle^MH&=&\sum_{i=1}^m[e_ie_i(H)-(\nabla^M_{e_i}e_i)(H)]\;\;\;({\rm
\;and\;by\;Gauss\;formula})\\\notag
&=&\sum_{i=1}^m[e_ie_i(H)-(\nabla_{e_i}e_i)(H)+h(e_i,e_i)\,\xi\,(H)]\\\notag
&=&-\sum_{i=1}^ma_i^2k_i^2k_{i-1}^2k_m[(2-m)ff'f''+f^2f''']\\\notag&=&-(1-k_m^2)k_m[(2-m)ff'f''+f^2f'''],
\end{eqnarray}
where in obtaining the last equality we have used the identity
\begin{equation}
\sum_{i=1}^ma_i^2k_i^2k_{i-1}^2=(1-k_m^2)
\end{equation}
which can be proved by mathematical induction on $m\ge 2$.\\

Substituting (\ref{RiN}), (\ref{H}), (\ref{RiT}), (\ref{A2}),
(\ref{gradH}), (\ref{LapH}), and $A({\rm grad}_g\,H)=H\,{\rm
grad}_g\,H$ into biharmonic equation (\ref{eq1}) we conclude that
the isometric immersion $\varphi$ is biharmonic if and only if
\begin{equation}\label{Keq}
\begin{cases}
-(1-k_m^2)f^2f^{'''}-[3-m+(2m-3)k_m^2]ff^{'}f^{''}+m(1+k_m^2)(f')^3 =0,\\
( m-4)ff'f''a_i=0, \;\;\; i=1, 2, \ldots, m.
\end{cases}
\end{equation}
The second equation of (\ref{Keq}), and hence the system (\ref{Keq})
itself, can be solved by considering the following three
cases.  \\
{\bf Case 1.}  $f'=0$ (which implies $H=-k_mf'=0$) gives the trivial
solution. In this case, $\varphi$ is actually totally geodesic since its image is a hyperplane in a space that is homothetic to a Euclidean space.\\
{\bf Case 2.}  $a_iff''=0, i=1, 2, \ldots, m$ (which, together with
(\ref{gradH}), implies ${\rm grad}_g\,H=0$). In this case the first
equation of (\ref{Keq}), with the aid of $\triangle^MH=0$ and
(\ref{LapH}), can be reduced to
\begin{equation}\label{f4}
[1+(m-1)k_m^2]ff^{''}-m(1+k_m^2)(f^{'})^2=0.
\end{equation}
If $ff''=0$, then Equation (\ref{f4}) reduces to $f^{'}=0$, which
gives the trivial solution again, i.e., the hypersurfaces are
minimal. If $ff''\ne 0$, then all $a_i=0, i=1, 2, \ldots, m$, and
hence $k_m=1$. Thus, Equation (\ref{f4}) reduces to
$ff''-2(f^{'})^2=0$ which has solutions $f(z)=\frac{1}{Az+B}$, where $A, B$ are constants.\\
{\bf Case 3.} $m=4$. In this case, the biharmonic equation
(\ref{Keq}) reduces to
\begin{eqnarray}\notag
-(1-k_4^2)f^2f'''-(5k_4^2-1)ff'f''+4(1+k_4^2)(f')^3=0,
\end{eqnarray}
from which we obtain Equation (\ref{single}).\\

Summarizing the above results we obtain the theorem.
\end{proof}

\section{The generalized Chen's conjecture on biharmonic submanifolds is false }

\indent In this section, we will show that Equation (\ref{single})
has many solutions including counter examples to the generalized
Chen's conjecture.
\begin{lemma}\label{LM1}
Let $ A>0,\; B>0,\;c $ be constants,
$\mathbb{R}^5_{+}=\{(x_1,\ldots, x_4, z)\in \r^5: z>0\}$ be the
upper-half space, and $f:\mathbb{R}^5_{+}\longrightarrow \r,\;\;
f(x_1,\ldots, x_4,z)=f(z)=(Az+B)^{t}.$  Then, for any $t\in
(0,\;1/2)$ and any $(a_1, a_2, a_3, a_4)\in
S^3\left(\sqrt{\frac{2t}{1-2t}}\;\right)$, the isometric immersion
\begin{equation}\label{4D}
\varphi : \mathbb{R}^4\longrightarrow
\left(\mathbb{R}^5_{+},h=f^{-2}(z)[\sum_{i=1}^4{\rm d}{x_i}^{2}+{\rm
d}{z}^{2}]\right)
\end{equation}
 with $\varphi(x_1,\ldots,
x_4)=(x_1,\ldots, x_4,\sum_{i=1}^{4}a_ix_i+c)$ is proper biharmonic
into the conformally flat space.
\end{lemma}
\begin{proof}
We try to find special solutions of Equation (\ref{single}) which
have the form $f(z)=(Az+B)^{t}$. In this case, we have
$f'=tA(Az+B)^{t-1}$,\;$f''=t(t-1)A^2(Az+B)^{t-2}$,\;$f'''=t(t-1)(t-2)A^3(Az+B)^{t-3}$.
Substituting these into Equation (\ref{single}) and using the
assumption that $A, B > 0$ we have
\begin{equation*}
(t-1)(t-2)\sum_{i=1}^4a_i^2+(4-\sum_{i=1}^4a_i^2)t(t-1)-4t^2(2+\sum_{i=1}^4a_i^2)=0,
\end{equation*}
which is equivalent to
\begin{equation}\label {algE}
\sum_{i=1}^4a_i^2=\frac{2t}{1-2t}.
\end{equation}

Solving the inequality $\frac{2t}{1-2t}> 0$ we conclude that for any
$t\in (0,\;1/2)$ and $(a_1, a_2, a_3, a_4)\in
S^3\left(\sqrt{\frac{2t}{1-2t}}\;\right)$ will solve the equation
(\ref{algE}) and hence the biharmonic equation (\ref{single}).  From
this we obtain the lemma.
\end{proof}
\begin{example} Let $ A>0,\; B>0,\; c $ be constants and let $t=\frac{1}{6}$ and $(\frac{\sqrt{2}}{4},
\frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{4})\in
S^3\left(\frac{\sqrt{2}}{2}\;\right)$. Then, by Lemma \ref{LM1}, we
have a proper biharmonic isometric immersion
$\varphi:\mathbb{R}^4\longrightarrow
(\mathbb{R}^5_{+},h=(Az+B)^{-\frac{1}{3}}(\sum_{i=1}^4{\rm
d}{x_i}^{2}+{\rm d}{z}^{2}))$ with $\varphi(x_1,\ldots,
x_4)=(x_1,\ldots, x_4,\frac{\sqrt{2}}{4}(x_1+x_2+x_3+x_4)+c)$.
\end{example}

\begin{lemma}\label{LM2}
For constant $A>0,\; B>0$ and $t\in (0,1)$, the conformally flat
space $(\mathbb{R}_{+}^{5},h=(Az+B)^{-2t}(\sum_{i=1}^4{\rm
d}{x_i}^{2}+{\rm d}{z}^{2}))$ has negative sectional curvature.
\end{lemma}
\begin{proof}
Let $f(z)=(Az+B)^t$. Then, as in the proof of Theorem \ref{f1}, we
use ${\bar e}_i=f(z)\partial_i, i=1,\ldots, 5$ to denote the
orthonormal frame on
$(\mathbb{R}_{+}^{5},h=(Az+B)^{-2t}(\sum_{i=1}^4{\rm
d}{x_i}^{2}+{\rm d}{z}^{2}))$. Let $P$ be a plane section at any
point and suppose that $P$ is spanned by an orthonormal basis $X,
Y$. Then, we have $X=\sum_{i=1}^{5}a_i \bar e_i, Y=\sum_{i=1}^{5}b_i
\bar e_i$. Using sectional  curvature relation (\ref{QL}) and the
fact that the sectional curvature $\bar K(p)$ of
$(\mathbb{R}_{+}^{5},\bar h=\sum_{i=1}^4{\rm d}{x_i}^{2}+{\rm
d}{z}^{2})$ vanishes identically we find the sectional curvature of
the conformally flat space to be
\begin{eqnarray*}\notag
K(P)&=&(h(\nabla_{X}\nabla \sigma,X)+h(\nabla_{Y}\nabla
\sigma,Y))+(|\nabla\sigma|^{2}-(X\sigma )^{2}-(Y\sigma)^{2})\\\notag
&=&X(X\sigma)+Y(Y\sigma)-(\nabla_XX)(\sigma)-(\nabla_YY)(\sigma)+(|\nabla\sigma|^{2}-(X\sigma
)^{2}-(Y\sigma)^{2}),
\end{eqnarray*}
where $\sigma=\ln\,f(z)$. A straightforward computation gives

\begin{eqnarray*}\notag
X\sigma&=&\sum_{i=1}^{5}a_i \bar e_i(\sigma)=a_5f',\\\notag
X(X\sigma)&=&\sum_{i=1}^{5}a_i \bar e_i(a_5f')=\sum_{i=1}^{5}a_i
\bar e_i(a_5)f'+a_5^2ff'',
\\\notag
\nabla_XX&=&\nabla_{\sum_{i=1}^{5}a_i \bar e_i}(\sum_{j=1}^{5}a_j
\bar e_j)\\\notag &=&\sum_{i=1}^{5}a_i\bar e_i(\sum_{j=1}^{5}a_j)
\bar e_j+\sum_{i=1}^{4}a_i^2f'\bar e_5-\sum_{i=1}^{4}a_ia_5f'\bar
e_i\\\notag (\nabla_XX)(\sigma)&=&\sum_{i=1}^{5}a_i\bar
e_i(a_5)f'+\sum_{i=1}^{4}a_i^2(f')^2.
\\\notag
X(X\sigma)-(\nabla_XX)(\sigma)&=&a_5^2ff''-\sum_{i=1}^{4}a_i^2(f')^2.
\end{eqnarray*}
Similarly, we have
\begin{eqnarray*}\notag
Y(Y\sigma)&=&b_5f', (\nabla_YY)(\sigma)=\sum_{i=1}^{5}b_i\bar
e_i(b_5)f'+\sum_{i=1}^{4}b_i^2(f')^2\\\notag
Y(Y\sigma)-(\nabla_YY)(\sigma)&=&b_5^2ff''-\sum_{i=1}^{4}b_i^2(f')^2,
\end{eqnarray*}
from which we have
\begin{eqnarray*}\notag
K(P)&=&X(X\sigma)+Y(Y\sigma)-(\nabla_XX)(\sigma)-(\nabla_YY)(\sigma)+(|\nabla\sigma|^{2}-(X\sigma
)^{2}-(Y\sigma)^{2})\\\notag &=&(a_5^2+b_5^2)ff''-(f')^2.
\end{eqnarray*}

For $f=(Az+B)^{t}$, we have
$f'=tA(Az+B)^{t-1},f''=t(t-1)A^2(Az+B)^{t-2}$, so
\begin{eqnarray*}\notag
K(P)&=&(a_5^2+b_5^2)ff''-(f')^2\\\notag &=&
A^2(Az+B)^{t-2}[(a_5^2+b_5^2)t(t-1)-t^2],
\end{eqnarray*}
which is strictly negative since $[(a_5^2+b_5^2)t(t-1)-t^2]<0$
 for $0<t<1$, and $A^2(Az+B)^{t-2}>0$ for $z>0$. From this we obtain the
 Lemma.
\end{proof}
Combining Lemma \ref{LM1} and Lemma \ref{LM2} we have
\begin{theorem}\label{MT2}
Let $ A>0,\; B>0,\;c $ be constants,
$\mathbb{R}^5_{+}=\{(x_1,\ldots, x_4, z)\in \r^5: z>0\}$ be the
upper-half space, and $f:\mathbb{R}^5_{+}\longrightarrow \r,\;\;
f(z)=(Az+B)^{t}$ . Then, for any $t\in (0,\;1/2)$ and any $(a_1,
a_2, a_3, a_4)\in S^3\left(\sqrt{\frac{2t}{1-2t}}\;\right)$, the
isometric immersion
\begin{equation}\label{5D}
\varphi : \mathbb{R}^4\longrightarrow
\left(\mathbb{R}^5_{+},h=f^{-2}(z)[\sum_{i=1}^4{\rm d}{x_i}^{2}+{\rm
d}{z}^{2}]\right)
\end{equation}
 with $\varphi(x_1,\ldots,
x_4)=(x_1,\ldots, x_4,\sum_{i=1}^{4}a_ix_i+c)$ gives a proper
biharmonic hypersurface into the conformally flat space with
strictly negative sectional curvature. These provide infinitely many
counter examples to the generalized Chen's conjecture on biharmonic
submanifolds.
\end{theorem}
The following corollary can be used to construct proper biharmonic
submanifolds of any codimension in a nonpositively curved manifold.
\begin{corollary}
For any positive integer $k$, there exists a proper biharmonic
submanifold of codimension $k$ in a nonpositively curved space.
Thus, the generalized Chen's conjecture is false.
\end{corollary}
\begin{proof}
Let \begin{equation}\label{4D} \varphi : \mathbb{R}^4\longrightarrow
\left(\mathbb{R}^5_{+},h=f^{-2}(z)[\sum_{i=1}^4{\rm d}{x_i}^{2}+{\rm
d}{z}^{2}]\right)
\end{equation}
 with $\varphi(x_1,\ldots,
x_4)=(x_1,\ldots, x_4,\sum_{i=1}^{4}a_ix_i+c)$ be one of the proper
biharmonic hypersurface given in Theorem \ref{MT2} and $\psi:
\mathbb{R}^n\longrightarrow \mathbb{R}^n\times
\mathbb{R}^{k-1}\equiv (\mathbb{R}^{n+k-1}, h_0)$ with
$\psi(y)=(y,0)$ be the totally geodesic embedding of a subspace into
a Euclidean space. Then, the isometric embedding $\phi:
\mathbb{R}^4\times \mathbb{R}^n\longrightarrow
\left(\mathbb{R}^5_{+}\times \mathbb{R}^{n+k-1}, h+h_0\right)$ with
$\phi(x,y)=(\varphi(x), \psi(y))$ gives a submanifold of codimension
$k$. Since $\phi$ is biharmonic with respect to each variable
separately and it is proper biharmonic with respect to $x$-variable
by Theorem \ref{MT2}, we can use Proposition 2.1 in \cite{Ou2} to
conclude that $\phi$ is a proper biharmonic embedding. Thus, the
image of $\phi$ provides a proper biharmonic submanifold of
codimension $k$. Since, by Lemma \ref{LM2}, the conformally flat
space $(\mathbb{R}_{+}^{5},h=(Az+B)^{-2t}(\sum_{i=1}^4{\rm
d}{x_i}^{2}+{\rm d}{z}^{2}))$ has negative sectional curvature and
the Euclidean space $(\mathbb{R}^{n+k-1}, h_0)$ has zero curvature,
their product $\left(\mathbb{R}^5_{+}\times \mathbb{R}^{n+k-1},
h+h_0\right)$ gives a space of nonpositive curvature. Thus, we
obtain the corollary.
\end{proof}
\begin{ack}
Both authors are grateful to Cezar Oniciuc for a careful reading and
checking the computations in the paper. We especially want to thank
him for pointing out a mistake in the computation of the Laplacian
of the mean curvature of the hypersurface in the previous version.
\end{ack}