\begin{document}

\title{Topological Aspects of Wave Propagation}


\author{Carlos Valero Valdes\\
Departamento de Matematicas Aplicadas y Sistemas\\
Universidad Autonoma Metropolitana-Cuajimalpa\\
Mexico, D.F 01120, México}


\date{27 March 2014}

\maketitle
\global\long\def\CC{\mathbb{C}}


\global\long\def\RR{\mathbb{R}}


\global\long\def\map{\rightarrow}


\global\long\def\mult{\mathcal{M}}


\global\long\def\EE{\mathcal{E}}


\global\long\def\OO{\mathcal{O}}


\global\long\def\FH{\mathcal{F}}


\global\long\def\CH{\mathcal{H}}


\global\long\def\tangent{T}


\global\long\def\cotangent{\tangent^{*}}


\global\long\def\conormal{\mathcal{C}}


\global\long\def\SS{\mathcal{S}}


\global\long\def\KK{\mathcal{K}}


\global\long\def\NN{\mathcal{N}}


\global\long\def\sym#1{\hbox{S}^{2}#1}


\global\long\def\symz#1{\hbox{S}_{0}^{2}#1}


\global\long\def\proj#1{\hbox{P}#1}


\global\long\def\SO{\hbox{SO}}


\global\long\def\GL{\hbox{GL}}


\global\long\def\U{\hbox{U}}


\global\long\def\tr{\hbox{tr}}


\global\long\def\ZZ{\mathbb{Z}}


\global\long\def\der#1#2{\frac{\partial#1}{\partial#2}}


\global\long\def\covder{\hbox{D}}


\global\long\def\diff{d}


\global\long\def\dero#1{\frac{\partial}{\partial#1}}

\begin{abstract}
In the context of wave propagation on a manifold $X$, the characteristic
functions are real valued functions on cotangent bundle of $X$ that
specify the allowable phase velocities of the waves. For certain classes
of differential operators (e.g Maxwell\textasciiacute{}s Equations)
the associated characteristic functions have singularities. These
singularities account for phenomena like conical refraction and the
transformation of longitudinal waves into transversal ones (or viceversa).
For a specific class of differential operators on surface, we prove
that the singularities of the characteristic functions can be accounted
from purely topological considerations. We also prove that there is
a natural way to desingularsize the characteristic functions, and
observe that this fact and Morse Theory establishes a specific connection
between singularities and critical points of these functions. The
relation between characteristic functions and differential operators
is obtained through what is known as the symbol of the operator. We
establish a connection between these symbols and holomorphic vector
fields, which will provide us with symbols whose characteristic functions
have interesting singularity sets. 
\end{abstract}

\section{Introduction}

In an appropriate context, certain aspects of wave propagation on
a Riemannian manifold $X$ can be described by means of what are known
as \emph{characteristic functions} $\{\lambda_{i}:S(\cotangent X)\rightarrow\RR\}_{1\leq i\leq k}$,
where $S(\cotangent X)=\{\xi\in\cotangent X|<\xi,\xi>=1\}$ and the
metric in $\cotangent X$ is the one induced by that in $\tangent X$.
For a covector $\xi\in S(\cotangent X)$, the set $\{\lambda_{i}(\xi)\}_{1\le i\leq k}$
consists of the set the \emph{allowable wave speeds} in the direction
of $\xi$. For certain systems of hyperbolic differential operators
it is a well know fact that the associated characteristic functions
have singular points (by which we mean that they are not differentiable
at these points). For example, each of the characteristic functions
associated to Maxwell's equations in biaxial crystals have four singularities.
These singularities account for the phenomenon of conical refraction
discovered by Hamilton (see \cite[pg. 571]{kn:born}). For hyperbolic
partial differential equations which correspond to generic non-homogeneous
media the existence of singularities in the characteristic functions
account for a phenomena known as wave transformation, in which longitudinal
waves convert into transversal waves or vice-versa (see \cite[Ch. 8]{kn:arnold}
and \cite{kn:braam}). It is then of interest to find methods for
detecting singularities of the characteristic functions associated
to general linear hyperbolic operators of the form $P:C^{\infty}(E)\rightarrow C^{\infty}(F)$,
where $C^{\infty}(E)$ and $C^{\infty}(F)$ are the spaces of sections
of the vector bundles $E$ and $F$ over space-time $X\times\RR$.
For certain constant coefficients differential operators it is well
know that the singularities of the characteristic functions occur
for topological reasons (see for example \cite{kn:john,kn:lax}).
We are interested in generalizing these results to the case of operators
of vector bundles. More concretely, we will consider the following
problems.
\begin{problem}
For a hyperbolic differential operator $P:C^{\infty}(E)\rightarrow C^{\infty}(F)$,
can we relate the existence of singularities of its characteristic
functions to the characteristic classes of $E$ and $F$?
\begin{problem}
Is there a ``natural'' way desingularise the characteristic functions? 
\end{problem}
\end{problem}
As we have explained before, the points at which the characteristic
function are non-smooth explain interesting physical phenomena. Other
points of interests are the critical points of the characteristic
functions, i.e points at which their derivative vanish. Assuming $X$
is compact, among the critical values of the characteristic functions
are the extremum values (i.e maximum and minimum) of wave fronts velocities,
which are of physical interest. By giving answers to the above problems
and using Morse theory we will be able to establish a specific connection
between of singularities and critical points of the characteristic
functions.

The purpose of this paper is to solve the the above problems in a
very specific case: that of fist order hyperbolic operators with symmetric
symbols, and acting on a rank two orientable vector bundle over a
surface. The reason for restricting to this case is that it already
provides us with interesting mathematics. Furthermore, in this case
we will be able to provide complete answers to the above two problems,
and this will pave the way for the more general cases. The restriction
to operators with symmetric symbols can be justified on the grounds
that they appear in most of the physical problems derived from variational
principles (see \cite[Ch. 8]{kn:arnold}). 

For the rest of the paper we will let $X$ stand for a riemmanian
compact, connected and oriented surface (whose metric we will denote
by $<,>_{X}$). The cotangent bundle $\cotangent X$ inherits a metric
from that of $\tangent X$, defined by letting the map $v\mapsto<v,\cdot>$
be an isometry . We will use the same notation for the metric in $\tangent X$
and that in $\cotangent X$. We will also assume that $F$ is an orientable
rank two vector bundle over $X$ with a metric (that we will denote
by $<,>_{F}$). The differential operators that interest us have the
form 
\[
P=\left(\frac{\partial}{\partial t}+L\right),
\]
where $L:C^{\infty}(F)\rightarrow C^{\infty}(F)$ is a differential
operator of order one with symmetric symbol. The symbol associated
to $L$ is an homomorphism $\sigma:\cotangent X\rightarrow\sym F$
from the bundle $\cotangent X$ to the bundle of symmetric linear
operators in $F$ (see Section 2). The outline of the paper is as
follows:

\emph{In} \emph{section 2}: We recall how to construct the characteristic
function associated to the differential operator $P$ by means of
the symbol of $L$. 

\emph{In} \emph{section }3: We find topological obstructions to the
the existence of non-singular characteristic functions (see Proposition
\ref{pro:SingCharFunctions} and its Corollary). We spend some time
proving that two apparently unrelated notions of genericity for symbols
are actually the same (see Propositions \ref{pro:TransversalityEquivalence}
and its Corollaries), and use this result to show that generically
the set of singularities of the characteristic functions form torus
knots on $S(\cotangent X)$. 

\emph{In section }4: We show how to desingularise the characteristic
functions associated to a symbol and use this to relate the singularities
and critical points of these functions (see Proposition \ref{pro:DesingularizationNearMultSet}).

\emph{In section} 5: We show how to construct symbols from holomorphic
vector fields, and use this to explore what kind of torus knots can
appear as singularities of the characteristic functions.

\emph{Notation}. Unless otherwise stated we will use the following
conventions. For any bundle $E$ over a manifold $M$ we will always
denote the projection map from $E$ to $M$ by $\pi:E\rightarrow M$,
where it will be clear from the context which bundle does $\pi$ has
as its domain. If $E$ is a bundle with a metric we will denote this
metric by $<,>_{E}$ but for the case where $E=\tangent X$  in which
case we will use the notation $<,>_{X}$. For elements $v\in E$ and
$v\in\tangent X$ we will their norms as $|v|_{E}$ and $|v|_{X}$.
The metric in $\tangent X$ induces a metric in $\cotangent X$ which
we will also denote by $<,>_{X}$.


\section{Motivation - The Characteristic Equation}

For a linear differential operator of degree one $L:C^{\infty}(F)\rightarrow C^{\infty}(F)$
, consider the asymptotic partial differential equation 
\[
\left(\frac{\partial}{\partial t}+L\right)\left(u_{0}e^{is\varphi(x,t)}\right)=0\hbox{\, as\,\,}s\mapsto\infty,
\]
where $u_{0}$ is a section of $F$ and $\varphi$ is a smooth function
in $X\times\RR$ . The above equation leads to the the characteristic
equation (see \cite[pg. 31]{kn:guillemin} ) 
\begin{equation}
\det\left(\frac{\partial\varphi}{\partial t}I+\sigma(d_{x}\varphi)\right)=0,\label{eq:Characteristic Equation Matricial}
\end{equation}
where $I:F\rightarrow F$ is the identity morphism and $\sigma:\cotangent X\rightarrow\hbox{End}(F)$
is the principal symbol of $L$. Since $L$ is a differential operator
of degree one, for every $x\in X$ we have that $\sigma$ is a linear
map from $\cotangent_{x}X$ to $\hbox{End}(F_{x})$. We will assume
that for all $\xi\in\cotangent X$ the endomorphism $\sigma(\xi)$
is a symmetric operator with respect to the metric in $F$, i.e for
all $x\in X$ and $\xi\in\cotangent_{x}X$ we have that
\[
<\sigma(\xi)v,w>_{F}=<v,\sigma(\xi)w>_{F}\hbox{\,\, for\, all\,}v,w\in F_{x}.
\]
When the above condition holds we will say that $\sigma$ is symmetric
(being understood that the symmetry is with respect to the metric
in $F$). If we denote the bundle of symmetric endomorphisms of $F$
by $\sym F$, then $\sigma$ is a map of the form $\sigma:\cotangent X\rightarrow\sym F$.
In this case, the characteristic equation (\ref{eq:Characteristic Equation Matricial})
is equivalent to the equations 
\begin{equation}
\frac{\partial\varphi}{\partial t}(x,t)+\lambda_{\sigma,i}(d_{x}\varphi(x,t))=0\hbox{\, for\,\,}i=1,2,\label{eq:Characteristic Equation Scalar}
\end{equation}
where the functions $\lambda_{\sigma,1},\lambda_{\sigma,2}:\cotangent X\rightarrow\RR$
are the eigenvalue functions of $\sigma$, i.e for each $\xi\in\cotangent X$
we have that $\lambda_{\sigma,1}(\xi)$ and $\lambda_{\sigma,2}(\xi)$
are the two real eigenvalues of the symmetric operator $\sigma(\xi)$.
From now on we will assume that we have chosen $\lambda_{\sigma,1}$
and $\lambda_{\sigma,2}$ so that $\lambda_{\sigma,1}\leq\lambda_{\sigma,2}$.
The functions $\lambda_{\sigma,1}$ and $\lambda_{\sigma,2}$ are
positively homogeneous of degree one in the fibre variable, i.e $\lambda_{\sigma,i}(r\xi)=r\lambda_{\sigma,i}(\xi)$
for $r>0$, and their physical interpretation is as follows. Consider
a time-dependent curve $x:\RR\rightarrow X$ satisfying $\varphi(x(t),t)=\hbox{constant}$,
so that
\[
d_{x}\varphi(x(t),t)\dot{x}(t)+\frac{\partial\varphi}{\partial t}(x(t),t)=0,
\]
where $\dot{x}$ is the derivative of $x$ with respect to to $t$.
From this last equation and (\ref{eq:Characteristic Equation Scalar})
we obtain
\[
d_{x}\varphi(x(t),t)\dot{x}(t)=\lambda_{\sigma,i}(d_{x}\varphi(x(t),t)).
\]
We conclude that for a point $x\in X$ traveling on the wave front
\begin{equation}
W_{t}^{c}=\{x\in X|\varphi(x,t)=c\}\label{eq:Wave Front}
\end{equation}
the component of $\dot{x}(t)$ normal to the front (know as \emph{phase
velocity}) is equal to $\lambda_{\sigma,i}(d_{x}\varphi(x,t)/|d_{x}\varphi(x,t)|_{X})$
(assuming $d_{x}\varphi(x,t)\not=0$). For this reason, we will refer
to $\lambda_{\sigma,1}$ and $\lambda_{\sigma,2}$ as the \emph{characteristic
velocity functions} of $\sigma$. These functions can become non-smooth
at certain points.

\begin{figure}
\includegraphics[scale=0.5]{FresnelSigmaA}

\caption{\label{fig:Fresnel-surface-of}Fresnel surface of a bi-axial crystal}
\end{figure}

\begin{example}
\label{MaxwellFresnel}For Maxwells equations on a crystal (see \cite[pg. 571]{kn:born})
the symbol $\sigma:(\RR^{3})^{*}\rightarrow\hbox{End}(\RR^{6})$ in
the characteristic equation \ref{eq:Characteristic Equation Matricial}
is given by 
\[
\sigma(\xi)(E,B)=(\xi\times B,-\xi\times(\epsilon^{-1}E))
\]
where $E,B\in\RR^{3}$ correspond to the electric and magnetic fields,
the dielectric tensor $\epsilon$ is a $3\times3$ positive definite
symmetric matrix which describes the properties of the crystal, and
$\times$ is the standard cross product in $\RR^{3}$ (notice we have
identified $(\RR^{3})^{*}$ with $\RR^{3})$. Figure \ref{fig:Fresnel-surface-of}
shows the Fresnel surface
\[
\mathcal{F}_{\epsilon}=\{\tau\xi|\xi\in S^{2}\hbox{\, and\,}\det(\tau I+\sigma(\xi))=0\}\hbox{\, where\,\,}S^{2}=\{\xi\in(\RR^{3})^{*}|<\xi,\xi>=1\},
\]
for the case in which $\epsilon$ has non-multiple eigenvalues. In
this case, the crystal associated to $\epsilon$ is said to be bi-axial
(e.g aragonite) and the four singularities of $\mathcal{F}_{\epsilon}$
account for a phenomenon known as conical refraction.
\end{example}

\section{Symbols and Their Multiplicities}


\subsection{Preliminaries}

From now on and for rest of the paper we will assume that $F$ is
an orientable real vector bundle over a riemmanian compact, connected
and oriented surface $X$. Motivated by the discussion in Section
2, we introduce our basic objects of study.
\begin{defn}
A\emph{ symbol} on a bundle $F\rightarrow X$ is a morphism $\sigma:\cotangent X\rightarrow\sym F$.
\end{defn}
The bundle $\sym F$ has a riemmanian metric given by
\[
(A,B)=\frac{1}{2}\tr(AB)\hbox{\, for\, all\,}x\in X\hbox{\, and\,}A,B\in\sym{F_{x}}.
\]
This metric induces a norm function on $\sym F$, which for an element
$A\in\sym F$ we will denote by $||A||$. 
\begin{lem}
\label{lem:EigenvaluesFormula}The eigenvalue functions of a symbol
$\sigma$ are given by the formulas
\begin{eqnarray}
\lambda_{\sigma,1} & = & \frac{1}{2}\tr(\sigma)-||\sigma_{0}||,\label{eq:LamdasFormula}\\
\lambda_{\sigma,2} & = & \frac{1}{2}\tr(\sigma)+||\sigma_{0}||\label{eq:LambdasFormula2}
\end{eqnarray}
where $\sigma_{0}=\sigma-\frac{1}{2}\tr(\sigma)I$ is the traceless
part of $\sigma.$ \end{lem}
\begin{proof}
Let $\xi\in\cotangent X$. A direct computation shows that the eigenvalues
of $\sigma_{0}(\xi)$ are $-||\sigma_{0}(\xi)||$ and $||\sigma_{0}(\xi)||$.
The eigenvalues of $\sigma(\xi)=\frac{1}{2}\tr(\sigma(\xi))I+\sigma_{0}(\xi)$
are the solutions of the equation
\[
\det\left(\left(\lambda-\frac{1}{2}\tr(\sigma(x))\right)I-\sigma_{0}(x)\right)=0,
\]
and hence the result.
\end{proof}
If we define spherisation $S(\cotangent X)$ of $\cotangent X$ as
the set
\[
S(\cotangent X)=\{\xi\in\cotangent X|<\xi,\xi>_{X}=1\},
\]
then then both $\sigma$ and its eigenvalue functions $\lambda_{\sigma,1}$
and $\lambda_{\sigma,2}$ are completely determined by the values
they take on $S(\cotangent X)$.
\begin{defn}
The multiplicity set of a symbol $\sigma$ is the set
\[
\mult_{\sigma}=\{\xi\in S(\cotangent X)|\lambda_{\sigma,1}(\xi)=\lambda_{\sigma,2}(\xi)\}.
\]

\end{defn}
By lemma \ref{lem:EigenvaluesFormula} we have that
\[
\mult_{\sigma}=\{\xi\in S(\cotangent X)|\sigma_{0}(\xi)=0\}.
\]
We will let $\symz F$ stand for the bundle of traceless elements
in $\sym F$. The following result characterize (generically) the
multiplicity set of $\sigma$ as the set at which the eigenvalue functions
are non-smooth.
\begin{prop}
\label{pro:singularitiesequationforfresnel}Consider symbol $\sigma$
and let $\sigma_{0}$ be its traceless part. If $\sigma_{0}|S(\cotangent X)$
is transversal to the zero section of $\symz F$ then functions $\lambda_{\sigma,1}|S(\cotangent X)$
and $\lambda_{\sigma,2}|S(\cotangent X)$ are non-smooth over the
set $\mult_{\sigma}$.\end{prop}
\begin{proof}
From the transversality assumption on $\sigma_{0}$ it follows that
function $\xi\mapsto||\sigma_{0}(\xi)||$ is non-differentiable for
points $\xi\in S(\cotangent X)$ such that $\sigma_{0}(\xi)=0$, i.e
over points $\xi$ in the set $\mult_{\sigma}$. The result then follows
from formulas \ref{eq:LamdasFormula} and \ref{eq:LambdasFormula2}.
\end{proof}

\subsection{The Multiplicity Set}

In this section we will find obstructions for $\mult_{\sigma}$ to
be empty (see the next Proposition and its Corollary) and we will
see that if $\mult_{\sigma}$ is non-empty then it is generically
the intersection of $S(\cotangent X)$ with line bundle $\KK_{\sigma}$
over a one dimensional sub-manifold $\SS_{\sigma}$ of $X$ (see the
discussion after Corollary \ref{cor:SingCharFunctions}).
\begin{prop}
\label{pro:SingCharFunctions}Consider a symbol $\sigma$ on the bundle
$F\rightarrow X$, and let $e(F),e(\tangent X)\in H^{2}(X,\RR)$ denote
the Euler classes of $F$ and $\tangent X$, respectively. If $e(\tangent X)-2e(F)\not=0$
in $H^{2}(X,\RR)$ then $\mult_{\sigma}\not=\emptyset$.\end{prop}
\begin{proof}
Consider a rotation matrix 
\[
R_{\theta}=\left(\begin{array}{cc}
\cos(\theta) & -\sin(\theta)\\
\sin(\theta) & \cos(\theta)
\end{array}\right)\in\SO(2)
\]
and
\[
A=\left(\begin{array}{cc}
a & b\\
b & -a
\end{array}\right)\in\symz{\RR^{2}}.
\]
It easy to see that 
\[
R_{\theta}AR_{\theta}^{T}=\left(\begin{array}{cc}
p & q\\
q & -p
\end{array}\right),
\]
where
\[
\left(\begin{array}{c}
p\\
q
\end{array}\right)=\left(\begin{array}{cc}
\cos(2\theta) & -\sin(2\theta)\\
\sin(2\theta) & \cos(2\theta)
\end{array}\right)\left(\begin{array}{c}
a\\
b
\end{array}\right).
\]
This means that the transition functions of the bundle $\symz F$
are the same as those of the bundle $F\otimes_{\CC}F$, so these bundles
must be isomorphic. If $\mult_{\sigma}$ were empty then we would
have that $\sigma_{0}(\xi)\not=0$ for all $\xi\in S(\cotangent X)$,
which means that $\sigma_{0}$ is an isomorphism between $\cotangent X$
and $\symz{(\tangent X)}$. Hence $\tangent X$ and $F\otimes_{\CC}F$
are isomorphic, which implies that.
\[
e(\tangent X)=e(F\otimes_{\CC}F)=2e(F).
\]
\end{proof}
\begin{cor}
\label{cor:SingCharFunctions}If the genus of $X$ is different from
one, then the multiplicity set of any symbol $\sigma$ on $\tangent X$
is non-empty.\end{cor}
\begin{proof}
Follows from the previous proposition and the fact that 
\[
\int_{X}(e(\tangent X)-2e(TX))=2(g(X)-1)\not=0,
\]
where $g(X)\not=1$ stands for the genus of $X$.\end{proof}
\begin{defn}
We will say that a symbol $\sigma$ over $F\rightarrow X$ is traceless
if $\sigma(\xi)\in\symz F$ for all $\xi\in\cotangent X$.
\end{defn}
We will now give a more detailed description of the multiplicity set.
The symbol $\sigma$ can be considered as a section of the bundle
$\hbox{Hom}(\cotangent X,\hbox{End}(F))$, and we will denote such
a section as $\hat{\sigma}:X\rightarrow\hbox{Hom}(\cotangent X,\hbox{End}(F))$
\begin{defn}
Let $\sigma$ by a traceless symbol on $F\rightarrow X$. We define
the singular set $\SS_{\sigma}$ and kernel bundle $\KK_{\sigma}$
by 
\begin{eqnarray*}
\SS_{\sigma} & = & \{x\in X|\dim(\ker(\hat{\sigma}(x)))>0\},\\
\KK_{\sigma} & = & \{\xi\in\cotangent_{x}X|x\in\SS_{\sigma}\hbox{\, and\,}\sigma(\xi)=0\}.
\end{eqnarray*}
For a general (not necessarily traceless) symbol $\sigma$ we define
$\SS_{\sigma}=\SS_{\sigma_{0}}$ and $\KK_{\sigma}=\KK_{\sigma_{0}}$.
\end{defn}
Since we have that
\[
\mult_{\sigma}=S(\cotangent X)\cap\KK_{\sigma_{0}},
\]
it is then of interest to find conditions under which $\SS_{\sigma}$
is a smooth submanifold of $X$ and $\KK_{\sigma}$ is a smooth bundle
over $\SS_{\sigma}$. To study this problem, observe that $\hbox{Hom}(\cotangent X,\symz F$)
stratifies as follows
\[
\hbox{Hom}(\cotangent X,\symz F)=\bigcup_{i=0}^{2}[\hbox{Hom}(\cotangent X,\symz F)]_{i}
\]
where 
\[
[\hbox{Hom}(\cotangent X,\symz F)]_{i}=\{A\in\hbox{Hom}(\cotangent X,\symz F)|\dim(\ker(A)=i\}.
\]
We then have that for a traceless symmetric symbol $\sigma$ 
\[
\SS_{\sigma}=\hat{\sigma}^{-1}\left(\bigcup_{i=1}^{2}[\hbox{Hom}(\cotangent X,\symz F]_{i}\right).
\]
The manifold $S(\cotangent X)$ has dimension 3 and $[\hbox{Hom}(\cotangent X,\symz F]_{i}$
is a smooth submanifold of $\hbox{Hom}(\cotangent X,\symz F$ of codimension
$i^{2}$ (see \cite[pg. 28]{kn:arnold4}). Hence, tranversality of
$\hat{\sigma}$ to $[\hbox{Hom}(\cotangent X,\symz F]_{2}$ and $[\hbox{Hom}(\cotangent X,\symz F]_{1}$
imply that 
\[
\hat{\sigma}^{-1}\left([\hbox{Hom}(\cotangent X,\symz F]_{2}\right)=\emptyset
\]
and that 
\[
\SS_{\sigma}=\hat{\sigma}^{-1}\left([\hbox{Hom}(\cotangent X,\symz F]_{1}\right)
\]
is a smooth one dimensional submanifold of $X$. 
\begin{prop}
\label{pro:TransversalityEquivalence}If $\sigma$ is a traceless
symmetric symbol $\sigma$ then $\hat{\sigma}$ is transversal to
$[\hbox{Hom}(\cotangent X,\symz{(\tangent X)}]_{1}$ and $[\hbox{Hom}(\cotangent X,\symz{(\tangent X)}]_{2}$
iff $\sigma|S(\cotangent X)$ is transversal to the zero section of
$\symz F$.\end{prop}
\begin{proof}
The proof is local. We can locally trivialize the bundles $\cotangent X$
and $F$ by choosing local orthonormal frames over appropriate open
sets $U\subset X$. In these trivializations the map $\sigma$ can
be written in the form
\[
\sigma(x,\xi)=M(x)\xi
\]
where 
\[
M(x)=\left(\begin{array}{cc}
a_{1}(x) & b_{1}(x)\\
a_{2}(x) & b_{2}(x)
\end{array}\right)\hbox{\, and\,\,\,}\xi=\left(\begin{array}{c}
\xi_{1}\\
\xi_{2}
\end{array}\right).
\]
We have that
\[
d\sigma_{0}(x,\xi)(\dot{x},\dot{\xi})=(dM(x)\dot{x})\xi+M(x)\dot{\xi}.
\]
The condition $\sigma_{0}(x,\xi)=0$ (for $\xi_{1}^{2}+\xi_{2}^{2}=1$)
is equivalent to the existence scalars $\alpha_{1},\alpha_{2}\in\RR$
such that 
\begin{equation}
(a_{i},b_{i})=\alpha_{i}(-\xi_{2},\xi_{1})\label{eq:sigma0EqualsZeroEquation}
\end{equation}
The condition that $\xi$ is unitary implies that $\dot{\xi}=\dot{\beta}(-\xi_{2},\xi_{1})$
for a scalar $\dot{\beta}\in\RR$. Using these identities we obtain
\[
d\sigma_{0}(x,\xi)(\dot{x},\dot{\xi})=\left(\begin{array}{c}
\dot{\gamma}_{1}\\
\dot{\gamma}_{2}
\end{array}\right)+\dot{\beta}\left(\begin{array}{c}
\alpha_{1}\\
\alpha_{2}
\end{array}\right)
\]
where
\[
dM(x)\dot{x}=\left(\begin{array}{cc}
\dot{a}_{1} & \dot{b}_{1}\\
\dot{a}_{2} & \dot{b}_{2}
\end{array}\right)\hbox{\, and\,\,}\dot{\gamma}_{i}=\dot{a}_{i}\xi_{1}+\dot{b}_{i}\xi_{2}.
\]
We conclude that the transversality condition $\sigma_{0}$ means
that there exists a $\dot{x}\in\RR^{2}$ such that 
\begin{equation}
\det\left(\begin{array}{cc}
\dot{\gamma}_{1} & \dot{\gamma}_{2}\\
\alpha_{1} & \alpha_{2}
\end{array}\right)\not=0.\label{eq:TransvesalityEqsForSigma0}
\end{equation}
Let $f=\det(M)=a_{1}b_{2}-a_{2}b_{1}$. The condition $\hat{\sigma}_{0}(x)\in\cup_{i=1}^{2}[\hbox{Hom}(\cotangent X,\symz{(\tangent X)}]_{i}$
is equivalent to the condition $f(x)=0$. The transversality conditions
on $\hat{\sigma}_{0}$ are equivalent to $0\in\RR$ being a regular
value of $f$ , i.e for each point $x\in f^{-1}(0)$ there must exists
$\dot{x}$ such that 
\[
df(x)\dot{x}=\dot{a}_{1}b_{2}+a_{1}\dot{b}_{2}-\dot{a}_{2}b_{1}-a_{2}\dot{b}_{1}\not=0
\]
If $f(x)=0$ then we must have that $\ker(M(x))>0.$ This means that
there exists $(\xi_{1},\xi_{2})$ such that $\xi_{1}^{2}+\xi_{2}^{2}=1$
and such that (\ref{eq:sigma0EqualsZeroEquation}) holds. A simple
computation shows that
\[
df(x)\dot{x}=\det\left(\begin{array}{cc}
\dot{\gamma}_{1} & \dot{\gamma}_{2}\\
\alpha_{1} & \alpha_{2}
\end{array}\right).
\]
Hence, the condition $df(x)\dot{x}\not=0$ is the same as (\ref{eq:TransvesalityEqsForSigma0}).
\end{proof}
In the space of traceless symbols the condition of $\sigma|S(\cotangent X)$
being transversal to the zero section of $\symz F$ is clearly a stable
condition, but not so clearly a generic condition since $\sigma$
lives within the class of maps from $\cotangent X$ to $\symz F$
that are morphisms (i.e linear maps on each fiber). 
\begin{cor}
\label{cor:GenericitySymbol0}In the space of traceless symbols the
condition of $\sigma|S(\cotangent X)$ being transversal to the zero
section of $\symz F$ is a generic condition.\end{cor}
\begin{proof}
That the condition is open follows from the general results regarding
the stability of the transversality of a map to a given smooth manifold.
That the condition is dense follows from Proposition \ref{pro:TransversalityEquivalence}
and the density of the condition: $\hat{\sigma}_{0}$ is transversal
to the stratified manifold $\cup_{i=1}^{2}$$[\hbox{Hom}(\cotangent X,\symz{(\tangent X)}]_{i}$
(see \cite[Ch. 2]{kn:arnold4}).\end{proof}
\begin{cor}
\label{cor:MultLineBundle}If $\sigma$ is a traceless symbol such
that $\sigma|S(\cotangent X)$ is transversal to the zero section
of $\symz F$, we have that $\SS_{\sigma}$ is a smooth one dimensional
submanifold of $X$ and $\KK_{\sigma}$ is a smooth real line bundle
over $\SS_{\sigma}$.\end{cor}
\begin{proof}
The smoothness of $\SS_{\sigma_{0}}$ follow immediately from the
Proposition \ref{pro:TransversalityEquivalence}. Also by Proposition
\ref{pro:TransversalityEquivalence} we have that for all $x\in\SS_{\sigma_{0}}$
the kernel of $\hat{\sigma}_{0}(x)$ is one dimensional, i.e the fibers
$\KK_{\sigma_{0}}$ are one dimensional vector spaces. The transversality
condition on $\sigma_{0}$ implies that $\mult_{\sigma}$ is a smooth
submanifold of $S(\cotangent X)$. From this and the the face that
$\mult_{\sigma}=S(\cotangent X)\cap\KK_{\sigma_{0}}$ we obtain $\KK_{\sigma_{0}}$
is a smooth real line bundle over $\SS_{\sigma_{0}}$.
\end{proof}
Consider then a generic symbols $\sigma$. Over every connected component
$C$ of $\SS_{\sigma}$ the multiplicity set over $C$, which we will
denote by $\mult_{\sigma}|C$, is a one dimensional compact submanifold
of the torus $S(\cotangent X)|C$. The projection $\pi:\cotangent X\rightarrow X$
when restricted to $\mult_{\sigma}|C$ is two to one, which intuitively
this means that $\mult_{\sigma}$ ``winds'' twice around the ``$C$-direction''
of the torus $S(\cotangent X)|C$. The winding number of $\mult_{\sigma}|C$
in the ``fiber-direction'' is given by
\begin{equation}
m_{C}=\frac{1}{2\pi}\int_{\mult_{\sigma}|C}a_{C}^{*}(d\theta)\in\ZZ\label{eq:WindingFormula}
\end{equation}
where $T:C\rightarrow\tangent C\subset TX$ is the unit tangential
vector field to $C$ with counter-clockwise orientation, the map $a:\mult_{\sigma}|C\rightarrow S^{1}$
is given by $a_{C}(\xi)=\exp(i\xi(T(\pi(\xi)))$ and $d\theta\in H^{1}(S^{1},\RR)$
is the angular form. The parity of $m_{C}$ determines whether the
real line bundle $K_{\sigma}|C$ is trivial or not. More concretely,
$\mult_{\sigma}|C$ is connected if $m_{C}$ is odd and it consists
of exactly two connected components if $m_{C}$ is even. 
\begin{example}
If $X=S^{2}$ then $\cotangent X$ is homeomorphic to the three dimensional
projective space $\RR P^{3}$. Over every connected component $C$
of $\mult_{\sigma}$ such that $m_{C}$ odd, we have that $\mult_{\sigma}|C$
is a torus knot of the type $(2,m_{C})$ with ambient space $\RR P^{3}$.
(see \cite[pg. 216]{kn:kosniowski} for a definition torus knots with
ambient space $\RR^{3}$.\end{example}
\begin{problem}
\label{Integer realisation}Can any integer appear as a number of
the form $m(C)$?

Using the results in the next section we will be able to give an answer
to this problem for the case when $X$ is the two dimensional sphere.
\end{problem}

\section{Desingularising Characteristic Velocities}

In this section we will develop tools for desingularising the eigenvalue
functions $\lambda_{\sigma,1}$ and $\lambda_{\sigma,2}$ of the previous
sections. We start by generalizing some of the notions introduced
earlier. Let $E$ be a rank 2 vector bundle over a differentiable
connected manifold $M$, and assume that $E$ is endowed with a metric
$<,>_{E}$. We will denote by $\sym E$ the set of symmetric morphisms
of $E$ (symmetric with respect to the metric $<,>_{E}$). For a given
section $s:M\rightarrow\sym E$ its traceless part $s_{0}:M\rightarrow\symz E$
is given by $s_{0}=s-(1/2)\tr(s)I$. The bundle $\sym E$ has a Riemmanian
metric given by 
\begin{equation}
(A,B)=\frac{1}{2}\tr(AB)\hbox{\,\, for\, any\,\,}p\in M\hbox{\, and\,\,}A,B\in\sym{E_{p}}.\label{eq:metric0}
\end{equation}
The eigenvalues of $s$ are then given by
\[
\lambda_{s,i}=\frac{1}{2}\tr(s)+(-1)^{i}||s_{0}||\hbox{\, for\,\,}i=1,2,
\]
where the norm is the one induced by the metric in $\symz E$. The
multiplicity set $\mult_{s}$ associated to $s$ is given by

\[
\mult_{s}=\{p\in M|\lambda_{s,0}(p)=\lambda_{s,1}(p)\}=\{p\in M|s_{0}(p)=0\}.
\]
Recall that the projectivisation $\proj E$ of a real vector bundle
$E$ is the space of its real one-dimensional subspaces. For point
$p\in M$, we will denote a given line in a fiber $E_{p}$ by $l_{p}$.
\begin{defn}
\label{def:SpaceOfEigenlines}For a given section $s:M\map S^{2}E$
we define its \emph{space of eigenlines} $\EE_{s}\subset\proj E$
by 
\begin{equation}
\EE_{s}=\bigcup_{p\in M}\{l_{p}\in\proj E|l_{p}\hbox{\, is\, spanned\, by\, a\, non-zero\, eigenvector\, of\,\,}s(p)\}.\label{eq:desingularisation}
\end{equation}
 The \emph{eigenvalue function} of $s$ is defined as the function
$\lambda_{s}:\EE_{s}\map\RR$ given by 
\begin{equation}
\lambda_{s}(l_{p})=\hbox{eigenvalue\, of\,\,}s(p)\hbox{\, corresponding\, to\,\,}l_{p}\label{eq:eigenvaluefunctions}
\end{equation}

\end{defn}
The projection map $\pi:\proj E\map M$ restricts to a projection
map $\pi_{s}:\EE_{s}\map M$, that when restricted to $\EE_{s}-\pi^{-1}(\mult_{s})$
is a double covering map onto $M-\mult_{s}$. For points $p\in\mult_{s}$
we have that $\pi_{s}^{-1}(p)=\proj{E_{p}}$. Since $s$ and $s_{0}$
have the same eigenspaces, we have that $\EE_{s}=\EE_{s_{0}}$. The
set $\pi_{s}^{-1}(\mult_{s})$ can be characterized in terms of $\lambda_{s_{0}}:\EE_{s}\rightarrow\RR$
as follows.
\[
\pi_{s}^{-1}(\mult_{s})=\lambda_{s_{0}}^{-1}(0).
\]

\begin{prop}
\label{prop:DesingProposition}Let $E$ be a rank $2$ riemmanian
bundle real over a manifold $M$ and $s:M\map\sym E$ be a smooth
section such that its traceless part $s_{0}$ is transversal to the
zero section of $\symz E$. We have that $\EE_{s}$ is a smooth submanifold
of $\proj E$, the functions $\lambda_{s}$ and $\lambda_{s_{0}}$
are a smooth functions, and $0\in\RR$ is a regular value of $\lambda_{s_{0}}$.\end{prop}
\begin{proof}
see \cite[Theorem 5]{kn:mocs}.\end{proof}
\begin{cor}
\label{cor:DesingTubular}The set $\mult_{s}$ has a tubular neighborhood
$\NN_{s}$ such that $\pi_{s}^{-1}(\NN_{s})$ is diffeomorphic to
$\proj{(E}|\mult_{s})\times(-\epsilon,\epsilon)$ (for some $\epsilon>0$).\end{cor}
\begin{proof}
Since $0\in\RR$ is a regular value of $\lambda_{s_{0}}$ there exists
and $\epsilon>0$ such that $\lambda_{s_{0}}^{-1}(-\epsilon,\epsilon)$
is diffeomorphic to $\lambda_{s_{0}}^{-1}(0)\times(-\epsilon,\epsilon)=\proj{(E}|\mult_{s})\times(-\epsilon,\epsilon).$
If we choose a positive$\epsilon$ small enough we can then ensure
that $\pi_{s}(\lambda_{s_{0}}^{-1}(-\epsilon,\epsilon))$ is a tubular
neighborhood of $\mult_{s}$.\end{proof}
\begin{example}
Some simple calculations show that the Fresnel surface $\mathcal{F}_{\epsilon}$
defined in Example \ref{MaxwellFresnel} can be given as
\[
\mathcal{F}_{\epsilon}=\{(\lambda_{s,1}(\xi))^{1/2}\xi|\xi\in S^{2}\}\cup\{(\lambda_{s,2}(\xi))^{1/2}\xi|\xi\in S^{2}\},
\]
where $\lambda_{s,1}\leq\lambda_{s,2}$ are the eigenvalue functions
of the section $s:S^{2}\rightarrow\sym{(\tangent S}^{2})$ defined
by the formula.
\[
<s(x)v,w>=<\epsilon^{-1}v,w>.
\]
The singularities of $\mathcal{F}_{\epsilon}$ coincide with the non-smoothness
points of the functions $\lambda_{s,1}$ and $\lambda_{s,2}$, i.e
with the zero set of the section $s_{0}:S^{2}\rightarrow\symz{(\tangent S^{2})}$.
From the proof of Proposition \ref{pro:SingCharFunctions} we have
that 
\[
\int_{s^{2}}e(\symz{(\tangent S^{2})}=\int_{s^{2}}e(\tangent S^{2}\otimes_{\CC}\tangent S^{2})=2\int_{s^{2}}e(\tangent S^{2})=4.
\]
This last computations is the topological explanation of the four
singularities of $\mathcal{F}_{\epsilon}$. It is also easy to verify
that for a biaxial crystal (i.e $\epsilon$ has non-multiple eigenvalue)
the section $s_{0}$ is transversal to $\symz{(\tangent S^{2})}$
at each of the four points in $\mult_{s}$. The tubular neighborhood
$\NN_{s}$ (described in the previous corollary) is given by $\cup_{i=1}^{4}D_{i}$
where the family $\{D_{i}\}_{i=1,\ldots,4}$ consists of four disjoint
circles in $S^{2}$. We then have that $\EE_{s}-\pi_{s}(\NN_{s})$
is the same as two disjoint copies of $S^{2}-\NN_{s}$ (see part A
of the figure). By the previous corollary $\pi^{-1}(\NN_{s})$ consists
of four cylinders. Part B of the figure illustrates the projection
map $\pi_{s}$ overs one of the disks $D_{i}$. By joining the two
copies of $S^{2}-\cup_{i=1}^{k}D_{i}$ along this four cylinders we
obtain a surface of genus 3 (see part C of the figure).
\end{example}
\begin{figure}
\includegraphics[scale=0.5]{Desingularization}\label{Flo:Desingularization}\caption{Construction of space of eigenlines for a generic symmetric tensor
on the sphere with four isolated multiple points}
\end{figure}
The critical points of $\lambda_{s}$ on $\EE_{s}-\pi_{s}^{-1}(\mult_{s})$
are the inverse image under $\pi_{s}$ of the critical points of $\lambda_{s,1}$
and $\lambda_{s,2}$. It remains to study the critical points of $\lambda_{s}$
over the set $\pi_{s}^{-1}(\mult_{s}).$ The condition for $\lambda_{s}$
to have a critical point on $\pi_{s}^{-1}(\mult_{s})$ will be given
in terms of a Riemmanian metric on the co-normal bundle of $\mult_{\sigma}$
in $X$. More concretely, we have the following result.
\begin{prop}
\label{pro:CriticalsAtMultSet}Consider a section $s:X\map\sym E$
such as the one in Proposition \ref{prop:DesingProposition}, and
let $s_{\RR}=(1/2)\tr(s).$ A necessary condition for an element $l_{p}\in\pi_{s}^{-1}(\mult_{s})$
to be a critical point of $\lambda_{s}$ is that $ds_{\RR}(p)\in\conormal(\mult_{s})$,
where 
\[
\conormal(\mult_{s})=\{\alpha\in\tangent_{p}^{*}M|p\in\mult_{s}\hbox{\, and\,\,}\alpha(v)=0\hbox{\,\, for\, all\,\,}v\in\tangent_{p}\mult_{s}\}.
\]
There exists a metric in $g_{\mult_{s}}$ in $\conormal(\mult_{s})$
such if for $p\in\mult_{s}$ we have that $ds_{\RR}(p)\in\conormal(\mult_{s})$
, then $\lambda_{s}$ has $0$,$1$ or $2$ critical points over the
set $\pi_{s}^{-1}(p)$ depending on wether \textup{$||\diff s_{\RR}(x)||_{\mult_{s}}<1$,
$||\diff s_{\RR}(x)||_{\mult_{s}}=1$ or $||\diff s_{\RR}(x)||_{\mult_{s}}>1$,
}\textup{\emph{respectively}}\textup{. }\textup{\emph{If a critical
points of $\lambda_{s}$ over $\pi^{-1}(\mult_{s})$ is of the Morse
type, then it has to be of index 1.}}\end{prop}
\begin{proof}
see \cite[Proposition 11]{kn:mocs}.\end{proof}
\begin{rem}
Observe that the necessary condition for $\lambda_{s}$ to have a
critical point at $l_{p}\in\pi_{s}^{-1}(\mult_{s})$, is simply that
$s_{\RR}|\mult_{s}$ (the restriction of $s_{\RR}$ to $\mult_{s}$)
has a critical point on $p$. This will always happens at certain
points if $\mult_{s}$ is a compact manifold.
\end{rem}
To apply the previous results to a symbol $\sigma:\cotangent X\rightarrow\sym F$
consider the section $s_{\sigma}:S(\cotangent X)\rightarrow\sym{(\pi^{*}F)}$
(where $\pi:S(\cotangent X)\rightarrow X$ is the natural projection)
defined by 
\[
s_{\sigma}(\xi)v=\sigma(\pi(\xi))\tilde{\pi}(v)\hbox{\, for\,}v\in\pi^{*}F
\]
where $\tilde{\pi}$ is the natural projection $\tilde{\pi}:\pi^{*}F\rightarrow F$. 
\begin{defn}
For a symbol $\sigma:\cotangent X\rightarrow\sym F$ we define
\[
\EE_{\sigma}=\EE_{s_{\sigma}}\subset\proj{(\pi^{*}F)}\hbox{\, and\,\,}\lambda_{\sigma}=\lambda_{s_{\sigma}}:\EE_{\sigma}\rightarrow\RR.
\]
There is a natural projection map $\pi_{\sigma}:\EE_{\sigma}\rightarrow S(\cotangent X)$
given by $\pi_{\sigma}=\pi_{s_{\sigma}}$. Applying the \ref{prop:DesingProposition}
to $s_{\sigma}$ we obtain the following result.\end{defn}
\begin{prop}
\label{pro:DesingularizationNearMultSet}For a generic morphism $\sigma:\cotangent X\rightarrow\sym F$
we have that $\EE_{\sigma}$ is a smooth manifold and $\lambda_{\sigma}$
is a smooth function. Furthermore, if the multiplicity set $\mult_{\sigma}$
is non-empty then it is the disjoint union of set $S_{1},\ldots,S_{m}\subset S(\cotangent X)$
each one of them diffeomorphic to a circle. For each $1\leq i\leq m$
we have that each $S_{i}$ has a tubular neighborhood $\NN_{i}$ such
that for an appropriate $\epsilon>0$ the set $\pi_{\sigma}^{-1}(\NN_{i})$
is diffeomorphic to $S^{1}\times S^{1}\times(-\epsilon,\epsilon)$.\end{prop}
\begin{proof}
We have the $s_{\sigma}:S(\cotangent X)\rightarrow\sym{(\pi^{*}F)}$
defined above is transversal to the zero section of $\symz{(\pi^{*}F)}$
iff $\sigma_{0}:\cotangent X\rightarrow\symz F$ is transversal to
the zero section of $\symz F$. By Corollary \ref{cor:GenericitySymbol0}
we have that this last condition is generic. The first two assertions
of the proposition follow from this, the fact that $\mult_{\sigma}=\sigma_{0}^{-1}(0)$
and Proposition \ref{prop:DesingProposition}. From Corollary \ref{cor:DesingTubular}
we have that for each $1\leq i\leq n$ we can find a tubular neighborhood
of $S_{i}$ such that $\pi_{\sigma}^{-1}(\NN_{i})$ is diffeomorphic
to $\proj{(\pi^{*}F}|S_{i})\times(-\epsilon,\epsilon)$. Since $F$
is orientable we have that $\pi^{*}F$ is orientable and hence $\pi^{*}F|S_{i}$
is a trivial bundle. From this we conclude that $\proj{(\pi^{*}F}|S_{i})$
is diffeomorphic to $S^{1}\times S^{1}$. 
\end{proof}
For symbols $\sigma$ such that $\lambda_{\sigma}$ is a Morse function,
this theorem and the cell decomposition of $\EE_{\sigma}$ provided
by the critical points of $\lambda_{\sigma}$ establishes the connection
between the critical points of wave speeds and the ``conical-refraction''
type behavior induced by multiplicity set $\mult_{\sigma}$. This
can be expressed in terms homology groups of $\EE_{\sigma}$ by using
the same techniques as in \cite[pgs. 20,28-31]{kn:morse}. 


\section{Symbols on the Riemann Sphere}


\subsection{Algebraic preliminaries}

A linear map from $(\RR^{2})^{*}$ to $\symz{\RR^{2}}$ can be seen
as an element $\symz{\RR^{2}\otimes\RR^{2}}.$ An orthonormal change
of coordinates in $\symz{\RR^{2}\otimes\RR^{2}}$ is given in terms
of the action of a rotation 
\[
R=\left(\begin{array}{cc}
\cos(\theta) & -\sin(\theta)\\
\sin(\theta) & \cos(\theta)
\end{array}\right)
\]
which acts on elements $A\otimes v\in\symz{\RR^{2}\otimes\RR^{2}}$
as
\[
R\cdot(A\otimes v)=RAR^{T}\otimes Rv.
\]
We will identify both $\symz{\RR^{2}}$and $\RR^{2}$ with $\CC$
by letting
\[
\left(\begin{array}{cc}
p & q\\
q & -p
\end{array}\right)\sim p+iq\hbox{\, and\,\,}(v_{1},v_{2})\sim v_{1}+iv_{2}.
\]
By Proposition \ref{pro:SingCharFunctions} the above action can be
seen as the action of $\exp(i\theta)\in\CC-\{0\}$ on $(p+iq)\otimes(v_{1}+iv_{2})$
given by
\[
\exp(i\theta)\cdot((p+iq)\otimes(a+ib))=(\exp(i2\theta)(p+iq))\otimes(\exp(i\theta)(v_{1}+iv_{2}))
\]
This action is better understood in the following basis 
\begin{eqnarray*}
g_{1} & = & \frac{1}{\sqrt{2}}(1\otimes1+i\otimes i),\\
g_{2} & = & \frac{1}{\sqrt{2}}(i\otimes1-1\otimes i),\\
g_{3} & = & \frac{1}{\sqrt{2}}(1\otimes1-i\otimes i),\\
g_{4} & = & \frac{1}{\sqrt{2}}(i\otimes1+1\otimes i),
\end{eqnarray*}
since a simple calculation shows that
\begin{eqnarray*}
\exp(i\theta)\cdot g_{1} & = & \cos(\theta)g_{1}+\sin(\theta)g_{2}\\
\exp(i\theta)\cdot g_{2} & = & -\sin(\theta)g_{1}+\cos(\theta)g_{2}\\
\exp(i\theta)\cdot g_{3} & = & \cos(3\theta)g_{3}+\sin(3\theta)g_{4}\\
\exp(i\theta)\cdot g_{4} & = & -\sin(3\theta)g_{3}+\cos(3\theta)g_{4}.
\end{eqnarray*}
We conclude that if  we have that

\[
\text{\ensuremath{\alpha}}g_{1}+\beta g_{2}+\gamma\cdot g_{3}+\delta\cdot g_{4}=\exp(i\theta)\cdot(ag_{1}+bg_{2}+cg_{3}+dg_{4})
\]
then 
\begin{eqnarray*}
\alpha+i\beta & = & \exp(i\theta)(a+ib)\\
\gamma+i\delta & = & \exp(i3\theta)(c+id).
\end{eqnarray*}
The morphism $\sigma=ag_{1}+bg_{2}+cg_{3}+dg_{4}:(\RR^{2})^{*}\rightarrow\symz{\RR^{2}}$
acts on cotangent vectors $\xi=(\xi_{1},\xi_{2})$ as follows

\begin{equation}
\sigma(\xi)=\left(\begin{array}{cc}
p & q\\
q & -p
\end{array}\right)\xi_{1}+\left(\begin{array}{cc}
r & s\\
s & -r
\end{array}\right)\xi_{2}\label{eq:SymbolCoordinatesFormula}
\end{equation}
where
\begin{eqnarray}
p+iq & = & \frac{1}{\sqrt{2}}((a+ib)+(c+id))\nonumber \\
r+is & = & \frac{i}{\sqrt{2}}((a+ib)-(c+id)).\label{eq:SigmaInTermsOfABCD}
\end{eqnarray}
Inversely, if we are given $\sigma$ as in \ref{eq:SymbolCoordinatesFormula}
then the corresponding coefficients $a,b,c,d$ must satisfy 
\begin{eqnarray*}
a+ib & = & \frac{1}{\sqrt{2}}((p+iq)-i(r+is))\\
c+id & = & \frac{1}{\sqrt{2}}((p+iq)+i(r+is)).
\end{eqnarray*}
From the above formulas it is easy to see that the $\sigma:(\RR^{2})^{*}\rightarrow\symz{\RR^{2}}$
is invertible iff and only 
\[
\det\left(\begin{array}{cc}
p & r\\
q & s
\end{array}\right)=|a+ib|^{2}-|c+id|^{2}\not=0,
\]
where $|z|$ is the standard norm of a complex number $z\in\CC$.
The above discussion proves the following.
\begin{prop}
Let $F$ be an $\SO(2)$ vector bundle of rank $2$ over a manifold
$M$. We have that $\hbox{Hom}(F^{*},\symz F)$ is isomorphic to $F\oplus(F\otimes_{\CC}F\otimes_{\CC}F)$. 
\end{prop}
In particular, to any quadruple of sections $V,V_{1},V_{2},V_{3}:M\rightarrow F$
we can an associate a section $\sigma(V,V_{1},V_{2},V_{3}):M\rightarrow\hbox{Hom}(F^{*},\symz F)$
that corresponds to the section $V+V_{1}\otimes V_{2}\otimes V_{3}$.
We can write a local expression for $\sigma(V,V_{1},V_{2},V_{3})$
by considering orthonormal vector fields $e_{1},e_{2}:U\rightarrow F$
over an open set $U\subset M$, and such that $e_{2}=ie_{1}$ (multiplications
by $i$ in $F$ is well defined since it is a $\SO(2)=\hbox{U}(1)$
bundle). We can write (for $k=1,2,3)$
\[
V=ve_{1},V_{k}=v_{k}e_{1}\hbox{\, for\, smooth\, function\,\,}v,v_{k}:U\text{\ensuremath{\rightarrow}}\CC
\]
so that
\[
V+V_{1}\otimes V_{2}\otimes V_{3}=ve_{1}+(v_{1}v_{2}v_{3})e_{1}\otimes_{\CC}e_{1}\otimes_{\CC}e_{1}.
\]
By formulas \ref{eq:SigmaInTermsOfABCD} we the have that $\sigma=\sigma(V,V_{1},V_{2},V_{3})$
is given by the formulas

\begin{eqnarray}
<\sigma(e_{1})e_{1},e_{1}>+i<\sigma(e_{1})e_{1},e_{2}> & = & \frac{1}{\sqrt{2}}(v+v_{1}v_{2}v_{3})\label{eq:HolomorphicToSymbol0}\\
<\sigma(e_{2})e_{1},e_{1}>+i<\sigma(e_{2})e_{1},e_{2}> & = & \frac{i}{\sqrt{2}}(v-v_{1}v_{2}v_{3}),\label{eq:HolomorphicToSymbol}
\end{eqnarray}
Observe that we have identified $F^{*}$ with $F$ by using the metric
in $<,>$ in $F$.


\subsection{Holomorphic fields and symbols}

Consider the unit sphere 
\[
S^{2}=\{p\in\RR^{3}|<p,p>=1\}
\]
with the metric it inherits as a subspace of $\RR^{3}$. We can parametrize
the regions $D_{i}=S^{2}-\{0,0,(-1)^{i+1}\}$ (for $i=1,2$) using
the stereographic projection maps $\varphi_{i}:\RR^{2}\rightarrow D_{i}$
given by 
\[
\varphi_{i}(x,y)=(1+x^{2}+y^{2})^{-1}(2x,(-1)^{i+1},2y,(-1)^{i+1}(x^{2}+y^{2}-1)).
\]
If we identify $\RR^{2}$ with $\CC$ in the natural way, then $\varphi_{2}\circ\varphi_{1}^{-1}:\CC-\{0\}\rightarrow\CC-\{0\}$
is given by $\varphi_{1}\circ\varphi_{2}^{-1}(z)=z^{-1}$. If we denote
coordinates in $D_{1}$ by $z=x+iy$ and those in $D_{2}$ by $w=u+iv$,
then change of variables formula is $w=z^{-1}$ and hence

\[
\dero x=-z^{-2}\dero u.
\]
The metric in $S^{2}$ is such that if we let 
\begin{eqnarray*}
e_{1}=\frac{1+|z|^{2}}{2}\dero x & , & e_{2}=\frac{1+|z|^{2}}{2}\dero y\\
f_{1}=\frac{1+|w|^{2}}{2}\dero u & , & f_{1}=\frac{1+|w|^{2}}{2}\dero w,
\end{eqnarray*}
then $e_{1},e_{2}$ are orthonormal with $e_{2}=ie_{1}$, and similarly
for $f_{1},f_{2}$. If $V,V_{1},V_{2},V_{3}$ are smooth vector fields
in $S^{2}$ then we can write 
\[
V=v\dero x,V_{k}=v_{k}\dero x\hbox{\, for\, smooth\, a\, function\,\,}v,v_{k}:\CC\rightarrow\CC
\]
and if $\sigma=\sigma(V,V_{1},V_{2},V_{3})$ is the symbol corresponding
to $V+V_{1}\otimes V_{2}\otimes V_{3}$ then from formulas \ref{eq:HolomorphicToSymbol0}
and \ref{eq:HolomorphicToSymbol} we obtain that 
\begin{eqnarray*}
<\sigma(e_{1})e_{1},e_{1}>+i<\sigma(e_{1})e_{1},e_{2}> & = & \frac{1}{\sqrt{2}}(\lambda(|z|)v(z)+(\lambda(|z|))^{3}s(z))\\
<\sigma(e_{2})e_{1},e_{1}>+i<\sigma(e_{2})e_{1},e_{2}> & = & \frac{i}{\sqrt{2}}(\lambda(|z|)v(z)-(\lambda(|z|))^{3}s(z)),
\end{eqnarray*}
where 
\[
s(z)=v_{1}(z)v_{2}(z)v_{3}(z)\hbox{\, and\,\,\,}\lambda(r)=\frac{2}{1+r^{2}}
\]
 In the $w$ variables the above equation becomes
\begin{eqnarray*}
<\sigma(f_{1})f_{1},f_{1}>+i<\sigma(f_{1})f_{1},f_{2}> & = & -\frac{1}{\sqrt{2}}(\lambda(|w|)w^{2}v(w^{-1})+\lambda^{3}(|w|)w^{6}s(w^{-1}))\\
<\sigma(f_{2})f_{1},f_{1}>+i<\sigma(f_{2})f_{1},f_{2}> & = & -\frac{i}{\sqrt{2}}(\lambda(|w|)w^{2}v(w^{-1})-\lambda^{3}(|w|)w^{6}s(w^{-1})).
\end{eqnarray*}
We conclude that singular set of $\sigma$ on the $z$ variables is
then given by
\begin{equation}
\SS_{z}=\{z\in\CC|(|v(z)|=\lambda^{2}(|z|)|s(z)|\}\label{eq:SingularSetSphereZ}
\end{equation}
and over the $w$ variables is given by
\begin{equation}
\SS_{w}=\{w\in\CC||w^{2}v_{0}(w^{-1})|=\lambda^{2}(|w|)|w^{6}s(w^{-1})|\}.\label{eq:SingularSetSphereW}
\end{equation}
We are now ready to construct traceless symmetric symbols from holomorphic
vector fields on $S^{2}$. Recall the following simple result.
\begin{lem}
Consider a quadratic polynomial $P(z)=a_{0}+a_{1}z+a_{2}z^{2}$. The
holomorphic vector field 
\[
V_{P}(z)=P(z)\dero x
\]
extends to a holomorphic vector field over the whole sphere $S^{2}$.\end{lem}
\begin{proof}
We only need to check that the $V_{P}$ is smooth at the point at
infinity. By the change of variables formula for vector fields, we
have that 
\[
P(z)\dero x=-w^{2}P(w^{-1})\dero u=-(a_{2}+a_{1}z+a_{0}w^{2})\dero u
\]
which is holomorphic at $0$. Hence $V_{P}$ is smooth at infinity. 
\end{proof}
Given a second degree polynomial $P$ and a sixth degree polynomial
$S=P_{1}P_{2}P_{2}$ (where the $P_{i}$\textasciiacute{}s are also
second degree polynomials) we will let 
\[
\sigma_{P,S}=\sigma(V_{P,}V_{P_{1}},V_{P_{2}}V_{P_{3}}).
\]
We will now study the symbols $\sigma_{m,n}=\sigma_{z^{n},z^{m}}$
To compute the singular set of $\sigma_{m,n}$ we will need the following
result. 
\begin{lem}
\label{lem:RadiusComputation}For $0\leq m\leq2$ and $0\leq n\leq6$
let
\[
Z_{m,n}=\{r\in\RR|r^{m}-(\lambda(r))^{2}r^{n}=0\}.
\]
We have that 
\[
Z_{m,0}=Z_{0,n}=\{1\}\hbox{\, iff\,\,\,}n\not=1,3
\]
and
\[
Z_{0,1}=\{\alpha,1\},Z_{0,3}=\{1,1/\alpha\}
\]
where $\alpha$ is the only real root of $r^{3}+r^{2}+3r-1$. Furthermore,
for $0<m\leq2$ and $0<n\leq6$ we have that 
\[
Z_{m,n}=\left\{ \begin{array}{cc}
Z_{m-n,0}\cup\{0\} & \hbox{if\,\,\,\,}n\leq m\\
Z_{0,n-m}\cup\{0\} & \hbox{if\,\,\,\,}m\leq n
\end{array}\right..
\]
\end{lem}
\begin{proof}
Compute!\end{proof}
\begin{prop}
For $0\leq m\leq2$ and $0\leq n\leq6$, the singular set $\SS_{m,n}$
of the symbol $\sigma_{m,n}$ is given by
\[
\SS_{m,n}=\left\{ \begin{array}{cc}
\{z\in\CC||z|\in Z_{m,n}\} & \hbox{if\,\,\,\,}m=2\hbox{\, or\,\,}n=6\\
\{z\in\CC||z|\in Z_{m,n}\}\cup\{\infty\} & \hbox{if\,\,\,\,}m<2\hbox{\, and\,\,}n<6
\end{array}\right..
\]
The traceless symbol $\sigma_{m,n}$ is transversal to the zero section
of $\symz{(\tangent S^{2})}$ at $\mult_{\sigma_{m,n}}|S^{1}$, where
$\{z\in\CC||z|=1\}$, iff $n-m\not=2$.\end{prop}
\begin{proof}
For the first part apply Lemma \ref{lem:RadiusComputation} to formulas
\ref{eq:SingularSetSphereZ} and \ref{eq:SingularSetSphereW}. The
determinant function of $\sigma:\cotangent S^{2}\rightarrow\symz{(\tangent S^{2})}$
is given at $|z|=r$ is by
\[
h(r)=(\lambda(r)r^{m})^{2}-(\lambda(r)^{3}r^{n})^{2}.
\]
 Is easy to verify that
\[
\der hr(1)=2(2+m-n).
\]
From this and Proposition \ref{pro:TransversalityEquivalence} we
obtain the result of the second part of this proposition.
\end{proof}
The above results shows that the only symbols everywhere transversal
to the zero section of $\symz{(\tangent S^{2})}$ are $\sigma_{0,6}$
and $\sigma_{2,0}$. We will now study winding numbers $m_{S^{1}}$
defined in \ref{eq:WindingFormula} associated to the multiplicity
sets 
\[
\mult_{m,n}=\mult_{\sigma_{m,n}}\subset S(\cotangent S^{2})
\]
over the circle $S^{1}=\{z\in\CC||z|=1\}.$
\begin{prop}
For $\sigma=\sigma_{m,n}$ we have that $m_{S^{1}}=n-m$. \end{prop}
\begin{proof}
At the point $e^{i\theta}\in S^{1}$ we have that
\[
\sigma_{m,n}(v_{1}e_{1}+v_{2}e_{2})=(pv_{1}+rv_{2})e_{1}+(qv_{1}+sv_{2})e_{2},
\]
where	 
\begin{eqnarray*}
p+iq & = & \frac{1}{\sqrt{2}}(e^{im\theta}+e^{in\theta})\\
r+is & = & \frac{i}{\sqrt{2}}(e^{im\theta}-e^{in\theta}).
\end{eqnarray*}
Hence $v_{1}e_{1}+v_{2}e_{2}$ is in $\ker(\sigma_{m,n})$ (at the
point $e^{i\text{\ensuremath{\theta}}}$) iff
\[
e^{im\theta}(v_{1}+iv_{2})+e^{in\theta}(v_{1}-iv_{2})=0.
\]
If we choose $v_{1}+iv_{2}$ to be of unit norm then we can write
$v_{1}+iv_{2}=e^{i\varphi}$, and the above equation leads to the
solutions
\begin{eqnarray*}
\varphi_{1} & = & \frac{1}{2}(n-m)\theta+\frac{\pi}{2}\\
\varphi_{2} & = & \varphi_{1}+\pi,
\end{eqnarray*}
and hence the result of the proposition.\end{proof}
\begin{rem}
The above results gives an answer to the problem \ref{Integer realisation}
when $0\leq m\leq2$ and $0\leq n\leq6$. These restrictions on $m,n$
can be removed by using partitions of unity to smooth out the symbols
at $0$ and $\infty$.
\end{rem}
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