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\newcommand{\slaz}[0]{\setminus \{0\}}
\newcommand {\wt}[0]{\widetilde}
\newcommand {\setR}[0]{\mathbb{R}}
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\author[Bucataru, Dahl]{%$\,$\\
  Ioan Bucataru\inst{1} \and
  Matias Dahl\inst{2}}
\institute[]{
  \inst{1}%
  A.I. Cuza University\\
  Romania
  \and
  \inst{2}%
  University College London\\
  UK}


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\date[Finsler Workshop 2009 --- Debrecen]{"Al. Myller" Centennial Conference --- June 21-26, 2010 Ia\c{s}i}
%\subject{Theoretical Computer Science}

\title{Descending maps between slashed tangent bundles}
%\date{May 29 -- 2009, Debrecen}

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\begin{document}

\begin{frame}
\vspace{1cm}

\begin{center}

{\LARGE \color[rgb]{0.2,0.1,1}
Descending Maps Between Slashed \\[3mm]
Tangent Bundles}
\\[3mm]
\end{center}
\vspace{0.3cm}

\begin{center}
%\raggedright 
{\bf Matias Dahl} \\[1mm]
{\small University College London, UK}
\vspace{0.5cm}

{\small joint work with }
\vspace{0.5cm}

%\raggedright 
{\bf Ioan Bucataru} \\[1mm]
{\small A.I. Cuza University, Romania}
\vspace{1cm}

%\begin{center}
{\small 
``Al. Myller'' Centennial Conference --- June 21-26, 2010 Ia\c{s}i
}
\end{center}
\end{frame}
%\frame{\maketitle}


\begin{frame}

{\Large \color[rgb]{0.20, 0.318, 0.82}  
Mathematical setting}
\vspace{0.2cm}
\begin{itemize}
\item $M$ is a manifold
%\item $TM$ is the tangent bundle
\item $TM\slaz$ is the \textbf{slashed tangent bundle}
\end{itemize}
\vspace{0.3cm}
{\Large \color[rgb]{0.20, 0.318, 0.82}   The Problem}
\vspace{0.2cm}
\begin{itemize}
\item Suppose $F$ is a diffeomorphism
$$
 F\colon TM\slaz\to TM\slaz.
$$
Characterize those $F$ that can be written as 
$$
 F = D\phi|_{TM\slaz}.
$$
for a diffeomorphism $\phi\colon M\to  M$.
$\,$\\
$\,$\\
When $\phi$ exists, one say that $F$ \textbf{descends}.
%\pause
%\item
%\textbf{Note:} Uniqueness result for inverse problems in anisotropic media = existence result of isometry
\end{itemize}
\end{frame}

\begin{frame}
{\Large Canonical involution in the second tangent bundle}
\textbf{Definition:}
Let $M$ be a manifold. Then the \textbf{canonical involution} 
is the diffeomorphism 
$$
  \kappa\colon \,\,TTM \,\,\to\,\, TTM
$$ 
that is locally given by
$$
  \kappa(x,y,X,Y)\,\,=\,\,(x,X,y,Y).
$$
$\,$\\
{\large \textbf{Note:}}
\begin{itemize}
\item $\kappa^2 = \operatorname{identity}$.
%\item The following diagram commutes:
%\begin{eqnarray*}
%\begin{xy}
%\xymatrix@C=2.0pc @R=3.0pc{
%%\xymatrix{
%  TTM\ar@{<->}[rr]^{\kappa} &  & TTM\ar[dl]^{D\pi_{TM\to M}} \\
%& TM \ar@{<-}[ul]^{\pi_{TTM\to TM}}& }
%\end{xy}
%\end{eqnarray*}
\end{itemize}
\end{frame}
 

\begin{frame}
\begin{block}{First main theorem:}
If $M$ is a manifold and $F$ is a diffeomorphism 
$$
  F\colon TM\slaz\,\,\to\,\, T M\slaz,
$$
then the following are equivalent:
\begin{enumerate}
\item[(i)] There exists a diffeomorphism $\phi\colon M\to M$ such that
\vspace{-0.2cm}
$$
   F\,\, =\,\, D\phi|_{TM\slaz}.
$$
\vspace{-0.8cm}
\item[(ii)] $DF\,\,=\,\,\kappa\circ DF\circ \kappa$ %on $T(TM\slaz)\cap \kappa(T(TM\slaz))$.
\end{enumerate}
\end{block}
\end{frame}

\begin{frame}
{\Large A related result}

\begin{block}{\textbf{Theorem [Robbin-Weinstein-Lie]:}
}
Let $F$ be a diffeomorphism
%\vspace{-0.2cm}
$$
  F\colon T^\ast M\,\,\to\,\, T^\ast M.
$$ 
Then the following are equivalent:
\begin{enumerate}
\item[(i)]  $F=\phi^\ast$ for a diffeomorphism $\phi\colon M\to M$.
\item[(ii)] $F^\ast \theta = \theta$. 
\end{enumerate}
\end{block}

$\,$\\
$\,$\\

{\large \textbf{Here:}}
\begin{itemize}
\item $\phi^\ast$ = \textbf{pullback of} $\phi$,
$\phi(x,\xi) = \left((\phi^{-1})^i(x), \pd{(\phi^{-1})^i}{x^a}\xi_i\right)$ 

\item $\theta$ = \textbf{canonical $1$-form}\, $\theta\in \Omega^1(T^\ast M)$,
$\theta = \xi_i dx^i$ 
\end{itemize}
\end{frame}

\begin{frame}
\textbf{Suppose:} $DF= \kappa\circ DF\circ \kappa$.\\
\vspace{0.1cm}
\textbf{Claim:} There exists a map $\phi\colon M\to M$ such that $F=D\phi|_{TM\slaz}$.
\pause

\vspace{1mm}
\textbf{Proof:} Let locally $
  F(x,y) = \left(F_1(x,y), F_2(x,y)\right).
$
\vspace{-0.1cm}
Then
\vspace{-0.1cm}
\begin{eqnarray*}
  DF(x,y,X,Y)& & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad \,
\end{eqnarray*}
\vspace{-0.9cm}
\begin{eqnarray*}
 &=& \left(F_1(x,y), F_2(x,y), \pd{F_1}{x^a}(x,y)X^a + \pd{F_1}{y^a}(x,y)Y^a,\right.\\
              & & \quad\quad\quad\quad\quad\quad\quad\quad \left.\pd{F_2}{x^a}(x,y)X^a + \pd{F_2}{y^a}(x,y)Y^a\right),
\end{eqnarray*}
\pause
\vspace{-0.5cm}
\begin{eqnarray*}
 \kappa\circ DF\circ\kappa(x,y,X,Y) \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\,
\end{eqnarray*}
\vspace{-0.9cm}
\begin{eqnarray*}
  &=& \left(F_1(x,X), \pd{F_1}{x^a}(x,X)y^a + \pd{F_1}{y^a}(x,X)Y^a,F_2(x,X),\right. \\
              & & \quad\quad\quad\quad\quad\quad\quad\quad \left.\pd{F_2}{x^a}(x,X)y^a + \pd{F_2}{y^a}(x,X)Y^a\right).
\end{eqnarray*}
\vspace{-0.1cm}
\pause
\begin{itemize}
\item First components: $F_1(x,y)=F_1(x,X)$. \pause 
Thus $F_1(x,y)$ does not depend on $y$. Set 
$\phi(x)=F_1(x,y)$.
\pause
\item Second components: $F_2(x,y) = \pd{\phi}{x^a}(x)y^a$. Thus $F=D\phi|_{TM\slaz}$.
\end{itemize}
\end{frame}



%\begin{frame}
%{\Large A second problem}
%Let manifold $M$ have two Riemann metrics.
%Find sufficient conditions on a diffeomorphism
%$$
%  F\colon TM\slaz\,\,\to\,\, T M\slaz
%$$
%such that $F=D\phi|_{TM\slaz}$ for an isometry 
%map $\phi\colon M\to M$.
%\end{frame}

\begin{frame}
\vspace{-0.2cm}
\begin{block}
{\textbf{Second main theorem:} }
Suppose $M$ is a connected manifold with $\dim M\ge 2$ and $F$ is a diffeomorphism
$$
  F\colon TM\slaz\,\,\to\,\, TM\slaz.
$$
If $M$ has two complete Riemann metrics $g$ and $\wt g$ such that
\begin{enumerate}
\item[(i)] $g$ has a \textbf{trapping hypersurface} $\Sigma\subset M$ $\Leftrightarrow$ Every geodesic of $g$ intersects $\Sigma$
\item[(ii)] for all $p\in \Sigma$, 
\vspace{-3mm}
\begin{eqnarray*}
  g_{ij}(p) &=& \wt g_{ij}(p),   \quad \Gamma^i_{jk}(p) = \wt \Gamma^i_{jk}(p), \quad\quad\quad\quad\quad \\
  DF(\xi) &=& \xi, \,\quad \quad\quad\quad \quad\quad\,\,\,\xi\in T(T_pM\slaz)
\end{eqnarray*}
\item[(iii)] If $J\colon I\to TM\slaz$ is a Jacobi field for $g$ then \\
$F\circ J\colon I\to TM\slaz$ is a Jacobi field for $\wt g$. 
\end{enumerate}
Then there exists a diffeomorphism $\phi\colon M\to M$ such that 
$F = D\phi|_{TM\slaz}$ and $\phi$ is an isometry.
\end{block}
\end{frame}

\end{document}

