
  \usepackage[utf8]{inputenc}
  \usepackage[usenames,dvipsnames]{xcolor}
  \usepackage{amsthm}
  \usepackage{yhmath,amsfonts,amssymb,amsmath,MnSymbol, mathrsfs,amsbsy}
  \usepackage{mdframed}
  \usepackage[all]{xy}
  \usepackage[english]{babel}
  \usepackage[colorlinks=true, linkcolor=black,citecolor=black]{hyperref}
  \usepackage{fullpage, version}
  \usepackage{geometry}
  \usepackage[toc, page]{appendix} 
  \usepackage{graphicx}
  \usepackage{enumerate}
  \usepackage{tkz-euclide}
  \usepackage{ulem}

  \usetkzobj{all}

  \def\Z{\mathbb{Z}}
  \def\H{\mathbb{H}}
  \def\M{\mathcal{M}}
  \def\s{\mathfrak S}
  \def\v{\mathfrak v}
  \def\Riem{\mathbf{Riem}}
  \def\riem{\mathrm{Riem}}
  \def\W{\mathbf W}
  \def\Ric{\mathbf{Ric}}
  \def\stab{\mathrm{Stab}}
  \def\ric{\mathrm{Ric}}
  \def\g{\bold g}
  \def\U{\mathcal U}
  \def\ab{ \mathrm{ab}}
  \def\Hom{\mathrm{Hom}}
\def\V{\mathcal V}
\def\F{\mathcal F}
\def\Ric{\mathbf{Ric}}
\def\III{\mathrm{III}}
\def\II{\mathrm{II}}
\def\I{\mathrm{I}}
\def\C{\mathscr{C}}
\def\R{\mathbb{R}}
\def\E{\mathbf E}
\def\G{\mathbf G}
\def\S{\mathcal S}
\def\N{\mathbb N}
\def\T{\mathcal{T}}
\def\rot{\mathbf{rot}}
\def\div{\mathbf{div}}
\def\grad{\overrightarrow{\mathbf{grad}}}
\def\d{\mathrm{d}}
\def\S{\mathcal{S}}
\def\K{\mathbf K}
\def\D{\mathcal{D}}
\def\SO{\mathrm{SO}}
\def\E{\mathbb{E}}
\def\BTZ{{\E_{0}^{1,2}}}
\def\sing{\mathrm{Sing}}
\def\RR{\mathcal{R}}
\def\mass{X_{mass}}
\def\isom{\mathrm{Isom}}
\def\graph{\mathrm{Graph}}
\newcommand{\fonction}[5]{\displaystyle#1:\begin{array}{l|rcl}
& \displaystyle #2 & \longrightarrow & \displaystyle #3 \\
    & \displaystyle #4 & \longmapsto & \displaystyle #5 \end{array}}
\newcommand{\fonctionn}[4]{\displaystyle\begin{array}{l|rcl}
& \displaystyle #1 & \longrightarrow & \displaystyle #2 \\
    & \displaystyle #3 & \longmapsto & \displaystyle #4 \end{array}}


\newtheorem{theo}{\bf{Theorem}}
\newenvironment{boxedtheo}
  {\begin{mdframed}\begin{theo}}
  {\end{theo}\end{mdframed}}
\newtheorem{lem}{Lemma}[section]
\newtheorem{cor}[lem]{Corollary}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{defi}[lem]{Definition}

\newtheorem{example}[lem]{Example}
\newtheorem{rem}[lem]{Remark}

\newenvironment{lemma}[1][Lemma]{\begin{trivlist}

\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{theorem}[1][theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newtheorem{conjecture}{Conjecture}


