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\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Vector fields}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}

\begin{exo}[Flow box]
Let $M$ be a compact manifold and let $X$ be a vector field on $M$. Let $p \in M$ be such that $X(p) \neq 0$, prove that there exist local coordinates around $p$ such that $X = \deron{}{x_1}$.

Are the orbits of $X$ submanifolds of $M$?
\end{exo}

\begin{exo}[Transitivity of the group of diffeomorphisms]
\begin{enumerate}
\item Let $a$ and $b$ in the open ball $B=\{x \in \R^n \mid \Norm{x} < 1\}$. Prove that there exists a diffeomorphism $f: \R^n \to \R^n$ such that $f(a)=b$ and $f = \id$ on $\R^n \setminus B$.
\item Let $M$ be a connected manifold, prove that it is path connected.
\item If moreover $\dim(M)\geq 2$, prove that the natural action of the group of diffeomorphisms of $M$ is $k$-transitive for any $k \in \N^*$.
\item What happens if $\dim(M) =1$?
\end{enumerate}
\end{exo}

\end{document}