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\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Tangent spaces, tangent maps}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}

\begin{exo}[Tangent space of a submanifold]
Describe the tangent space $T_pM \subset T_pN$ of the submanifold $M$ in $N$ at a point $p$, for each of the four characterizations of a submanifold.
\end{exo}

\begin{exo}[Tangent space of a product]
Let $M$ and $N$ be two smooth manifolds, find a natural isomorphism between $T_{(p,q)}\left(M \times N \right)$ and $T_pM \times T_qN$.
\end{exo}

\begin{exo}[Computation of a differential]
Compute the differential of $\overline{F}:\T^2 \to \S^2$ defined as the quotient of the map from $\R^2$ to $\S^2$:
\begin{equation*}
F :(x,y) \mapsto (\cos(2\pi x)\cos(2\pi y),\cos(2\pi x)\sin(2\pi y),\sin(2\pi x)).
\end{equation*}
On which set is $\overline{F}$ a local diffeomorphism?  Is $\overline{F}$ restricted to this domain a global diffeomorphism?
\end{exo}


\begin{exo}
Is it possible to immerse a compact manifold $M$ of dimension $n>0$ in $\R^n$?
\end{exo}

\end{document}