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\begin{document}

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

\begin{exo}[The sphere]
\begin{enumerate}
\item \label{q1} Prove that $\S^n = \left\{ x \in \R^{n+1} \mvert \Norm{x}_2=1 \right\}$ is a smooth manifold of dimension $n$, by building a smooth atlas.
\item The Euclidean sphere $\S^n$ is also a manifold as a submanifold of $\R^{n+1}$ (see previous exercise sheet, exercise~4). Show that the differentiable structure built in question~\ref{q1} is the same as the one induced by $\R^{n+1}$.
\item What happens if we replace the Euclidean norm $\Norm{\cdot}_2$ by another norm $\Norm{\cdot}$?
\end{enumerate}
\end{exo}

\begin{exo}[Product manifolds]
Let $M$ and $N$ be smooth manifolds, show that $M \times N$ is a smooth manifold. What is its dimension?
\end{exo}

\begin{exo}[The torus]
\begin{enumerate}
\item Build a differentiable structure on $\T^n = \R^n / \Z^n$ such that the canonical projection $p : \R^n \to \T^n$ is a local diffeomorphism.
\item In the previous exercise sheet, we built an homeomorphism from $\T^n$ to $(\S^1)^n \subset \C^n$ (exercise~5). Prove that this is in fact a diffeomorphism.
\end{enumerate}
\end{exo}

\begin{exo}[The projective space]
Recall that $\R \P^n$ is defined as $\left(\R^{n+1} \setminus \{0\}\right) / \R^*$, where $\R^*$ acts on $\R^{n+1}$ by dilation. As a set, $\R\P^n$ is the set of lines in $\R^{n+1}$.

Let $x = (x_0,\dots,x_n) \in \R^{n+1} \setminus \{0\}$, we denote by $(x_0 : \dots : x_n)$ its class in $\R \P^n$.
\begin{enumerate}
\item Let $i \in \{0,\dots,n\}$. Show that $U_i = \left\{(x_0 : \dots : x_n) \in \R \P^n \mvert x_i \neq 0 \right\}$ is open in $\R \P^n$, and that it is homeomorphic to $\R^n$ via the canonical projection $p:\R^{n+1}\setminus \{0\} \to \R\P^n$.
\item Show that $\R \P^n$ is a smooth manifold.
\item Show that $p$ is smooth.
\item Show that the restriction of $p$ to $\S^n$ is a local diffeomorphism.
\end{enumerate}
\end{exo}

\end{document}