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\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Submanifolds, partitions of unity}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}

\begin{exo}[Veronese embedding]
Recall that $\R\P^n$ is defined as $(\R^{n+1}\setminus\{0\})/\sim$, where $\sim$ is the colinearity equivalence relation. If $x=(x_0,\dots,x_n) \in \R^{n+1}\setminus \{0\}$, we denote by $(x_0:\dots:x_n) \in \R\P^n$ the line spanned by $x$. Since $(x_0:\dots:x_n) = (\lambda x_0:\dots: \lambda x_n)$ for any $\lambda \in \R^*$, these are called homogeneous coordinates.

Let $h : \R\P^2 \to \R\P^5$ be defined by $(x:y:z) \longmapsto (x^2:y^2:z^2:xy:yz:zx)$, check that $h$ is well-defined and prove that it is an embedding.
\end{exo}

\begin{exo}[Global equation]
Let $M$ be a smooth compact connected hypersurface in $\R^n$. We admit that $\R^n \setminus M$ has exactly two connected component, one bounded and one unbounded. Prove that there exists a global regular equation for $M$, that is there exists $f : \R^n \to \R$ smooth, such that $M = f^{-1}(0)$ and $d_xf$ does not vanish on $M$.
\end{exo}

\begin{exo}[Retraction of sublevels]
Let $M$ be a compact manifold and $f : M \to \R$ be a smooth function. We denote by $M_a$ the sublevel $\{x \in M \mid f(x)< a\}$. Let $a,b \in \R$ be such that $]a-\epsilon,b+\epsilon[$ does not contain any critical value of $f$, for some positive $\epsilon$. Prove that $M_a$ and $M_b$ are diffeormorphic.
\end{exo}

\end{document}