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\begin{document}

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\lhead{Differential Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Riemannian submanifolds}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}


\begin{exo}
Let $(M,g)$ be a Riemannian manifold and let $f:M\to \R$ be a smooth map. We assume that $f$ vanishes transversally, that $d_xf \neq 0$ whenever $f(x) = 0$. Let us denote by $Z$ the hypersurface $f^{-1}(0)$ and by $\II$ its second fundamental form.
\begin{enumerate}
\item Show that for any $x \in Z$, $d_xf \circ \II_x = -\nabla_x^2f_{|T_xZ}$.
\item Express $\II$ only in terms of $f$.
\end{enumerate}
\end{exo}

\begin{exo}
Recall that if $(M,g)$ is a Riemannian surface, then its Gauss curvature at $p$ is $\kappa(p) = K(T_pM)$, the sectional curvature of $T_pM$.
\begin{enumerate}
\item Is there a metric $g$ on $\S^2$ such that $\kappa$ takes a negative value at some point?
\item Is there a metric $g$ on the torus $\T^2$ such that $\kappa$ does not vanish?
\item Is there a $2$-dimensional submanifold (without boundary) $M$ of $\R^3$ such that $\kappa$ vanishes everywhere?
\item Same question assuming that $M$ is not a plane.
\item Same question assuming that $M$ is compact without boundary. (\emph{Hint:} Use Sard's theorem: the set of critical values of a smooth map between smooth manifolds has measure~$0$.)
\item Is there a compact surface without boundary $M$ in $\R^3$ such that $\kappa$ is negative everywhere?
\end{enumerate}
\end{exo}

\end{document}