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\begin{document}

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


\begin{exo}[Standard metrics]
Let $g_0$ denote the standard Riemannian metric on $\R^n$, that is for all $x \in \R^n$, $(g_0)_x$ is the standard Euclidean inner product.
\begin{enumerate}
\item Are the translations isometries of $(\R^n,g_0)$? Define a natural Riemannian metric on the flat torus $\T^n = \R^n/\Z^n$.
\item Let $i:\S^n \to \R^{n+1}$ denote the canonical inclusion, we set $g = i^*(g_0)$. Do elements of the orthogonal group $O_{n+1}(\R)$ induce isometries of $(\S^n,g)$? Define a natural Riemannian metric on the real projective space $\R\P^n$.
\end{enumerate}
\end{exo}

\begin{exo}[Dimension~$1$]
\begin{enumerate}
\item Let $M$ be a connected smooth manifold without boundary of dimension $1$. Are any two Riemannian metrics on $M$ conformal to one another?
\item Same question if $\dim(M) \geq 2$.
\item What are the connected Riemannian manifolds without boundary of dimension $1$ up to isometries?
\end{enumerate}
\end{exo}

\begin{exo}[Hyperbolic spaces]
Let $B= -dx^0\otimes dx^0 + dx^1\otimes dx^1 + \dots + dx^n\otimes dx^n$ denote the standard Lorentz form on $\R^{n+1}$. Let $\mathcal{H}^n$ denote the set $\{x \in \R^{n+1} \mid B(x,x) = -1, x_0 >0\}$. We also denote by $\H^n$ the half-space $\{x \in \R^n \mid x_n >0\}$ and by $\D^n$ the unit open ball in $\R^n$.
Prove that the following are Riemannian manifolds that are isometric to one another:
\begin{itemize}
\item $\mathcal{H}^n$ endowed with the restriction of $B$,
\item $\H^n$ with the metric $\frac{1}{\norm{x_n}^2} \sum_{i=1}^n dx^i \otimes dx^i$,
\item $\D^n$ with the metric $\frac{4}{(1-\Norm{x}^2)^2} \sum_{i=1}^n dx^i \otimes dx^i$
\end{itemize}
\end{exo}

\begin{exo}[Hyperbolic half-plane and Poincaré disc]
In this exercise, we simply denote by $\H$ the hyperbolic half-plane $\H^2$ and by $\D$ the hyperbolic disc $\D^2$ (also called \emph{Poincaré disc}).
\begin{enumerate}
\item Check that conformal diffeomorphisms of $\D$ (resp.~$\H$) preserving the orientation are biholomorphisms.
\item Describe the conformal diffeomorphisms of $\D$ (resp.~$\H$). Which one are isometries?
\end{enumerate}
\end{exo}

\end{document}