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\begin{document}

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


Let $(x_1,\dots,x_n)$ be local coordinates on an open subset of a Riemannian manifolds $(M,g)$, let $(g_{ij}(x))$ denote the matrix of $g$ in these coordinates and let $(g^{ij}(x))$ denote its inverse. Let $(\Gamma_{ij}^k(x))$ denote the Christoffel symbols associated with the Levi--Civita connection of $(M,g)$ in these coordinates. We recall that, for any $i,j$ and $k \in \{1,\dots,n\}$:
\begin{equation*}
\Gamma_{ij}^k = \frac{1}{2}\sum_{l=1}^n g^{kl}\left(\deron{g_{il}}{x_j}+\deron{g_{jl}}{x_i}-\deron{g_{ij}}{x_l}\right).
\end{equation*}

\begin{exo}[Image of a geodesic]
Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds and let $f:M_1 \to M_2$ be a smooth map.
\begin{enumerate}
\item If $f$ is an isometric diffeomorphism, is the image of a geodesic of $M_1$ a geodesic of $M_2$?
\item Same question if $f$ is a conformal diffeomorphism.
\item Same question if $f$ is an isometric embedding.
\end{enumerate}
\end{exo}

\begin{dfn}
Let $(M,g)$ be a Riemannian manifold. We say that two maximal geodesics are \emph{parallel} if they are either disjoint or equal up to reparametrization.
\end{dfn}

\begin{exo}[Model spaces]
\begin{enumerate}
\item Let us consider $\R^n$ with is canonical Euclidean metric.
\begin{enumerate}
\item What are the geodesics?
\item Compute the exponential map at any point $p \in \R^n$.
\item What is its injectivity radius?
\item Are there closed geodesics?
\item Are all geodesics closed?
\item Is the image of a geodesic a submanifold of the ambient space?
\item Let $\gamma$ be a geodesic and let $p \in \R^n \setminus \im(\gamma)$. How many geodesics passing through $p$ and parallel to $\gamma$ are there?
\end{enumerate}
\item Let $\alpha_1,\dots,\alpha_n >0$, same questions for $\T_\alpha^n = \R^n / (\alpha_1\Z \oplus \dots \oplus \alpha_n\Z)$ with the metric induced by the Euclidean one on $\R^n$.
\item Same questions on $\S^n$ with the metric induced by the Euclidean one on $\R^{n+1}$.
\item Same questions on the Poincaré disc $\D$ with the metric $g_\D := \frac{4}{(1-x^2-y^2)^2} (\dx x^2 + \dx y^2)$
\item Same questions on the upper half-plane $\H$ with the metric $g_\H := \frac{1}{y^2} (\dx x^2 + \dx y^2)$.
\end{enumerate}
\end{exo}

\begin{exo}[Normal coordinates]
Let $(M,g)$ ba a Riemannian manifold and let $p \in M$.
\begin{enumerate}
\item Can we find local coordinates $(x_1,\dots,x_n)$ centered at $p$ such that the matrix $(g_{ij}(x))$ of $g$ in these coordinates satisfies $(g_{ij}(x)) = I_n + O(\Norm{x})$, where $I_n$ is the identity matrix of size $n$? Don't forget to make sense of the $O(\Norm{x})$.
\item Show that in the normal coordinates centered at $p$ we have: $(g_{ij}(x)) = I_n + O(\Norm{x}^2)$, and $\Gamma_{ij}^k(x)=O(\Norm{x})$ for any $i,j$ and $k$.
\item Can we do better? ($I_n + O(\Norm{x}^3)$, constant equal to $I_n$, \dots)
\end{enumerate}
\end{exo}

\end{document}