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\begin{document}

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


\begin{exo}[Derivatives at a vanishing point]
Let $E \to M$ be a vector bundle and let $\nabla$ and $\tilde{\nabla}$ be two connections on $E$. Let $s \in \Gamma(E)$ and $x \in M$ be such that $s(x) = 0$. Compare $\nabla_xs$ and $\tilde{\nabla}_xs$.
\end{exo}

\begin{exo}
Let $E \to M$ be a vector bundle eqquiped with a connection $\nabla$. Let $x_0 \in M$ and $y \in E_{x_0}$, is there a local (resp.~global) section $s \in  \Gamma(E)$ such that $s(x_0)=y$ and $\nabla_{x_0}s = 0$?
\end{exo}

\begin{exo}
Let $s$ be a smooth section of the vector bundle $E \to M$ and let $x_0 \in M$. Is there a connection $\nabla$ on $E$ such that $\nabla s$ vanishes on some neighborhood of $x_0$ in $M$?
\end{exo}

\begin{exo}[Christoffel symbols]
\begin{enumerate}
\item Let $(M,g)$ be a Riemannian manifold. We denote by $(x_1,\dots,x_n)$ local coordinates on a open subet $U$ of $M$ and by $G=(g_{ij})$ the matrix of $g$ in these coordinates. Let $(g^{kl})$ denote the coefficients of $G^{-1}$. Check that the Christoffel symbols $(\Gamma_{ij}^k)_{1\leq i,j,k \leq n}$ of the Levi--Civita connection $\nabla$ of $(M,g)$ are symmetric in $(i,j)$. Prove that for any $i,j,k \in \{1,\dots,n\}$ we have:
\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*}
\item Recall that the half-plane model of the hyperbolic plane is $\H^2 :=\{(x,y) \in \R^2 \mid y >0\}$ endowed with the metric $g_{(x,y)} := \frac{1}{y^2} (\dx x \otimes \dx x + \dx y \otimes \dx y)$. Compute the covariant derivatives of $\deron{}{x}$ and $\deron{}{y}$ for the Levi--Civita connection.
\item Recall that the Poincaré disc is the unit open disc $\D^2 \subset \R^2$ endowed with the metric $g_{(x,y)} := \frac{4}{(1-x^2-y^2)^2} (\dx x \otimes \dx x + \dx y \otimes \dx y)$. Compute the covariant derivatives for the Levi--Civita connection of the vector fields $\deron{}{r}$ and $\deron{}{\theta}$ associated with the polar coordinates on $\D^2 \setminus \{0\}$.
\end{enumerate}
\end{exo}

\end{document}