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\begin{document}

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

\begin{exo}[Warm-up]
Let $g_0$ denote the Euclidean metric of $\R$, we define another Riemannian metric $g$ on $\R$ by $g_x := e^{-x^2} (g_0)_x$ for any $x\in \R$. Let $\nabla_0$ (resp.~$\nabla$) denote the Levi--Civita connection on $T\R$ associated with $g_0$ (resp.~$g$). Compute $(\nabla_0) \deron{}{x}$ and $\nabla \deron{}{x}$.
\end{exo}

\begin{exo}[Dual connection]
Let $\nabla$ be a connection on a vector bundle $E \to M$. We still denote by $\nabla$ the induces connections on bundles of the form $E \otimes \dots \otimes E \otimes E^* \otimes \dots \otimes E^* \to M$.
\begin{enumerate}
\item Compute $d(\alpha(X))$ where $\alpha \in \Gamma(E^*)$ and $X \in \Gamma(E)$.
\item Compute $\nabla \Id$ where $\Id : x \mapsto \Id_{E_x}$ is a section of $\End(E) \to M$.
\end{enumerate}
\end{exo}

\begin{exo}[Hessian and torsion]
Let $M$ be a manifold and let $f:M \to \R$ be smooth.
\begin{enumerate}
\item In a chart $(U,\varphi)$, we can compute the second diffenrential of $f_\varphi := f \circ \varphi^{-1}$. How do chart transitions act on $D^2f_\varphi$? Can we define an intrinsic notion of second differential of $f$ that would read as $D^2f_\varphi$ in the chart $(U,\varphi)$, for any such chart?
\item Let us now assume that $TM$ is endowed with a connection $\nabla$. Let $\nabla^2f$ be the section of $T^*M\otimes T^*M$ defined by:
\begin{equation*}
\forall x \in M,\ \forall u,v \in T_xM, \quad \nabla^2_xf(u,v) := \left(\nabla_u(df)\right)\cdot v.
\end{equation*}
Let $X$ and $Y$ be two vectors fields on $M$, prove that $\nabla^2f(X,Y)=X\cdot(Y\cdot f) - (\nabla_XY)\cdot f$.
\item Give an necessary and sufficient condition on $\nabla$ ensuring that, for all $f \in \mathcal{C}^\infty(M)$, for all $x \in M$, $\nabla^2_xf$ is symmetric.
\end{enumerate}
\end{exo}

\end{document}