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\begin{document}

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


\begin{dfn}
A \emph{metric} on a vector bundle $E\to M$ is a section $h \in \Gamma(E^* \otimes E^*)$ such that, for all $x \in M$, $h_x$ is an inner product on $E_x$.
\end{dfn}

\begin{dfn}
A connection $\nabla$ on a vector bundle $E \to M$ eqquiped with a metric $h$ is said to be \emph{compatible} with $h$ if, for any $s_1,s_2 \in \Gamma(E)$, $dh(s_1,s_2) = h(\nabla s_1,s_2)+h(s_1,\nabla s_2)$.
\end{dfn}

In the sequel, let $p:E \to M$ be a rank $r$ vector bundle and let $n$ denote the dimension of $M$.

\begin{dfn}
For any $y \in E$, we denote $V_y = \ker d_yp$. Then, $V \to E$ is a rank $r$ sub-bundle of $TE\to E$ and is called the \emph{vertical sub-bundle} of $TE$.
\end{dfn}

\begin{dfn}
An \emph{horizontal sub-bundle} of $TE$ is a sub-bundle $H \to E$ of $TE \to E$ such that, for any $y \in E$, $H_y \otimes V_y = T_yE$.
\end{dfn}

\begin{rems}
\begin{itemize}
\item Note that $H$ has rank $n$ and that for any $y \in E$, $V_y = T_y(E_x)$ is canonically isomorphic to $E_x$, where $x=p(y)$.
\item In the literature, horizontal sub-bundles are called \emph{Ehresmann connections}, we don't use this terminology in order to avoid confusing horizontal sub-bundles with connections (in the sense of the course).
\end{itemize}
\end{rems}

For any $\lambda \in \R$, let $M_\lambda : E \to E$ denote the fiberwise multiplication by $\lambda$.

Let $\Delta : M \to M\times M$ be defined by $\Delta(x)=(x,x)$. Then $\Delta^* (E \times E) \to M$ is the space
\begin{equation*}
\left\{(x,y,y') \in M \times E \times E \mvert p(y)=x=p(y')\right\} \simeq \left\{(y,y') \in E \times E \mvert p(y)=p(y')\right\}
\end{equation*}
with the natural projection. We denote by $A:\Delta^* (E \times E) \to E$ the fiberwise addition of $E$.

\begin{dfn}
We say that an horizontal sub-bundle $H$ of $TE$ is \emph{linear} if:
\begin{itemize}
\item for any $y,y' \in E$ such that $p(y)=p(y')$ we have:
\begin{equation*}
d_{(y,y')}A \left((H_y \times H_{y'}) \cap T_{(y,y')}\Delta^*(E\times E)\right) = H_{A(y,y')};
\end{equation*}
\item for any $\lambda \in \R$ and $y \in E$ we have $d_yM_\lambda(H_y) = H_{M_\lambda(y)}$.
\end{itemize}
\end{dfn}

The main goal of the following exercise is to prove that the choice of a linear horizontal sub-bundle of $TE$ is equivalent to the choice of a connection on $E$.

\begin{exo}
Let $p:E \to M$ be a vector bundle of rank $r$ on a $n$-dimensional basis.
\begin{enumerate}
\item Let $H \to E$ be a linear horizontal sub-bundle of $TE$. Define a connection $\nabla$ on $E$ associated with $H$.\\
\emph{Hint:} consider the projection onto $V_y$ along $H_y$.
\item Conversely, assume that $E$ is eqquiped with a connection $\nabla$. Define a linear horizontal sub-bundle $H$ of $TE$ associated with $\nabla$.\\
\emph{Hint:} consider the image of $d_xs$, where $s \in \Gamma(E)$ is such that $\nabla_xs=0$.
\item Recall that if $s(x)=0$ then $\nabla_xs$ does not depend on $\nabla$. What does it mean in terms of the associated linear horizontal sub-bundles?
\item Let $h$ be a metric on $E$ and let $\nabla$ be a connection on $E$ compatible with $h$. What does it mean for the associated linear horizontal sub-bundle?
\end{enumerate}
\end{exo}

\end{document}