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\begin{document}

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

\begin{exo}
Let $M$ be a smooth manifold. What are the global sections of $M \times \R^k \to M$, $TM \to M$ and $\bigwedge^k T^*M \to M$?
\end{exo}

\begin{exo}[Frames]
Let $E \to M$ be a smooth vector bundle of rank $k$. A \emph{local frame} for $E$ over the open subset $U \subset M$ is a family $(s_1,\dots,s_k)$ of smooth sections of $E_{\vert U} \to U$ such that, for any $x \in U$, $(s_1(x),\dots,s_k(x))$ is a basis of the fiber $E_x$.
\begin{enumerate}
\item Check that it is equivalent to give a local frame for $E$ over $U$ or a local trivialization $E_{\vert U} \simeq U \times \R^k$.
\item Give a necessary and sufficient condition on global sections of $E \to M$ for this bundle to be trivial.
\item Is $T\T^n \to \T^n$ trivial? Is $T\S^2 \to \S^2$ trivial?
\item Does any smooth vector bundle admit a non-zero smooth section? A non-vanishing smooth section?
\end{enumerate}
\end{exo}

\begin{exo}[Pullback]
\begin{enumerate}
\item Is the pullback of a trivial vector bundle trivial?
\item Let $\pi:\S^n \to \R\P^n$ be the canonical projection, compare $T\S^n \to \S^n$ and $\pi^*(T\R\P^n) \to \S^n$.
\end{enumerate}
\end{exo}

\begin{exo}[Sub-bundles]
Let $E \to M$ be a smooth vector bundle of rank $r$ and let $V \subset E$. We say that $V \to M$ is a \emph{sub-bundle} of $E \to M$ of rank $k$ if, for any $x \in M$, there exists a local trivialization $\phi:E_{\vert U} \simeq U \times \R^r$ of $E$ over a neighborhood $U$ of $x$ such that:
\begin{equation*}
\phi \left( E_{\vert U} \cap V \right) = U \times \left(\R^k \times \{0\} \right).
\end{equation*}
\begin{enumerate}
\item Check that a sub-bundle of a smooth vector bundle is a smooth vector bundle.
\item Let $N$ be a smooth submanifold of $M$ and let $i:N\to M$ be the inclusion. Is $TN$ a sub-bundle of $i^*(TM)$?
\end{enumerate}
\end{exo}

\begin{exo}[Group of line bundles]
\label{group}
\begin{enumerate}
\item Let $L \to M$ be a line bundle on a smooth manifold, is $L \otimes L^* \to M$ trivial?
\item Define an abelian group structure on the set of line bundles over $M$ up to isomorphism.
\end{enumerate}
\end{exo}

\begin{exo}[Tautological line bundle]
Let $J = \left\{(D,x) \in \R\P^n \times \R^{n+1} \mvert x \in D\right\}$, we say that $J \to \R\P^n$ is the \emph{tautological line bundle} over $\R\P^n$.
\begin{enumerate}
\item Check that $J$ is a sub-bundle of $\R\P^n \times \R^{n+1}$ of rank $1$. Is it trivial?
\item Let $L=J^*$, check that an homogeneous polynomial $P$ of degree $d$ in $(n+1)$ variables defines a global section of $L^{\otimes d} \to \R\P^n$.
\item For which $d \in \Z$ is $L^{\otimes d} \to \R\P^n$ trivial?
\end{enumerate}
\end{exo}

\begin{exo}[Line bundles on the sphere]
\begin{enumerate}
\item What are the line bundles over $\S^1$ up to isomorphism? Which one is $T\S^1$?
\item Describe the group structure defined in exercise~\ref{group} when $M = \S^1$.
\item What are the line bundles over $\S^n$ up to isomorphism when $n\geq 2$?
\end{enumerate}
\end{exo}

\end{document}