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\begin{document}

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


\begin{exo}
\begin{enumerate}
\item Let $f:t \mapsto e^t$ from $\R$ to $\R_+^*$ and let $\alpha = \frac{d x}{x}$, compute $f^*\alpha$.
\item Same question with $f:\R^2 \to \R^2$ defined by $(r,\theta) \mapsto (r\cos \theta, r \sin \theta)$ and $\alpha = d x \wedge d y$.
\end{enumerate}
\end{exo}

Let us recall the definition of orientability.
\begin{dfn}[Orientability]
A manifold $M$ of dimension $n$ is said to be \emph{orientable} if there exists a differential form $\omega \in \Omega^n(M)$ such that $\omega_x \neq 0$ for all $x$ in $M$. That is, for any $x \in M$, if $(v_1(x),...,v_n(x))$ is a basis of $T_x M$ then $\omega_x(v_1(x),...,v_n(x))\neq 0$. Such a form $\omega$ is said to be a volume form.
\end{dfn}

\begin{exo}[Spheres]
\begin{enumerate}
\item Let $d V = d x^0 \wedge \dots \wedge d x^n$ denote the standard volume form on~$\R^{n+1}$ (that is the $(n+1)$-form equal to the determinant at each point) and let $X$ be the radial vector field $:x \mapsto \sum x_i \deron{}{x_i}$.

What is $\omega = X \lrcorner d V$ (recall that $(Y \lrcorner \alpha) (Y_1,\dots,Y_p)= \alpha(Y,Y_1,\dots,Y_p)$)?
\item Show that $\omega$ is invariant under the action of $SO_{n+1}(\R)$.
\item Let $i:\S^n \to \R^{n+1}$ be the canonical embedding, Show that $i^*(\omega)$ is a volume form.
\end{enumerate}
\end{exo}


\begin{dfn}[Parallelizable]
We say that a manifold $M$ is \emph{parallelizable} if there exist $n$ vector fields $(X_1,\dots,X_n)$ on $M$ such that, for any $x \in M$, $(X_1(x),\dots,X_n(x))$ is a basis of $T_xM$.
\end{dfn}

\begin{exo}[Orientability]
\begin{enumerate}
\item Show that any parallelizable manifold is orientable.
\item Show that a product of two orientable manifolds is itself orientable.
\item Show that the tangent bundle of a manifold is an orientable manifold.
\end{enumerate}
\end{exo}


\begin{exo}[Torus]
Is the torus $\T^n$ orientable? If so, give an explicit volume form.
\end{exo}

\begin{exo}[Projective spaces]
\begin{enumerate}
\item Let $n \in \N$ and $f :\S^n \to \S^n$, $x \mapsto -x$. Is this map orientation-preserving?
\item Is the projectif space $\R\P^n$ orientable?
\end{enumerate}
\end{exo}
\end{document}