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    \begin{center}
    \textbf{ MATH534A, Problem Set 8, due Nov 13}
    \end{center}
    All problems are worth the same number of points.
    \begin{enumerate}
    \item Let $M,N$ be smooth manifolds of the same dimension, $F \colon M \to N$ be a smooth map, and let $A \subset M$ have measure $0$. Prove that $F(A)$ has measure $0$.
    \item Let $M$ be a smooth manifold, and let $A \subset M$ have measure $0$. Prove that $A \neq M$.
    \item Let $M$ be a smooth manifold of dimension $m$, and let $\pi \colon TM \to M$ be the natural projection, $\pi(p,v) = p$. For a chart $(U,\phi)$ on $M$, define $\hat \phi\colon \pi^{-1}(U) \to \R^m \times \R^m$ by
    $$
    \hat \phi(p, v) = (\phi(p), d_p\phi(v)).
    $$
    \begin{enumerate}
    \item Prove that there is a unique topology on $TM$ such that for any chart  $(U,\phi)$ on $M$ the set $\pi^{-1}(U) \subset TM$ is open, and $\hat \phi$ maps $\pi^{-1}(U)$ homeomorphically to an open subset of $\R^{2m}$. 
    \item Show that $TM$ with this topology is Hausdorff and second countable. Hence show that $TM$ endowed with charts $(\pi^{-1}(U), \hat \phi)$ is a smooth manifold.
    \end{enumerate}
    \item   \begin{enumerate}
    \item Let $F \colon M \to N$ be a smooth map. Consider the map $dF \colon TM \to TN$ given by $dF(p,v) = (F(p), d_pF(v))$. This map is sometimes called the \textit{global differential of $F$} (but it is also okay to call it the differential of $F$). Prove that this map is smooth.
    \item Let $F \colon M \to \R^n$ be a smooth map. Consider the map $TM \to \R^n$ given by $(p,v) \mapsto d_pF(v)$. Prove that this map is smooth.
          \end{enumerate}
    \item    \begin{enumerate}
    \item Let $F \colon M \to N$ be a diffeomorphism, and let $v$ be a smooth vector field on $M$. Prove that $F_*v$ is a smooth vector field on $N$.
    \item Prove that this is in general not true if $F$ is just a smooth bijection.\\
    \textbf{Hint:} Compute the pushforward of the vector field $\dbyd{x}$ by the map $\R\to \R$ given by $x \mapsto x^3$.
            \end{enumerate}
            \item One says that the tangent bundle to $M$ is \textit{trivial} if there exists a diffeomorphism $\xi \colon TM \to M \times \R^m$ (where $m$ is the dimension of $M$) which is of the form $\xi(p,v) = (p, \dots)$, and moreover the restriction of $\xi$ to $T_pM$ is a vector space isomorphism between $T_pM$ and $\{p\} \times \R^m$.
            \begin{enumerate}
    \item Prove that $M$ has trivial tangent bundle if and only if $M$ is \textit{parallelizable}, which means that it admits smooth vector fields $v_1, \dots, v_m$ which form a basis of $T_pM$ at every point.
    \item Prove that $\R^n$ and $S^1$ are parallelizable manifolds.
      \end{enumerate}
      \textbf{Remark:} We will see later in the course that not all manifolds are parallelizable. In particular, $S^2$ and $\RP^2$ are not (for $S^2$ this follows e.g. from \textit{hairy ball theorem}). Moreover, the tangent bundle for each of these manifolds is not diffeomorphic to the direct product of the manifold itself and $\R^2$. (This is stronger then non-triviality of the tangent bundle, because triviality means that there exists a diffeomorphism $TM \to M \times \R^m$ with additional properties.)
              \end{enumerate}
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