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\title{MAT 239: Homework 2}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item \textbf{Chain Rule:} Assuming the chain rule for maps between open subsets of $\R^n$ 
		$$f:\R^n\to \R^m, \; g:\R^m\to \R^k, \; d(g\circ f)_x = dg_{f(x)}\circ df_x,$$
		prove the chain rule for maps between smooth manifolds, i.e. if $X,Y,Z$ are smooth manifolds of dimension $m,n,k$ respectively and $f:X\to Y$ and $g:Y\to Z$ are smooth maps, prove that 
		$$d(g\circ f)_x = dg_{f(x)}\circ df_x$$
		 using parametrizations.
		
		\item \textbf{Vector fields on $S^2$:} Let 
		$$S^2=\{(x,y,z)\in \R^3 \mid x^2+y^2+z^2=1 \}.$$
		Prove that $\zeta: S^2\to TS^2$ defined by 
		$$\zeta((x,y,z)) = ((x,y,z), \langle z,z,-x-y \rangle )$$
		is a smooth vector field on $S^2$.
		
		\item \textbf{$TT^2$ is a trivial bundle:} Let $T^2=S^1\times S^1$ (you may embed in $\R^3$ or $\R^2\times \R^2=\R^4$ as in hw1).
		\begin{enumerate}
			\item Find two vector fields $\zeta_1$ and $\zeta_2$ on $T^2$, such that for each $x\in T^2$ $(\zeta_1(x), \zeta_2(x))$ is a basis for $T_xT^2$. (Define $\zeta_1$ and $\zeta_2$, check they are smooth vector fields, and check they give a basis at each point.)
			\item Show that the map
			$$\Psi: T^2\times \R^2 \to TT^2$$
			defined by
			$$\Psi(x,(a,b)) = (x,a\zeta_1(x)+b\zeta_2(x))$$
			is a diffeomorphism.
		\end{enumerate}
		When $TM$ is diffeomorphic to $M\times \R^n$, we say that $TM$ is a \emph{trivial bundle}.
		
	\end{enumerate}
	
	
\end{document}