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\title{MAT 239: Homework 3}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		
		\item Given two parametrizations, $\phi_1: U_1\to X$ and $\phi_2: U_2\to X$ whose images overlap in $X$, we can choose an orientation on $U_1\subset \R^n$ and an orientation on $U_2\subset \R^n$, each of which induces an orientation on the portion of $X$ in the image. 
		\begin{enumerate}
			\item Suppose the images of $\phi_1$ and $\phi_2$ overlap in $X$. Show that the following are equivalent:
			\begin{enumerate}
				\item The orientation induced by $d\phi_1$ is equivalent to the orientation induced by $d\phi_2$ on the tangent spaces $T_xX$ for points $x$ in the intersection of the images of $\phi_1$ and $\phi_2$.
				\item For the transition function $t=\phi_2^{-1}\circ \phi_1$, $dt_s$ takes the orientation on $U_1$ to an equivalent orientation on $U_2$.
				\item If $\det(dt_s)$ is positive, the orientation on $U_1$ is equivalent to the orientation on $U_2$ (as open subsets of $\R^n$), and if $\det(dt_s)$ is negative, the orientation on $U_1$ is inequivalent to the orientation on $U_2$ (as open subsets of $\R^n$).
			\end{enumerate}
			If any of these equivalent notions holds, we say that the orientations chosen on $\phi_1:U_1\to X$ and $\phi_2:U_2\to X$ are \emph{compatible}.
			\item Suppose $\phi_i: U_i\to X$ are parametrizations for $i=1,2,3$ such that $ \image(\phi_1), \image(\phi_2) \subset \image(\phi_3)$ and $\image(\phi_1)\cap \image(\phi_2)\neq \emptyset$. Show that if we choose orientations for $\phi_1$, $\phi_2$, and $\phi_3$, such that the $\phi_1$ orientation is compatible with the $\phi_2$ orientation and the $\phi_1$ orientation is compatible with the $\phi_3$ orientation, then the $\phi_2$ orientation is compatible with the $\phi_3$ orientation.
		\end{enumerate}
			
		\item Suppose $X$ and $Y$ are orientable manifolds. Prove that $X\times Y$ is orientable. Conclude that the $n$-torus $T^n=S^1\times \cdots \times S^1$ is orientable.
					
		\item Let $A$ be an $n\times n$ matrix and $b$ a vector in $\R^n$. Show that the map $f:\R^n\to \R^n$ defined by $f(x)=Ax+b$ is a smooth map with $df_x=A$ at every $x\in \R^n$. Conclude that $f$ is a diffeomorphism if and only if $\det(A)\neq 0$.
		
		\item Let $a\in \R$ be a constant, and define $g_a: \R \to S^1\times S^1$ by 
		$$g_a(t) = ( \cos t, \sin t, \cos(at), \sin(at) ).$$
		Prove that $g_a$ is an immersion. For which $a$ is $g_a$ injective and why? For which $a$ is $g_a$ an embedding and why?
		
		\item Let $f:X\to Y$ be a smooth function. Define $gr^f: X \to X\times Y$ by $gr^f(x) = (x,f(x))$. Prove that $gr^f$ is an embedding. Conclude that its image $graph(f)=gr^f(X)$ is a smooth submanifold of $Y$ which is diffeomorphic to $X$.
		
		\item \begin{enumerate}
			\item If $f$ and $g$ are immersions, show that $f\times g$ is an immersion.
			\item If $f$ and $g$ are submersions, show that $f\times g$ is a submersion.
			\item If $f$ and $g$ are immersions, show that $g\circ f$ is an immersion.
			\item If $f$ and $g$ are submersions, show that $g\circ f$ is a submersion.
		\end{enumerate}
		
		
	\end{enumerate}

\end{document}