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\title{MAT 239: Homework 1}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item \textbf{Stereographic projection:} Stereographic projection defines a diffeomorphism $p_N:S^2\setminus \{N\}\to \R^2$ where $N=(0,0,1)$ is the north pole as follows. Embed $\R^2$ in $\R^3$ as the points where $z=0$. Then for any point $(x_0,y_0,z_0)\in S^2\setminus \{N\}$, the line through $(x_0,y_0,z_0)$ intersects $\R^2$ at a unique point $p_N((x_0,y_0,z_0))$.
			\begin{enumerate}
				\item Find an explicit formula for $p_N((x_0,y_0,z_0))$.
				\item Find an explicit formula for $p_N^{-1}: \R^2\to S^2\setminus \{N\}$.
				\item Define an analogous projection $p_S:S^2\setminus\{S\} \to \R^2$, and compute the transition function $p_S\circ p_N^{-1}$ on the domain where they are commonly defined.
				\item Generalize stereographic projection to define a diffeomorphism $p_N^k: S^k\setminus\{N\} \to \R^k$.
			\end{enumerate}
		
		\item \textbf{Tangent space to the sphere:} Consider the $2$-sphere
		$$S^2 = \{(x,y,z)\in \R^3 \mid x^2+y^2+z^2=1 \}.$$
		\begin{enumerate}
			\item Determine the tangent space $T_{(x,y,z)}S^2$.
			\item Let $f:S^2\to \R$ be the restriction of the projection $f(x,y,z)=x$. Calculate 
			$$df_{(x,y,z)}: T_{(x,y,z)}S^2 \to T_x\R$$
			and determine at which points $(x,y,z)$ is this the zero map.
		\end{enumerate}
		
		
		\item \textbf{The $2$-Torus:} The $2$-torus $T^2$ is defined to be $S^1\times S^1$. We can realize this as a subset of $\R^N$ in different ways.
			\begin{enumerate}
				\item Viewing $S^1\subset \R^2$, defines $T^2=S^1\times S^1 \subset \R^4$. Define diffeomorphisms from enough open subsets of $\R^2$ to open subsets of $T^2\subset \R^4$ to cover $T^2$ to prove that it is a $2$-dimensional manifold.
				\item Consider the subset of points $T_{a,b}$ in $\R^3$ at distance $b$ from the circle of radius $a$ where $0<b<a$. Prove that $T^2$ is diffeomorphic to this subset.
				\item Determine the tangent space to $T_{a,b}$ at each point, presented as the span of two vectors. 
			\end{enumerate}
		
		\item \textbf{Manifolds with boundary:} 
			\begin{enumerate}
				\item Prove that the unit ball
					$$B^n = \{(x_1,\dots, x_n) \mid x_1^2+\cdots + x_n^2 \leq 1 \}$$
					is a smooth $n$-manifold with boundary.
				\item Prove that the unit square
					$$R = \{ (x,y) \mid 0\leq x,y \leq 1 \}$$
					is NOT a 2-manifold with boundary.
			\end{enumerate}
		
		\pagebreak
		
		\item \textbf{Projective spaces:} 
		\begin{enumerate}
			\item The \emph{real $n$-dimensional projective space}, $\rpn$ is the quotient of $\R^{n+1}\setminus 0$ by the equivalence relation $(x_1,\cdots, x_{n+1})\sim(\lambda x_1,\cdots, \lambda x_{n+1})$ for $\lambda\in \R\setminus 0$. A typical way to represent $\R P^n$ is to use \emph{homogeneous coordinates} as follows:
			$$\rpn = \{[x_1:x_2:\cdots: x_n, x_{n+1}] \mid (x_1,x_2,\cdots, x_n, x_{n+1})\in \R^{n+1}\setminus (0,0,\cdots, 0,0) \}$$
			where $[x_1:x_2:\cdots : x_n: x_{n+1}] = [\lambda x_1: \lambda x_2: \cdots : \lambda x_n : \lambda x_{n+1}]$.
			
			Prove that $\rpn$ is a manifold. Hint: show that the subsets of points with a representative where $x_i=1$ give a covering by coordinate charts.
			
			\item The complex projective spaces, $\cpn$, is defined similarly as the quotient of $\C^{n+1}\setminus 0$ by the equivalence relation $(z_1,\cdots, z_{n+1})\sim(\lambda z_1,\cdots, \lambda z_{n+1})$ for $\lambda\in \C\setminus 0$. It also can be expressed in homogeneous coordinates $[z_1:z_2:\cdots: z_{n+1}]=[\lambda z_1:\lambda z_2:\cdots:\lambda z_{n+1}]$ for $\lambda\in \C\setminus 0$. Show that $\cpn$ is a manifold. What is its (real) dimension?
		\end{enumerate}
		
		\item \textbf{Products:}
		\begin{enumerate}
			\item Suppose $f:X\to X'$ and $g:Y\to Y'$ are smooth maps. Define the product map $f\times g: X\times Y \to X'\times Y'$ by
			$$(f\times g)(x,y) = (f(x), g(y)).$$
			Prove that $f\times g$ is smooth and that $d(f\times g)_{(x,y)} = df_x \times dg_y$.
			\item Show that for any smooth manifolds $X$ and $Y$, 
			$$T_{(x,y)}(X\times Y) = T_xX \times T_xY.$$
			\item Show that the projection map $\pi: X\times Y \to X$ given by $\pi(x,y) = x$ is smooth and $d\pi_{(x,y)}(v,w) = v$ for $(v,w)\in T_xX\times T_yY = T_{(x,y)}(X\times Y)$.
		\end{enumerate}
		
		
		\item \textbf{Cut-off and bump functions:}
			\begin{enumerate}
				\item Consider the following function $\rho: \R\to \R$ defined by
				$$\rho(x) = \begin{cases} e^{-1/x^2} & x>0 \\ 0 & x\leq 0 \end{cases}$$
				Prove that $\rho$ is smooth.
				
				\item For $a<b$, define $\sigma_{a,b}(x) = \rho(x-a)\rho(b-x)$. Prove that $\sigma_{a,b}$ is a smooth function which is positive on $(a,b)$ and $0$ elsewhere.
				
				\item Define 
				$$\tau_{a,b}(x) = \frac{\int_x^b \sigma_{a,b}(u)du }{\int_a^b \sigma_{a,b}(u)du }.$$
				Prove that $\tau_{a,b}$ is a smooth function such that $\tau_{a,b}(x)=1$ for $x\leq a$, $0<\tau_{a,b}(x)<1$ for $a<x<b$, and $\tau_{a,b}(x)=0$ for $t\geq b$.
				
				\item Construct a smooth function on $\R^k$ that equals $1$ on points in the ball of radius $\varepsilon>0$, $0$ outside the ball of radius $\alpha>\varepsilon$, and is strictly between $0$ and $1$ in between.
				
			\end{enumerate}
		
	
	\end{enumerate}
	
	
\end{document}