\documentclass[11pt]{report}
 
\usepackage[margin=0.5in]{geometry} 
\usepackage{amsmath,amsfonts,amssymb,amsthm,hyperref,xy,tikz,mathtools,float}
\xyoption{all}
\xyoption{pdf}


\newtheorem{definition}{Definition}

 
\begin{document}

\large{2014 Fall Math 5041 \quad Assignment 6. \qquad Due: Dec 8, 2014}
\medskip

\large{Name: }
\bigskip

1. (a) Let $V$ be a vector space. Let $v_1, v_2, \ldots, v_k \in V$ be a finite collection of vectors. Show that
$$
		v_1 \wedge v_2 \wedge \ldots \wedge v_k \neq 0
$$
if and only if $v_1, v_2, \ldots, v_k$ are linearly independent.

\smallskip

(b) For any $k$-form $\alpha$ and any $l$-form $\beta$ on a manifold $M$, show that
$$
	\alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha = (-1)^{|\alpha||\beta|} \beta \wedge \alpha.
$$

\bigskip

2. We think of $V = \mathbb{R}^3$ as a vector space. Let $M$ be an open subset of $\mathbb{R}^3$. The standard inner product $\langle \cdot, \cdot \rangle$ on $\mathbb{R}^3$ defines an isomorphism
$$
	V \to V^*, \qquad v \mapsto \langle v, \cdot \rangle
$$
which in turn induces an isomorphism of spaces of sections
$$
	A : \Gamma(M, TM) \to \Omega^1(M), \qquad A(X) = \langle X, \cdot \rangle
$$
where we have used $TM = M \times V$ and $T^*M = M \times V^*$.

On the other hand, the standard volume form $\omega_0 = dx_1 \wedge dx_2 \wedge dx_3$ defines an isomorphism
$$
	V \to \wedge^2 V^*, \qquad v \mapsto \iota_v \omega_0 = \omega_0(v, \cdot)
$$
which also induces an isomorphism
$$
	B: \Gamma(M, TM) \to \Omega^2(M), \qquad B(X) = \iota_X \omega_0 = \omega_0(X, \cdot).
$$
Finally, we also have an isomorphism
$$
	C : C^\infty(M) \to \Omega^3(M), \qquad C(f) = f \omega_0.
$$

Show that the standard vector calculus notions of $\mathtt{grad}$, $\mathtt{curl}$ and $\mathtt{div}$ can be defined as follows:
\begin{enumerate}
	\item $\mathtt{grad}(f) = A^{-1}(df)$ for $f \in C^\infty(M)$.
	\item $\mathtt{curl}(X) = B^{-1}(d(A(X)))$ for $X \in \Gamma(M, TM)$.
	\item $\mathtt{div}(X) = C^{-1}(d(B(X)))$ for $X \in \Gamma(M, TM)$.
\end{enumerate}
In particular, you have proved that the following diagram commute.
\begin{equation}
	\begin{aligned}
	\xymatrix{
		C^\infty(M) \ar[d]_-{\cong} \ar[r]^-{\mathtt{grad}} & \Gamma(M, TM) \ar[r]^-{\mathtt{curl}} \ar[d]^-{A} & \Gamma(M, TM) \ar[r]^-{\mathtt{div}} \ar[d]^-{B} & C^\infty(M) \ar[d]^-{C}\\
		\Omega^0(M) \ar[r]^-{d} & \Omega^1(M) \ar[r]^-{d} & \Omega^2(M) \ar[r]^-{d} & \Omega^3(M)
	}
	\end{aligned}
\end{equation}

\bigskip

3. (a) Show that an inner product $g$ on a vector space $V$ defines an isomorphism $g^\sharp: V \to V^*$ with
$$
	(g^\sharp(v))(w) \coloneqq g(v, w)
$$
for all $v, w \in V$.

\smallskip
(b) Recall that a Riemannian metric $g$ on a manifold $M$ is a smooth section of the bundle $\mathtt{Sym}^2 T^*M \to M$ such that the bilinear map $g_x : T_xM \times T_xM \to \mathbb{R}$ is an inner product. If $g$ is a Riemannian metric on a manifold $M$, the pair $(M, g)$ is called a Riemannian manifold. Show that every manifold M admits a Riemannian metric.

\smallskip

Hint: Partition of unity.


\bigskip

4. Use the results in Problem 2. to recover the classical Stoke's theorem and the divergence theorem from the (generalized) Stoke's theorem.

\bigskip

5. Let $M$ be a manifold with boundary. Recall that a smooth retraction from $M$ to $\partial M$ is a smooth map $r: M \to \partial M$ that fixes $\partial M$. Show that if $M$ is orientable and compact, then there is no such retraction.

\smallskip

Remark: This, together with the fact that the closed unit ball is orientable, gives an alternative proof of the Brouwer's fixed point theorem. Why? Re-examine our original proof of Brouwer fixed point theorem!

\bigskip

6. (a) Let $M$ be a manifold, and let $L$ be an oriented compact regular submanifold of dimension $l$. Let $j: L \hookrightarrow M$ be the embedding. Show that
$$
		H^k(M) \to \mathbb{R}, \qquad [\omega] \mapsto \int_L j^*(\omega)
$$
is a linear map.

\smallskip

(b) Consider the 1-form
$$
	\alpha = \frac{-y}{x^2 + y^2} dx + \frac{x}{x^2 + y^2} dy
$$
on $M = \mathbb{R}^2 \setminus \{(0, 0)\}$. Show that $d\alpha = 0$ and $[\alpha] \neq 0 \in H^1_{dR}(M)$.




\end{document}