\documentclass[reqno,11pt]{amsart}
%\documentstyle[amssymb 12pt]{article}
  
  \usepackage[all]{xy}
  \usepackage{epsf,epsfig,amsfonts,graphicx,color}
  \usepackage{xypic}

   
 \def\R{\mathbb R}
\def\bP{\mathbb P}
\def\C{\mathbb C}
 \def \E {\mathbb E}
 \def \V {\mathbb V}
  \def \W {\mathbb W}
 \def \S {\mathbb S}
  \def \I {\mathbb I}
 \def \Z {\mathbb Z}
\def\N {\mathbb N}
\def\T {\mathbb T}
  
 \def\on{orthonormal }
\def \nbhd {neighborhood }
\def \nbhds {neighborhoods }
\def\Ri{Riemannian }
\def\Ri{ subRiemannian }

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{conj}{Conjecture}

\newtheorem{exer}{Exercise}[section]

\newcommand{\dd}[2] { {{\partial #1}   \over {\partial #2}} }

\usepackage{amsmath,amssymb}
\usepackage{graphicx}





\date{today}




\setlength{\textheight}{9.5in}

%\voffset=-1in

\begin{document}
\pagestyle{empty}
%\input{mssymb}



\begin{center}\vspace{1in}{\bf MATH 209, MANIFOLDS II, WINTER 2011}\\
\bigskip
{\bf Homework Assignment 2: More one-forms and integration}\\
\end{center}
\bigskip

 

\bigskip

 [Contact forms; integral surfaces; two-forms ].
  


PRELUDE:  You will need to have done   exercise 4 from  from HW 1, 
the exercise about $\int xdy$, in order to do the exercise to hand in.  
 

\bigskip

{\bf 1.}  [HAND IN]  An integral curve $c$  for a one-form $\alpha$ is a curve satisfying
$c^* \alpha = 0$.  Let  the curve $c(t) = (x(t),y(t),z(t))  , 0 \le t \le 1 $be an integral curve for $\alpha  = dz - ydx$. Suppose that the projection
$ C(t) = (x(t), y(t))$  of $c$  to the xy plane is a closed curve. Show that $z(1) ? z(0)$  is the area enclosed by $C$.
[Cf. problem 2 of HW 1] 

\bigskip
 
{\bf 2.}  [HAND IN] 
A non-vanishing one-form $\alpha$ on a manifold  $M$ 
defines a  hyperplane field $D \subset TM$ by
$D = \{ \alpha = 0  \} $, in other words $D_m = \{v \in T_m M:  \alpha(m) v = 0  \}  \subset T_m M$ for $m \in M$ .   An  integral
submanifold for this field is an embedded submanifold whose tangent spaces
at each point is contained in the hyperplane field at that point.

(a) Let $i: \Sigma \to M$ be the inclusion of a submanifold into $M$. Show that $\Sigma$ is an integral
submanifold if and only if $i^* \alpha = 0$. 
 
(b) {\bf Show  that the hyperplane field defined by 
$\alpha = dz - ydx$ on $M = \R^3$ admits no integral surfaces.} 

Hint:  Argue by contradiction.  Suppose it did and let $\Sigma$ be such a surface and $i$ the inclusion.
Argue that $\Sigma$ can be expressed as a graph $z = f(x,y)$ and then look at closed  curves lying on $\Sigma$
and  use problem {\bf 1} . 

 
\bigskip

{\bf 3.}  Let  $\alpha$ be a  non-vanishing one-form

a)  Show that if $\alpha$ admits an integral hypersurface then 
$d \alpha \equiv 0$ (mod   $\alpha$).    {\bf Hint:  let $i$ be the inclusion. Use $i^*d = d i^*$.} 

b) Show that a two-form $\omega$ is $0$ mod $\alpha$ if and only if $\omega \wedge \alpha = 0$.
\bigskip

{\bf 4.} Write $z = x + iy$  so as to identify $\C$  with $\R^2$  as usual. Express the real and imaginary parts of the complex one-form 
$dz/z$  in terms of $x, y, dx, dy$. Repeat, using
 polar coordinates $r, ?$  and $dr, d?$.



\bigskip
{\bf 5.}    For $\omega = dx \wedge dy + dz \wedge dw $ on $\R^4$
compute $\omega^2 : = \omega \wedge \omega$.
\bigskip

{\bf 6.}   Coordinatize $\R^{2n}$ by coordinates $x^1, \ldots , x^n, y_1, \ldots y_n$.
Set $\omega = \Sigma dx^i \wedge dy_i$.     Then $\omega^n = c dx^1 \wedge  dx^2 \ldots \wedge dx^n \wedge dy^1 \wedge \ldots dy^n$
for some constant $c$.  Find $c$.



\end{document} 
