\documentclass[12pt]{amsart}

\topmargin-0.7truein
\textwidth6.5truein
\textheight9.6truein
\oddsidemargin=0.0truein
\evensidemargin=0.0truein

\newcommand{\Z}{{\mathbb Z}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\RP}{{\mathbb {RP}}}
\newcommand{\CP}{{\mathbb {CP}}}

\newcommand{\Int}{{\operatorname{int}}}
\newcommand{\Hom}{{\operatorname{Hom}}}
\newcommand{\Mat}{{\operatorname{Mat}}}
\newcommand{\GL}{{\operatorname{GL}}}
\newcommand{\SO}{{\operatorname{SO}}}
\renewcommand{\O}{{\operatorname{O}}}
\newcommand{\im}{{\operatorname{im}}}


\renewcommand{\labelenumi}{{\bfseries\arabic{enumi}.}}
\renewcommand{\theenumii}{\roman{enumii}}
\renewcommand{\theenumiii}{\alph{enumiii}}
\renewcommand{\thepage}{}

\begin{document}

\title[Homework \#5]%
{{\large \quad MTH3400 --- Geometry 1 --- Spring 1997}\\[0.2in]
{\bfseries Homework \#5}}


%\author{Alexander I.~Suciu}

\maketitle

\begin{enumerate}

\item Prove that on any non-compact, connected differentiable manifold there 
exists an incomplete smooth vector field.

\vskip 0.2truein

\item Give an example of a submersion $f:M\to N$, and a vector 
field  $X\in \mathfrak{X}(M)$ which is not $f$-related to any 
 $Y\in \mathfrak{X}(N)$.  

\vskip 0.2truein

\item Construct $3$ linearly independent vector fields on $S^3$, 
and calculate their Lie brackets.

\vskip 0.2truein

\item
Let $H$  be the Heisenberg group
\[
H=\left\{ \pmatrix 1&x_{12}&x_{13}\\0&1&x_{23}\\0&0&1\endpmatrix
\bigm| x_{12}, x_{13}, x_{23} \in \R \right\} 
\]
%of upper-diagonal $3\times 3$ real matrices with $1$'s on the diagonal. 
This group has natural coordinates $(x_{12}, x_{13}, x_{23})$, 
and it acts on itself by left translations. 
Let  $v_{12},v_{13},v,_{23}$ be the left-invariant vector-fields on $H$, with 
values at the identity  $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$, respectively. Consider 
the $2$-dimensional distributions  $E$ and $F$ on $H$  generated by
$v_{12}, v_{13}$ and $v_{12}, v_{23}$, respectively. 
Show that  $E$ is integrable and $F$ is not.


\vskip 0.2truein

\item  Let  $M=\R^3\setminus \{0\}$.  Consider the following 
$2$-dimensional distributions on $M$:
\smallskip
\begin{enumerate}
\item $E=\cup_{v\in M}E_v$, with $E_v=\{w \in T_vM\mid w\perp v\}$;
\smallskip
\item $F=\cup_{v\in M}F_v$, with $F_v=\{(a,b,c) \in T_vM\mid (b,c,a)\perp v\}$.
\end{enumerate}
\smallskip
In each case, decide whether the distribution is integrable or not, and, if 
it is, describe the integral manifolds.

\end{enumerate}

\end{document}