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\begin{document}

\title[Homework \#4]%
{{\large \quad MTH3400 --- Geometry 1 --- Spring 1997}\\[0.2in]
{\bfseries Homework \#4}}


%\author{Alexander I.~Suciu}

\maketitle

\begin{enumerate}

\item
\begin{enumerate}
\item Show that $T^n$ is parallelizable, for all $n\ge 1$.
\smallskip
\item Show that $S^n\times S^k$ is parallelizable, for all 
$n$ odd, and $k\ge 1$.  
\smallskip
\item Show that $S^{4n-1}$ has $3$ independent vector fields, 
for all $n\ge 1$.  
\end{enumerate}

\vskip 0.2truein

\item Let $X\in \mathfrak{X}(\R)$ be the vector field 
\[
X= e^t \frac{d}{dt}\, .
\]
For $a\in \R$, compute the integral curve 
$s_a:(-\epsilon_a,\delta_a)\to \R$ to $X$ through $a$.  
(Be sure to specify its domain.) 
Find the flow determined by $X$.  Is $X$ a complete 
vector field?

\vskip 0.2truein

\item  Let $X, Y\in \mathfrak{X}(\R)$ be smooth 
vector fields given by
\begin{align*}
X&=x^2 \cos^2(\phi(x)) \frac{d}{dx}\\
Y&=x^2 \sin^2(\phi(x)) \frac{d}{dx}
\end{align*}
where $\phi:\R\to [0,\frac{\pi}{2}]$ is a smooth map such that 
$\phi\equiv 0$ on $[0,\frac{1}{5}]\cup [\frac{4}{5},1]$, 
$\phi\equiv \frac{\pi}{2}$ on $[\frac{2}{5},\frac{3}{5}]$, 
and $\phi(x+1)=\phi(x)$.  (Such a map exists by the smooth 
Urysohn lemma.) Prove that:
\begin{enumerate}
\item $X$ and $Y$ are complete vector fields.
 \smallskip\item $X+Y$ is not complete.
\end{enumerate}

\vskip 0.2truein

\item Let $\Phi:\R\times M\to M$ be a (global) flow.  
A subset $C\subset M$ is said to be {\it $\Phi$-invariant} 
if $\Phi_t(x)\in C$, for all $x\in C$, $t\in \R$. Let 
$R_x = \{\Phi_t(x)\}_{t\in \R}$ be the flow line 
through $x$.  Show that the closure $\overline{R}_x$ 
is a $\Phi$-invariant set.

\vskip 0.2truein

\item Given $A\in \Mat(n)$, view the right translation 
\[
R_A:\GL(n)\to \Mat(n)
\]
as a  vector field $R_A\in \frak{X}(\GL(n))$.  (The value 
of this vector field at $B\in \GL(n)$ is the matrix 
$BA\in \Mat(n)=T_B(\GL(n))$.)    For
\[
A=\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}
\]
find the (global) flow $\Phi: \R\times \GL(2) \to \GL(2)$ 
generated by $R_A$. 

\vskip 0.2truein

\item Problem 13 from Chapter 5 in Spivak's book.

\end{enumerate}

\end{document}