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\begin{document}

\title[Homework \#1]%
{{\large \quad MTH3400 --- Geometry 1 --- Spring 1997}\\[0.2in]
{\bfseries Homework \#3}}


%\author{Alexander I.~Suciu}

\maketitle

\begin{enumerate}

\item  Let $p:E\to M$ be a vector bundle.   
\begin{enumerate}
    \smallskip\item 
Show there is a map of vector bundles (over $M$)  
$f: M\times \R^n\to E$ 
which is surjective on each fiber, if and only if there
are sections $s_1,\ldots,s_n$ of $E$ over $M$ such that, for each
$x\in M$,  $s_1(x),\ldots,s_n(x)$ span the fiber $p^{-1}(x)$.
    \smallskip\item
Show there is a map of vector bundles (over $M$) 
$f: M\times \R^n\to E$ which is injective on each fiber, 
if and only if there are sections $s_1,\ldots,s_n$ of $E$ over $M$ 
such that, for each $x\in M$,  $s_1(x),\ldots,s_n(x)$ 
are linearly independent in the fiber $p^{-1}(x)$.
    \smallskip\item
Show that $E$ is isomorphic to the trivial rank $n$ bundle
$M\times \R^n\to M$ if and only if there exist sections
$s_1,\ldots, s_n$ of $E$ over $M$ such that, for each $x\in M$, 
$s_1(x),\ldots, s_n(x)$ form a basis for the fiber $p^{-1}(x)$.
\end{enumerate}

\vskip 0.2truein

\item   \begin{enumerate}
\item 
Let $u=x^2-y^3$, $v=3xy+y^2-x^2$.  For which $(a,b)$ in
$\R^2$ is there a neighborhood $U$ of $(a,b)$ such that $(U,(u,v))$
is a coordinate system?
 \smallskip\item  
For which real numbers $c$ is the locus $y^2-x(x-1)(x-c)=0$
a submanifold of $\R^2$? 
  \end{enumerate}

\vskip 0.2truein

\item  Let 
$\RP^2=
\{x=(x_1,x_2,x_3)\in \R^3 \mid x_1^2+x_2^2+x_3^2=1\}/x\sim -x$ 
be the real projective space, with the standard 
differentiable structure.  Let $\Phi:\RP^2 \to \R^5$ be given by
\[
\Phi([x]) = (x_1^2, x_2^2, x_1x_2, x_1x_3, x_2x_3)
\]
Show that $\Phi$ is a well-defined, smooth embedding.  Compute 
$\Phi_*:T(\RP^2)\to T(\R^5)$. 


\vskip 0.2truein

\item 
\begin{enumerate}
\item  
Prove that  the group $\SO(3)$  of orthogonal $3\times 3$ matrices 
with determinant $1$ is an embedded submanifold of the $9$-dimensional 
Euclidean space of all $3\times 3$ matrices.
 \smallskip\item  
Prove that $\SO(3)$ with the differentiable structure described above is
diffeomorphic to the real projective space $\RP^3$ with the standard 
differentiable structure.

%{\it Hint:} Represent an ortogonal transformation as a rotation 
%around an axis.
  \end{enumerate}

\vskip 0.2truein

\item  Describe the structure of  differentiable manifold on the complex 
projective space $\CP^n$ by explicitly defining coordinate charts and 
calculating transition functions.


\vskip 0.2truein

\item 

Let $M$ be a differentiable manifold and $f:M\to M$ a diffeomorphism. 
Consider the direct product  $M\times [0,1]$  with the identification of pairs 
of points $(f(x),0)$ and $(x,1)$, for all $x\in M$.  Let 
$M_f=M\times [0,1]/\sim$ be the quotient 
space (called the {\it mapping torus } of $f$, or, the 
{\it suspension construction } on $f$.)
\begin{enumerate}
\item  
Show that  $M_f$ possesses a natural structure of differentiable 
manifold.   
\smallskip\item
Show  that $M_f$ is a locally trivial fiber bundle with base 
$S^1$ and fiber $M$.
\smallskip\item  
Apply the suspension  construction to the following three cases:
\smallskip
\begin{enumerate}
\item  
$M=\R,\  \  f(x)=-x$.
\item  
$M=S^1,\ f(z)=-z$.
\item  
$M=S^1,\  f(z)=\overline{z}$.
\end{enumerate}
 \smallskip
Identify the resulting mapping tori.  In which of the three cases is the
bundle $M_f \to S^1$ trivial?  
\end{enumerate}

\end{enumerate}

\end{document}