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\begin{document}

\title[Homework \#1]%
{{\large \quad MTH3400 --- Geometry 1 --- Spring 1997}\\[0.2in]
{\bfseries Homework \#2}}


%\author{Alexander I.~Suciu}

\maketitle

\begin{enumerate}

\item Let $p\in S^n \subset \R^{n+1}$ and consider the tangent 
space at $p$, 
\[
T_p(S^n)=\{ \alpha^{\prime}(0) \mid \alpha:(-\epsilon,\epsilon)\to \R^{n+1} 
\text { is smooth, } \alpha(0)=p,\, \im(\alpha)\subset S^n\}.
\]
Prove that $T_p(S^n)$ is the linear subspace of $\R^{n+1}=T_p(\R^{n+1})$ 
consisting of all vectors $v \in \R^{n+1}$ such that $v\perp p$.

\vskip 0.25truein

\item  Let $U\subset \R^n$, $V\subset \R^m$ be open subsets, 
and $f:U \to V$ a smooth map.  
 \begin{enumerate}
    \smallskip\item 
Suppose there is a smooth map $g:V\to \R$ of rank $1$ 
such that $g\circ f\equiv 0$ on $U$.  Show that the 
rank $r$ of the Jacobian matrix $Jf$ is everywhere $<m$.  
    \smallskip\item 
Suppose that $f$ does have constant rank $r<m$ 
on $U$.  Show that, on some open subset $W\supset f(U)$, 
there is a smooth map $g:W\to \R$ of rank $1$ 
such that $g\circ f\equiv 0$ on $U$.
  \end{enumerate}

\vskip 0.25truein

\item  (See also Problem~33, p.~83, in Spivak's book.) 
Consider the {\it orthogonal group} 
\[
 \O(n)=\{A\in \GL(n)\mid A^{\top}\cdot A= I\},
\]
where $A^{\top}$ denotes the transpose of $A$, and $I$ is the 
identity $n\times n$ matrix.  Consider  the map $\Phi:\GL(n)\to \GL(n)$,  
defined by $\Phi(A)= A^{\top}\cdot A$.  Prove:

  \begin{enumerate}
   \smallskip\item Show that $\O(n)$ is a compact subspace of $\GL(n)$.  
    \smallskip\item $\Phi$ is smooth.
    \smallskip\item Relative to the standard identification $T_I(\GL(n)) = \Mat(n)$, 
the differential $d_I\Phi:T_I(\GL(n))\to T_I(\GL(n))$ is given by:
\[
d_I\Phi(A) = A^{\top} + A.
\]
   \item  The map $\Phi$ has constant rank $\frac{n(n+1)}{2}$.
    \smallskip\item  using the above, conclude that the orthogonal group 
$\O(n)\subset \GL(n)$ is a smooth, compact submanifold of 
dimension $\frac{n(n-1)}{2}$.
  \smallskip\item Show that the vector subspace $T_I(\O(n))\subset \Mat(n)$ 
is the space of skew-symmetric matrices.
  \end{enumerate}

\vskip 0.25truein

\item  Problem~34, pp.~83-84, in Spivak's book.

\end{enumerate}

\end{document}