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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Tangent spaces, immersions, submersions}}
\vskip 1cm
  Most problems here are from Guillemin-Pollack.

  \subsection*{Tangent spaces}

  
  
\begin{enumerate}[1.]
\item Find the tanget space, as a subspace of $\R^3$, of the
  paraboloid defined by
  $$x^2+y^2-z^2=a$$ at $(\sqrt{a},0,0)$ where $a>0$.

  
  \item Find the tangent space, as a subspace of the space of complex
    $n\times n$ matrices, of the unitary group $U(n)$ at $I\in U(n)$.  

  \end{enumerate}

  \subsection*{Immersions and local diffeomorphisms}
\begin{enumerate}[resume]
\item Let $f:\R\to\R$ be a local diffeomorphism. Prove that the image
  of $f$ is an open interval and $f$ maps $\R$ diffeomorphically onto
  this interval. Hint: The set of points $x$ where $f'(x)>0$
  [$f'(x)<0$] is open.

\item Construct a local diffeomorphism $f:\R^2\to\R^2$ that is not
  injective. 

\item \label{5} Suppose $f:X\to Y$ is an injective local diffeomorphism. Show
  that $f$ is a diffeomorphism onto its image.

\item Let $G:\R^2\to S^1\times S^1$ be defined by $G(s,t)=(e^{2\pi
  is},e^{2\pi it})$. It's a local diffeomorphism from the plane to the
  torus. Show that if $L$ is an irrational slope line given by the
  equation $t=as$ with $a$ irrational, then $G$ is injective on $L$.

\item Let $x_1,x_2,\cdots,x_n$ be the standard coordinate functions on
  $\R^n$, i.e. $x_i$ is the projection to the $i^{th}$ coordinate. Let
  $X\subset\R^n$ be a $k$-submanifold. Show that for every $x\in X$
  there are some $k$ coordinate functions $x_{i_1},\cdots,x_{i_k}$
  that form a local coordinate system around $x$. In other words, the
  projection to these $k$ coordinates is a chart around $x$.

\item (Inverse Function Theorem generalized) Let $f:X\to Y$ be a
  smooth map, $Z\subset X$ a submanifold, and assume that $f$ is
  injective on $Z$ and it is a local diffeomorphism at every point
  $z\in Z$. Show that $f$ maps a neighborhood of $Z$ diffeomorphically
  onto its image. Hint: Use Problem \ref{5}.

\end{enumerate}
  
\subsection*{Submersions}

\begin{enumerate}[resume]
  \item If $X$ is compact (and nonempty!) and $Y$ connected, then a
    submersion $f:X\to Y$ is surjective. Hint: Submersions send open
    sets to open sets.

  \item Let $P$ be a degree $m>0$ homogeneous polynomial in $k$ variables, meaning
    that $$P(tx_1,\cdots,tx_k)=t^m P(x_1,\cdots,x_k)$$
    Show that any $a\neq 0$ is a regular value of $P:\R^k\to\R$, so
    that $\{P(x)=a\}$ is a submanifold of $\R^k$.

    \item (Stack of records theorem) Let $f:X\to Y$ be smooth with $X$
      compact and $\dim X=\dim Y$. Let $y$ be a regular value of
      $f$. Show that there is a neighborhood $U$ of $y$ such that
      $f^{-1}(U)$ is a finite disjoint union $V_1\sqcup\cdots\sqcup
      V_N$ of open sets such that $f$ maps each $V_i$
      diffeomorphically to $U$. See the picture and hint in
      Guillemin-Pollack.

      \item Prove that the set of $2\times 2$ matrices of rank 1 is a
        3-dimensional submanifold of $M(2)=\R^4$. Hint: Consider
        $\det:M(2)\smallsetminus\{0\}\to\R$.

 \end{enumerate}
  
\subsection*{Miscellaneous}

\begin{enumerate}[resume]
  \item Consider the set $\C^{n+1}\smallsetminus\{0\}$ with the
    equivalence relation
    $$(X_0,X_1,\cdots,X_n)\sim (Y_0,Y_1,\cdots,Y_n)$$ if there is a
    nonzero complex number $\lambda$ such that $X_j=\lambda Y_j$ for
    all $j$. The quotient space is the {\it complex projective space}
    $\C P^n$. The equivalence class of $(X_0,X_1,\cdots,X_n)$ is
    usually denoted $[X_0:X_1:\cdots:X_n]$.
    Show that this is a manifold of dimension $2n$. More
    specifically, let $U_j=\{[X_0,X_1,\cdots,X_n]\mid X_j\neq 0\}$ and
    define $\phi_j:U_j\to\C^n$ by $$[X_0,X_1,\cdots,X_n]\mapsto
    \bigg(\frac{X_0}{X_j},
    \frac{X_1}{X_j},\cdots,\frac{X_{j-1}}{X_j},\frac{X_{j+1}}{X_j},
    \cdots,\frac{X_n}{X_j}\bigg)$$
    Show that these define an atlas.

  \item The {\it real projective space} $\R P^n$ is defined similarly,
    but the equivalence relation is on $\R^{n+1}\smallsetminus\{0\}$
    and $\lambda\in\R\smallsetminus\{0\}$. Show similarly that $\R
    P^n$ is a manifold of dimension $n$. Also show that $\R P^1$ is
    diffeomorphic to $S^1$. (It is also true that $\C P^1$ is
    diffeomorphic to $S^2$.)

    \item Show that $SL_n(\R)$ is path connected. Hint: From linear
      algebra, every matrix in $SL_n(\R)$ can be transformed to the
      identity $I$ by elementary row operations. Show that one can
      interpolate a path between any two consecutive such
      matrices. For example, adding the first row to the second can be
      interpolated by adding $t$ times the first row for $t\in [0,1]$.

\end{enumerate}

It is also true, by a similar argument, that $GL_n(\R)$ has two
components, but $GL_n(\C)$ is connected.
    
\end{document}
