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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Tangent Spaces, transversality, Lie bracket}}
\vskip 1cm
  


\subsection*{Tangent spaces and vector bundles}

\begin{enumerate}[1.]

\item Let $X\subset \R^N$ be a submanifold. Show that
  $$\{(x,v)\in X\times\R^N\mid v\in T_x(X)\subset\R^N\}$$ is a
  manifold diffeomorphic to $T(X)$. This is how G-P define the tangent
  bundle.

  \item Recall that $\R P^n$ can be viewed as the space of lines
    through 0 in $\R^{n+1}$. Let
    $$E=\{(x,\ell)\in \R P^n\times \R^{n+1}\mid x\in\ell\}$$
    Show that the projection $\pi:E\to \R P^n$ to the first coordinate
    is a line bundle (i.e. it is a 1-dimensional smooth vector
    bundle). Further show that this bundle is nontrivial.

  \item Show that $T(X\times Y)$ is diffeomorphic to $TX\times TY$.
    Hint: Define a natural map $T(X\times Y)\to TX\times TY$ and then
    check that it is a diffeomorphism in charts.

    \item Show that if a sphere $S^k$ admits a nowhere vanishing
      vector field then the antipodal map is homotopic to the
      identity.

      \item Show that every smooth map $S^1\to S^k$ is nullhomotopic
        if $k>1$. Note: This is also true for continuous maps. You can
        prove it using Problem \ref{homotopy} below, although I am not
        assigning it.

\end{enumerate}

\subsection*{Lie bracket}
\begin{enumerate}[resume]
\item Let $V=\frac \partial {\partial x} + y \frac \partial {\partial
  z}$ and $W=\frac \partial {\partial y}$ be two vector fields on
  $\R^3$. Compute $[V,W]$ and show that $[V,W](p)$ does not belong to
  the span of $V(p)$ and $W(p)$ for any $p\in\R^3$.
  
  \item A real vector space $V$ with a bilinear pairing
    $[\cdot,\cdot]:V\times V\to\R$ is called a {\it Lie algebra} if
    the pairing satisfies:
    \begin{enumerate}[(i)]
    \item (alternating) $[v,w]=-[w,v]$ for $v,w\in V$, and
      \item (Jacobi identity) $[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0$ for
        $u,v,w\in V$.
    \end{enumerate}
    Show that the vector space of all vector fields on a manifold $X$
    is a Lie algebra with respect to the Lie bracket operation. Note:
    This Lie algebra is infinite dimensional, but we will later see
    that the vector space of {\it left invariant} vector fields on a
    Lie group is a finite dimensional Lie algebra.

\end{enumerate}


\subsection*{Transversality}
\begin{enumerate}[resume]

  \item Let $Z\subset\R^3$ be a submanifold, closed as a
    subset. Assume $0\not\in Z$. Show that almost every 1-dimensional
    linear subspace $\ell\subset\R^3$ is transverse to $Z$. Hint: Show
    that $\R\times\R^3\to\R^3$, $(t,x,y,z)\to (tx,ty,tz)$ is
    transverse to $Z$. This is a special case of GP \#7 in the
    transversality section, but they forgot to assume $0\not\in
    Z$. Can you find a counterexample if $0\in Z$?

  \item Let $f:\R^n\to\R^n$ be smooth, $n>1$, $K\subset \R^n$ compact,
    $\epsilon>0$. Show that there is a smooth map $g:\R^n\to\R^n$
    which approximates $f$ in the sense that $\|g(x)-f(x)\|<\epsilon$
    for all $x\in K$ and so that $dg_x\neq 0$ for any
    $x\in\R^n$. Hint: Consider $\R^n\times M(n)\to M(n)$,
    $(x,A)\mapsto df_x+A$ and show it is transverse to $\{0\}$. What
    goes wrong if $n=1$?

  \item Let $U\subseteq\R^n$ be open, $f:U\to\R$ smooth. Show that for
    almost all $a=(a_1,\cdots,a_n)\in\R^n$ the map $f_a:U\to\R$,
    $f_a(x_1,\cdots,x_n)=f(x_1,\cdots,x_n)+a_1x_1+\cdots+a_nx_n$ is
    Morse.
    Note: This is the main input in the proof that smooth functions to
    $\R$ can be approximated by Morse functions. The general case
    involves Fubini's theorem. This problem is also from GP, but there
    is a typo there.

\end{enumerate}

\subsection*{Tubular neighborhoods}

\begin{enumerate}[resume]
\item \label{homotopy}
  Let $Y\subset \R^N$ be a compact submanifold. Show that there
    is some $\epsilon>0$ so that if two continuous maps $f,g:X\to Y$
    are $\epsilon$-close (i.e. $\|f(x)-g(x)\|<\epsilon$ for every
    $x\in X$), then $f$ and $g$ are homotopic, i.e. there is a
    continuous map $H:X\times [0,1]\to Y$ such that $H_0=f$,
    $H_1=g$. Moreover, if $f$ and $g$ are smooth, then $H$ can be
    taken to be smooth.

    \item Let $X\subset \R^N$ be a hypersurface, i.e. a codimension 1
      submanifold. A point $y\in\R^N$ is a {\it focal point} of $X$ if
      it is a critical value of the normal bundle map $h:N(X)\to\R^N$,
      $h(x,v)=x+v$. Compute the set of focal points of the
      parabola $$X=\{(x,x^2)\mid x\in\R\}\subset\R^2$$ Hint: I am
      getting $\{(-4t^3,3t^2+\frac 12)\mid t\in\R\}$.

\end{enumerate}

      
\end{document}
