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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Forms}}
\vskip 1cm
  


\subsection*{Miscellaneous}



\begin{enumerate}[1.]

\item Let $M$ be any $n$-manifold and $TM\to M$ its tangent bundle. Show
  that the total space is always orientable as a $2n$-manifold (but not
  always as a vector bundle). Hint: Show that the transition maps
  between standard charts are orientation preserving.

\item Recall that $\R P^3$ is an orientable manifold. Let $f_n:\R
  P^3\to\R P^3$ be given by
  $$[X_0:X_1:X_2:X_3]\mapsto [X_0^n:X_1^n:X_2^n:X_3^n]$$
  for $n\geq 1$. Compute the degree $deg(f_n)\in\Z$. Optional: If you are
  comfortable with complex manifolds and complex derivatives you could
  also do it for $\C P^3$ (or $\C P^k$). (All complex manifolds are
  orientable.) 

\item Let $V$ be the vector field in the plane $\R^2=\C$ defined by
  $V(z)=z^n$, $n\geq 1$. Thus 0 is the only singular point of $V$. Compute the
  index $ind_0V$.

\end{enumerate}

\subsection*{Exterior algebra}

\begin{enumerate}[resume]

  \item An alternating 2-tensor $T\in\Lambda^2(V^*)$ is {\it decomposable} if it can
    be written as $T=A\wedge B$ for 1-tensors $A,B$. Show that every
    alternating 2-tensor in $\Lambda^2((\R^3)^*)$ is decomposable.

    \item Show that the 2-tensor $\omega=dx_1\wedge dx_2+dx_3\wedge
      dx_4$ in $\Lambda^2((\R^4)^*)$ is not decomposable. Hint: Consider
      $\omega\wedge\omega$.

    \item \label{contraction}
      For $v\in V$ and $\omega\in\Lambda^k(V^*)$ define the {\it
        contraction} $v\intprod
      \omega\in\Lambda^{k-1}(V^*)$ (also called {\it interior
        product}) by
      $$v\intprod
      \omega(v_1,v_2,\cdots,v_{k-1})=\omega(v,v_1,v_2,\cdots,v_{k-1})$$
      \begin{enumerate}[(a)]
        \item Show that $v\intprod
          (w\intprod\omega)=-w\intprod(v\intprod\omega)$.
          \item Suppose $v_1,\cdots,v_n$ is a basis of $V$ and
            $\phi_1,\cdots,\phi_n$ is the dual basis of $V^*$. Show
            that
            $$v_i\intprod
            (\phi_{i_1}\wedge\phi_{i_2}\wedge\cdots\wedge\phi_{i_k})$$
            equals $$(-1)^{j-1}\phi_{i_1}\wedge\cdots\wedge\widehat{\phi_{i_j}}\wedge\cdots\wedge\phi_{i_k}$$
            if $i$ equals some $i_j$ and otherwise it equals 0. (The
            hat means omit that term.)
            \item If $\omega_1\in\Lambda^k(V^*)$ and
              $\omega_2\in\Lambda^l(V^*)$ prove the Leibnitz rule:
              $$v\intprod (\omega_1\wedge\omega_2)=(v\intprod
              \omega_1)\wedge\omega_2+(-1)^k\omega_1\wedge(v\intprod\omega_2)$$
              Hint: Use (b) and linearity of everything.
      \end{enumerate}
      


      \end{enumerate}
      
The significance here is that if $\omega$ is a $k$-form on a manifold
$X$ and $V$ is a vector field on $X$, then the pointwise contraction
$V\intprod\omega$ is a $(k-1)$-form and this operation will satisfy
the Leibnitz rule.

\subsection*{Forms}


\begin{enumerate}[resume]

  \item Let $f:\R^2\to\R$ be defined by $f(x,y)=xy-y^2$. Compute the
    pullback $f^*(\omega)$ of the 1-form $\omega=t~ dt$.

  \item Let $f:\R^2\to\R^2$ be defined by
    $f(x,y)=(xy^2+x,y^3-xy)$. Compute $f^*(\omega)$ and $f^*(\eta)$
    where $\omega=x~ dy+y~ dx$ and $\eta=dx~ \wedge dy$.

  \item Let $X$ be an oriented $n$-manifold and $-X$ the same manifold
    with opposite orientation. Let $\omega$ be a compactly supported
    $n$-form on $X$. Show that
    $$\int_{-X}\omega=-\int_X\omega$$

  \item Let $c:[a,b]\to X$ be a smooth curve and let $f:[c,d]\to
    [a,b]$ be a smooth map with $f(c)=a$ and $f(d)=b$. If $\omega$ is
    any 1-form on $X$, show that
    $$\int_a^b c^*(\omega)=\int_c^d (cf)^*(\omega)$$
    where by $\int_a^b$ we denote integral over $[a,b]$ with the usual
    orientation from $a$ to $b$.

  \item Let $\omega$ be a 1-form on a connected manifold $X$. Suppose
    that for every smooth map $f:S^1\to X$ the integral
    $$\int_{S^1}f^*(\omega)=0$$
    Show that there exists a smooth function $g:X\to\R$ such that
    $\omega=dg$. Hint: Fix a basepoint $x_0\in X$. Connectivity
    guarantees that for every $x\in X$ there is a smooth curve
    $c:[a,b]\to X$
    from $x_0$ to $x$. Define $g(x)=\int_a^b c^*(\omega)$. First show it
    doesn't depend on the choice of $c$.

  \item \label{normal}
    Let $S$ be a surface in $\R^3$ and $\vec{n}=(n_1,n_2,n_3)$ a
    unit norm vector field along $S$ perpendicular to the tangent
    space of $S$ at every point. Show that
    $$\omega=n_1~ dx_2\wedge dx_3+n_2~ dx_3\wedge dx_1+n_3~ dx_1\wedge
    dx_2$$
    is an area form on $S$, i.e. for any two vectors $v,v'\in T_pS$
    the value of $\omega(v,v')$ is (up to sign) the area of the
    parallelogram in $\R^3$ spanned by $v,v'$. Hint: you only need to
    check this on one pair of linearly independent vectors at every
    point.

  \item Using Problem \ref{normal} show that the 2-form
    $$\omega=x_1~ dx_2\wedge dx_3+x_2~ dx_3\wedge dx_1+x_3~ dx_1\wedge
    dx_2$$
    is the area form on $S^2$. 
  

\end{enumerate}



\end{document}
