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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Flows and Lie derivative}}
  \vskip 1cm

  

  \subsection*{Integral curves and flows}

  



\begin{enumerate}[1.]

\item Let $M$ be a manifold and $V,W$ two smooth vector fields on
  $M$. Suppose that $V$ and $W$ are equal outside a compact set. Also
  suppose that $V$ generates a global flow (i.e. integral curves for
  $V$ are defined on all of $\R$). Show that $W$ also generates a
  global flow.


\item \label{f-related} Let $f:M\to N$ be a smooth map, $V,W$ vector
  fields on $M,N$ respectively. We say that $V,W$ are {\it
    $f$-related} if for every $x\in M$ we have
  $df(V(x))=W(f(x))$. Assuming $V$ and $W$ are $f$-related prove that
  if $\alpha:(a,b)\to M$ is an integral curve for $V$ then
  $f\circ\alpha$ is an integral curve for $W$.

\item Let $M$ be a compact manifold without boundary, $f:M\to\R$
  smooth. Suppose that all points in $[a,b]\subset\R$ are regular
  values for $f$. Show that $f^{-1}(-\infty,a]$ and
    $f^{-1}(-\infty,b]$ are diffeomorphic. Hint: This is part of the
      Morse theorem. Use Problem \ref{f-related} and construct a
      vector field on $M$ so that the flow generated by it takes
      $f^{-1}(-\infty,a]$ to $f^{-1}(-\infty,b]$. The vector field on
          $M$ doesn't have to be $f$-related to a vector field on $\R$
          everywhere, only on $f^{-1}[a,b]$, and choosing a Riemannian metric on
          $M$ can be useful.

\end{enumerate}

\subsection*{Lie derivative}


Let $M=\R^2$, $X=x\frac{\partial}{\partial
  x}+xy\frac{\partial}{\partial y}$

\begin{enumerate}[resume]
  \item Let $\omega=y~ dx+x~ dy$. Compute the Lie derivative
    $L_X(\omega)$.

  \item Let $Y=xy\frac{\partial}{\partial
    x}+y\frac{\partial}{\partial y}$. Compute the Lie derivative $L_XY$.

\end{enumerate}

\subsection*{Identities}
There are many identities between Lie derivatives, Lie brackets of
vector fields, exterior derivative and interior product
$\intprod$. You will recall from homework 8 that if $\omega$ is a
$k$-form and $V$ a vector field then $V\intprod\omega$ is a
$(k-1)$-form defined by
$V\intprod\omega(X_1,\cdots,X_{k-1})=\omega(V,X_1,\cdots,X_{k-1})$. It
is convenient to denote by $i_X$ the operator $i_X\omega=X\intprod \omega$.
Examples of identities include $L_XY=[X,Y]$, Cartan's Magic Formula
$L_X=di_X+i_Xd$ and the identity
$d\omega(X,Y)=\frac 12(X\omega(Y)-Y\omega(X)-\omega([X,Y]))$ for vector fields
$X,Y$ and a 1-form $\omega$ from an earlier homework. Identities are
local and to prove them we can work in a chart.

A good strategy that simplifies the calculations is to use the fact
that $d,i_X,L_X$ satisfy the Leibnitz rule and you can try to use this
to show that if some identity involving a form $\omega$ is satisfied
for forms $\omega_1,\omega_2$ then it is satisfied for
$\omega=\omega_1\wedge\omega_2$. This reduces the problem to the case
when $\omega$ is a 0-form or $\omega=dx_i$. You can also usually
assume that one of the vector fields $X$ or $Y$ in the identity is
$\partial/\partial x_1$ (and also 0 but that case is usually
obvious). I think I didn't lose any factorials in the identities, but
if you think I did let me know.

\begin{enumerate}[resume]
  \item The goal of this Problem and the next is to prove the identity $$[L_X,i_Y]=i_{[X,Y]}$$ That is, for any
    form $\omega$,
    $$L_X(i_Y(\omega))-i_Y(L_X(\omega))=i_{[X,Y]}(\omega)$$
  Here show as suggested above that if this is true for
  $\omega_1,\omega_2$ then it's true for
  $\omega_1\wedge\omega_2$. They also say ``both sides are
  derivations'', in this case of degree $-1$ (the degree of a form
  goes down by 1).

  \item Finish the proof of the identity in the previous problem. You
    can assume $Y=\partial/\partial x_1$, $\omega=dx_i$ and $X=\sum_j
    a_j\frac {\partial}{\partial x_j}$, where $a_j$ are functions.

\item If $X,Y$ are vector fields and $\omega$ is a 1-form then
  $$(L_X\omega)(Y)=X(\omega(Y))-\omega([X,Y])$$
  Hint: Here the Leibnitz rule involves only $f\omega$, i.e. show that
  if the identity holds for $\omega$ then it holds for $f\omega$, and
  then the calculation is reduced to $\omega=dx_i$.
  
\end{enumerate}





\end{document}
