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\begin{center}\vspace{1in}{\bf MATH 209, MANIFOLDS II, WINTER 2011}\\
\bigskip
{\bf Homework Assignment I: One-forms and integration}\\
\end{center}
\bigskip



{\bf 1.} [HAND IN]  A coframe for a manifold is a collection of one-forms which
everywhere frames $T^*M$.  Suppose that $\{\theta^i \}_{i =1, \ldots, n} $ coframes $M$.  
Expand out $d \theta^i$ in terms of the coframe to define functions
$c^i _{jk}$ be  $d \theta^i = \Sigma c^i _{jk}  \theta^j \wedge \theta^k$. 

a) Show that if we impose $c^i _{jk} = -c^i _{kj}$ then these functions are uniquely determined by the coframe.

b)  Use $d^2 = 0$ to show  that if these functions are all constants then they are the  structure constants of some  Lie algebra. 

\vskip .3cm 

{\bf 2}  Prove the validity of  \begin{theorem} [Cartan's magic formula]  $$L_X = di_X + i_X d$$
as operators  on the algebra $\Omega$ of smooth forms on a manifold , where $L_X$ is the operation of Lie differentiation by a smooth vector field $X$
on that manifold. 
\end{theorem}
Use $L_X = \frac{d}{dt}|_{t = 0} \Phi_t ^*$
where $\phi_t$ is the flow of  $X$
 as  your  definition of the operator
$L_X: \Omega \to \Omega $.  For the purposes of this exercise, set $S_X = di_X + i_X d$. 

Step 0. Verify that $L_X f = S_X f$ for $f$ a zero form, which  is to say, a smooth function. 

Step 1.  Verify that $L_X d = d L_X$ and similarly that  $S_X d = d S_X$.  

Step 2.  Verify that both $L_X$ and $S_X $ are algebra homomorphisms:
that is, for any $\alpha, \beta \in \Omega$ we have that  $L_X (\alpha \wedge \beta) = L_X \alpha \wedge \beta + \alpha \wedge L_X \beta$
and likewise for $S_X$. 

Step 3.    Finish up.


\vskip .3cm 

{\bf 3}.  Prove the identities 
\begin{eqnarray}
L_X L_Y - L_Y L_X & = & L_{[X, Y]}\\
L_X i_Y - i_Y L_X & = & i_{[X, Y]}\
\end{eqnarray}
where $X, Y$ are smooth vector fields on a manifold and $L_X, L_Y, i_X, i_Y$
the corresponding operators on the algebra $\Omega$ of forms.







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