It turns out that there is a fantastic relationship between the interior product, the exterior derivative, and the Lie derivative.
It starts with the observation that for a function 
and a vector field 
, the Lie derivative is 
and the exterior derivative evaluated at is 
. That is, 
on functions.
Next we consider the differential 
of a function. If we apply 
to it, the nilpotency of the exterior derivative tells us that we automatically get zero. On the other hand, if we apply 
, we get 
, which it turns out is 
. To see this, we calculate
&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(df)\right](Y)-df(Y)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(df((\Phi_t)_*Y)-df(Y)\right)\\&=\frac{\partial}{\partial t}\left[(\Phi_t)_*Y\right](f)\bigg\vert_{t=0}\\&=\frac{\partial}{\partial t}Y(f\circ\Phi_t)\bigg\vert_{t=0}\\&=YXf\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5BL_X%28df%29%5Cright%5D%28Y%29%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28%5Cleft%5B%28%5CPhi_t%29%5E%2A%28df%29%5Cright%5D%28Y%29-df%28Y%29%5Cright%29%5C%5C%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28df%28%28%5CPhi_t%29_%2AY%29-df%28Y%29%5Cright%29%5C%5C%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7D%5Cleft%5B%28%5CPhi_t%29_%2AY%5Cright%5D%28f%29%5Cbigg%5Cvert_%7Bt%3D0%7D%5C%5C%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7DY%28f%5Ccirc%5CPhi_t%29%5Cbigg%5Cvert_%7Bt%3D0%7D%5C%5C%26%3DYXf%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[L_X(df)\right](Y)&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(df)\right](Y)-df(Y)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(df((\Phi_t)_*Y)-df(Y)\right)\\&=\frac{\partial}{\partial t}\left[(\Phi_t)_*Y\right](f)\bigg\vert_{t=0}\\&=\frac{\partial}{\partial t}Y(f\circ\Phi_t)\bigg\vert_{t=0}\\&=YXf\end{aligned}")
just as if we took 
and applied it to 
.
So on exact 
-forms, gives zero while gives 
. And on functions gives , while gives zero. In both cases we find that
and in fact this holds for all differential forms, which follows from these two base cases by a straightforward induction. This is Cartan’s formula, and it’s the natural extension to all differential forms of the basic identity 
on functions.
July 26, 2011
Posted by John Armstrong |
Differential Topology, Topology |
1 Comment
We have yet another operation on the algebra 
of differential forms: the “interior product”. Given a vector field 
and a 
-form 
, the interior product 
is the 
-form defined by
=\omega(X,X_1,\dots,X_{k-1})](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Ciota_X%5Comega%5Cright%5D%28X_1%2C%5Cdots%2CX_%7Bk-1%7D%29%3D%5Comega%28X%2CX_1%2C%5Cdots%2CX_%7Bk-1%7D%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\iota_X\omega\right](X_1,\dots,X_{k-1})=\omega(X,X_1,\dots,X_{k-1})")
That is, we just take the vector field and stick it into the first “slot” of a -form. We extend this to functions by just defining 
.
Two interior products anticommute: 
, which follows easily from the antisymmetry of differential forms. Each one is also clearly linear, and in fact is also a graded derivation of with degree -1:
where 
is the degree of 
. This can be shown by reducing to the case where and 
are wedge products of -forms, but rather than go through all that tedious calculation we can think about it like this: sticking into a slot of 
means either sticking it into a slot of or into one of . In the first case we just get 
, while in the second we have to “move the slot” through all of , which incurs a sign of 
, as asserted.
July 26, 2011
Posted by John Armstrong |
Differential Topology, Topology |
5 Comments
