The Lie Derivative on Forms
We’ve defined the Lie derivative 
of one vector field 
by another, 
. This worked by using the flow of to compare nearby points, and used the derivative of the flow to translate vectors.
Well now we know how to translate 
-forms by pulling back, and thus we can define another Lie derivative:
What happens if 
is a 
-form — a function 
? We check
&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(f)\right](p)-f(p)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(f\left(\Phi_t(p)\right)-f(p)\right)\\&=X_p(f)=Xf(p)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5BL_Xf%5Cright%5D%28p%29%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28%5Cleft%5B%28%5CPhi_t%29%5E%2A%28f%29%5Cright%5D%28p%29-f%28p%29%5Cright%29%5C%5C%26%3D%5Clim%5Climits_%7Bt%5Cto0%7D%5Cfrac%7B1%7D%7Bt%7D%5Cleft%28f%5Cleft%28%5CPhi_t%28p%29%5Cright%29-f%28p%29%5Cright%29%5C%5C%26%3DX_p%28f%29%3DXf%28p%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[L_Xf\right](p)&=\lim\limits_{t\to0}\frac{1}{t}\left(\left[(\Phi_t)^*(f)\right](p)-f(p)\right)\\&=\lim\limits_{t\to0}\frac{1}{t}\left(f\left(\Phi_t(p)\right)-f(p)\right)\\&=X_p(f)=Xf(p)\end{aligned}")
That is, the Lie derivative by acts on 
exactly the same as the vector field does itself.
I also say that the Lie derivative by is a degree-zero derivation of the algebra 
. That is, it’s a real-linear transformation, and it satisfies the Leibniz rule:
for any -form 
and 
-form 
. Linearity is straightforward, and given linearity the Leibniz rule follows if we can show
for 
-forms 
. Indeed, we can write and as linear combinations of such – and -fold wedges, and then the Leibniz rule is obvious.
So, let us calculate:
So we see how we can peel off one of the -forms. A simple induction gives us the general case.
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Posted by John Armstrong |
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