The Interior Product
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.
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July 26, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
Why is this post in the “point-set topology” category?
An errant click on a very tightly-spaced list.
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