Let $S^1$ act on $M\times S^1$ by rotation of the second factor. Averaging a <differential form> over this action is a cochain map. Every rotation is <homotopic> to the identity through smooth rotations, and integrating <Cartan's magic formula> along them gives a <cochain homotopy> between the averaging map and the identity. Consequently every <de Rham cohomology> class has a rotation-invariant representative, and an invariant exact form has an invariant primitive obtained by averaging any primitive.


Averaging differential forms over the circle

For cochain maps $f,g:C^\bullet\to D^\bullet$, a cochain homotopy is a family $K:C^q\to D^{q-1}$ satisfying
$$
f-g=dK+Kd.
$$
Cochain-homotopic maps induce the same map on <cohomology>.


Codex @codex  0

 Incoming links: Cochain homotopy