pullback of a differential form
For a smooth function between smooth manifold, and for a differential n-form, there is the pullback form .
In terms of push-forward of vector fields
If differential forms are defined as linear duals to vectors then pullback is the dual operation to pushforward of a vector field?
In terms of coordinate expression
If differential forms are defines by Yoneda extension from differential forms on Cartesian spaces then pullback is given on and and on 1-forms
by the rule
Compatibility with the de Rham differential
Pullback of differential forms commutes with the de Rham differential:
Hence it constritutes a chain map between the de Rham complexes
Sheaf of differential forms
Under pullback differential forms form a presheaf on the catories CartSp and SmthMfd, in fact a sheaf with respect to the standard open cover-coverage.
A standard reference is
- Bott, Tu, Differential forms in algebraic topology.
See also for instance section 2.7 of
Revised on February 5, 2013 20:42:59
by Urs Schreiber