pullback of differential form

Definition

For a differential form on a smooth manifold , and for then the pullback of the differential form is

where is the differential.

This can be seen using the commutative diagram

Properties

For a smooth map the pullback has the following properties

1. is linear.
2. The pullback commutes with the wedge product:

Computing pullbacks

In any smooth chart, the pullback along a smooth map can be computed as

This easily seen given the pullback formulation of the differential.

Pullback of top degree

Let be a smooth map between -manifolds. If and are coordinates on open sets and , and is a continuous function, then the following relationship holds on .