Let be a smooth map between manifolds and let be a smooth -form on . We have the natural push forward/total differential given by , where is a curve satisfying and , but this also gives a natural way to pull back differential forms from back to .
Define by , which totally doesn’t look like it’s doing much, but this is precisely what gives is the tools to integrate differential forms on manifolds.
For example, suppose I have a curve and a smooth 1-form on . Then
which essential means that we can integrate any 1-form on any manifold by looking at how it behaves in our local copy of on the manifold. Pretty neat, right?