Pullbacks of Differential Forms

Let f : M \to N be a smooth map between manifolds and let \phi be a smooth k-form on N. We have the natural push forward/total differential f_* = df : TM \to TN given by df(v_p) = (f \circ \alpha)'(0), where \alpha : I \to M is a curve satisfying \alpha(0) = p and \alpha'(0) = v, but this also gives a natural way to pull back differential forms from N back to M.
Define f^* : \Omega^k (N) \to \Omega^k (M) by f^*(\phi)(v_{p, 1}, \dots, v_{p, k}) = \phi(f_*(v_{p, 1}), \dots, f_*(v_{p, k})), 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 f : I \to M and a smooth 1-form \phi on M. Then

    \[ \int_{f(I)} \phi = \int_I f^*(\phi) = \int_I \phi(f'(t)) \, dt, \]

which essential means that we can integrate any 1-form on any manifold by looking at how it behaves in our local copy of \RR on the manifold. Pretty neat, right?

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