Editing Pullback (differential geometry) (section)

==Pullback of differential forms==
A particular important case of the pullback of covariant tensor fields is the pullback of [[differential form]]s. If <math>\alpha</math> is a differential <math>k</math>-form, i.e., a section of the [[exterior bundle]] <math>\Lambda^k(T^*N)</math> of (fiberwise) alternating <math>k</math>-forms on <math>TN</math>, then the pullback of <math>\alpha</math> is the differential <math>k</math>-form on <math>M</math> defined by the same formula as in the previous section:
<math display="block"> (\phi^*\alpha)_x(X_1,\ldots, X_k) = \alpha_{\phi(x)}(d\phi_x(X_1),\ldots, d\phi_x(X_k))</math>
for <math>x</math> in <math>M</math> and <math>X_j</math> in <math>T_xM</math>.

The pullback of differential forms has two properties which make it extremely useful.

# It is compatible with the [[wedge product]] in the sense that for differential forms <math>\alpha</math> and <math>\beta</math> on <math>N</math>, <math display="block">\phi^*(\alpha \wedge \beta)=\phi^*\alpha \wedge \phi^*\beta.</math>
# It is compatible with the [[exterior derivative]] <math>d</math>: if <math>\alpha</math> is a differential form on <math>N</math> then <math display="block">\phi^*(d\alpha) = d(\phi^*\alpha).</math>