Editing Pullback (differential geometry) (section)

==Pullback of cotangent vectors and 1-forms==

Let <math>\phi:M\to N</math> be a [[smooth map]] between [[smooth manifolds]]. Then the [[pushforward (differential)|differential]] of <math>\phi</math>, written <math>\phi_*</math>, <math>d\phi</math>, or <math>D\phi</math>, is a [[vector bundle morphism]] (over <math>M</math>) from the [[tangent bundle]] <math>TM</math> of <math>M</math> to the [[pullback bundle]] <math>\phi^*TN</math>. The [[dual space|transpose]] of <math>\phi_*</math> is therefore a bundle map from <math>\phi^*T^*N</math> to <math>T^*M</math>, the [[cotangent bundle]] of <math>M</math>.

Now suppose that <math>\alpha</math> is a [[section (fiber bundle)|section]] of <math>T^*N</math> (a [[differential form|1-form]] on <math>N</math>), and precompose <math>\alpha</math> with <math>\phi</math> to obtain a [[pullback bundle|pullback section]] of <math>\phi^*T^*N</math>. Applying the above bundle map (pointwise) to this section yields the '''pullback''' of <math>\alpha</math> by <math>\phi</math>, which is the 1-form <math>\phi^*\alpha</math> on <math>M</math> defined by
<math display="block"> (\phi^*\alpha)_x(X) = \alpha_{\phi(x)}(d\phi_x(X))</math>
for <math>x</math> in <math>M</math> and <math>X</math> in <math>T_xM</math>.