Editing Pullback (differential geometry) (section)

==Pullback of (covariant) tensor fields==
The construction of the previous section generalizes immediately to [[tensor|tensor bundle]]s of rank <math>(0,s)</math> for any natural number <math>s</math>: a <math>(0,s)</math> [[tensor field]] on a manifold <math>N</math> is a section of the tensor bundle on <math>N</math> whose fiber at <math>y</math> in <math>N</math> is the space of multilinear <math>s</math>-forms
<math display="block"> F: T_y N\times\cdots \times T_y N\to \R.</math>
By taking <math>\phi</math> equal to the (pointwise) differential of a smooth map <math>\phi</math> from <math>M</math> to <math>N</math>, the pullback of multilinear forms can be combined with the pullback of sections to yield a pullback <math>(0,s)</math> tensor field on <math>M</math>. More precisely if <math>S</math> is a <math>(0,s)</math>-tensor field on <math>N</math>, then the '''pullback''' of <math>S</math> by <math>\phi</math> is the <math>(0,s)</math>-tensor field <math>\phi^*S</math> on <math>M</math> defined by
<math display="block"> (\phi^*S)_x(X_1,\ldots, X_s) = S_{\phi(x)}(d\phi_x(X_1),\ldots, d\phi_x(X_s))</math>
for <math>x</math> in <math>M</math> and <math>X_j</math> in <math>T_xM</math>.