Editing Pullback (differential geometry) (section)

==Pullback by diffeomorphisms==
When the map <math>\phi</math> between manifolds is a [[diffeomorphism]], that is, it has a smooth inverse, then pullback can be defined for the [[vector field]]s as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold. The linear map
<math display="block">\Phi = d\phi_x \in \operatorname{GL}\left(T_x M, T_{\phi(x)}N\right)</math>

can be inverted to give
<math display="block">\Phi^{-1} = \left({d\phi_x}\right)^{-1} \in \operatorname{GL}\left(T_{\phi(x)}N, T_x M\right).</math>

A general mixed tensor field will then transform using <math>\Phi</math> and <math>\Phi^{-1}</math> according to the [[tensor product]] decomposition of the tensor bundle into copies of <math>TN</math> and <math>T^*N</math>. When <math>M=N</math>, then the pullback and the [[pushforward (differential)|pushforward]] describe the transformation properties of a [[tensor]] on the manifold <math>M</math>. In traditional terms, the pullback describes the transformation properties of the covariant indices of a [[tensor]]; by contrast, the transformation of the [[Covariance and contravariance of vectors|contravariant]] indices is given by a [[pushforward (differential)|pushforward]].