Pullback (differential geometry)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Suppose that ¿:M¿ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by ¿), and is frequently denoted by ¿*. More generally, any covariant tensor field - in particular any differential form - on N may be pulled back to M using ¿. When the map ¿ is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice-versa. In particular, if ¿ is a diffeomorphism between open subsets of Rn and Rn, viewed as a change of coordinates (perhaps between different charts on a manifold M), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject.