Pullback (differential geometry)
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High Quality Content by WIKIPEDIA articles! 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…mehr

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High Quality Content by WIKIPEDIA articles! 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.