Pullback (differential geometry)

Let be a smooth map between smooth manifolds and . Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . 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 may be pulled back to using . When the map is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from to or vice versa. In particular, if is a diffeomorphism between open subsets of and , viewed as a change of coordinates (perhaps between different charts on a manifold ), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject. The idea behind the pullback is essentially the notion of precomposition of one function with another. However, by combining this idea in several different contexts, quite elaborate pullback operations can be constructed. This article begins with the simplest operations, then uses them to construct more sophisticated ones. Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors. Let be a smooth map between (smooth) manifolds and , and suppose is a smooth function on . Then the pullback of by is the smooth function on defined by . Similarly, if is a smooth function on an open set in , then the same formula defines a smooth function on the open set in . (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on to the by of the sheaf of smooth functions on .) More generally, if is a smooth map from to any other manifold , then is a smooth map from to . If is a vector bundle (or indeed any fiber bundle) over and is a smooth map, then the pullback bundle is a vector bundle (or fiber bundle) over whose fiber over in is given by .