Editing Pullback (differential geometry) (section)

{{Short description|Mathematical operation}}
{{about|pullback operations in differential geometry, in particular, the pullback of [[differential form]]s and [[tensor (intrinsic definition)|tensor fields]] on [[smooth manifold]]s|other uses of the term in [[mathematics]]|pullback}}

Let <math>\phi:M\to N</math> be a [[smooth map]] between [[smooth manifold]]s <math>M</math> and <math>N</math>. Then there is an associated [[linear map]] from the space of [[One form|1-forms]] on <math>N</math> (the [[linear space]] of [[section (fiber bundle)|sections]] of the [[cotangent bundle]]) to the space of 1-forms on <math>M</math>. This linear map is known as the '''pullback''' (by <math>\phi</math>), and is frequently denoted by <math>\phi^*</math>. More generally, any [[Covariance and contravariance of vectors|covariant]] tensor field &ndash; in particular any [[differential form]] &ndash; on <math>N</math> may be pulled back to <math>M</math> using <math>\phi</math>.

When the map <math>\phi</math> is a [[diffeomorphism]], then the pullback, together with the [[Pushforward (differential)|pushforward]], can be used to transform any tensor field from <math>N</math> to <math>M</math> or vice versa. In particular, if <math>\phi</math> is a diffeomorphism between open subsets of <math>\R^n</math> and <math>\R^n</math>, viewed as a [[change of coordinates]] (perhaps between different [[Manifold#Charts|charts]] on a manifold <math>M</math>), then the pullback and pushforward describe the transformation properties of [[Covariance and contravariance of vectors|covariant and contravariant]] tensors used in more traditional (coordinate dependent) approaches to the subject.

The idea behind the pullback is essentially the notion of [[Pullback#Precomposition|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 functor|contravariant functors]].