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 to the space of 1-forms on. This linear map is known as the pullback, 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, then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional 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 turns several constructions in differential geometry into contravariant functors.
Pullback of smooth functions and smooth maps
Let be a smooth map between 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 .More generally, if is a smooth map from to any other manifold, then is a smooth map from to.
Pullback of bundles and sections
If is a vector bundle over and is a smooth map, then the pullback bundle is a vector bundle over whose fiber over in is given by.In this situation, precomposition defines a pullback operation on sections of : if is a section of over, then the pullback section is a section of over.
Pullback of multilinear forms
Let be a linear map between vector spaces V and W, and letbe a multilinear form on W. Then the pullback Φ∗F of F by Φ is a multilinear form on V defined by precomposing F with Φ. More precisely, given vectors v1, v2,..., vs in V, Φ∗F is defined by the formula
which is a multilinear form on V. Hence Φ∗ is a operator from multilinear forms on W to multilinear forms on V. As a special case, note that if F is a linear form on W, so that F is an element of W∗, the dual space of W, then Φ∗F is an element of V∗, and so pullback by Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself:
From a tensorial point of view, it is natural to try to extend the notion of pullback to tensors of arbitrary rank, i.e., to multilinear maps on W taking values in a tensor product of r copies of W, i.e.,. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from to given by
Nevertheless, it follows from this that if Φ is invertible, pullback can be defined using pushforward by the inverse function Φ−1. Combining these two constructions yields a pushforward operation, along an invertible linear map, for tensors of any rank.
Pullback of cotangent vectors and 1-forms
Let be a smooth map between smooth manifolds. Then the differential of, written,, or, is a vector bundle morphism from the tangent bundle of to the pullback bundle. The transpose of is therefore a bundle map from to, the cotangent bundle of.Now suppose that is a section of, and precompose with to obtain a pullback section of. Applying the above bundle map to this section yields the pullback of by, which is the 1-form on defined by
for in and in.
Pullback of (covariant) tensor fields
The construction of the previous section generalizes immediately to tensor bundles of rank for any natural number : a tensor field on a manifold is a section of the tensor bundle on whose fiber at in is the space of multilinear -formsBy taking equal to the differential of a smooth map from to, the pullback of multilinear forms can be combined with the pullback of sections to yield a pullback tensor field on. More precisely if is a -tensor field on, then the pullback of by is the -tensor field on defined by
for in and in.
Pullback of differential forms
A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If is a differential -form, i.e., a section of the exterior bundle of alternating -forms on, then the pullback of is the differential -form on defined by the same formula as in the previous section:for in and in.
The pullback of differential forms has two properties which make it extremely useful.
- It is compatible with the wedge product in the sense that for differential forms and on,
- It is compatible with the exterior derivative : if is a differential form on then
Pullback by diffeomorphisms
When the map between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold. The linear mapcan be inverted to give
A general mixed tensor field will then transform using and according to the tensor product decomposition of the tensor bundle into copies of and. When, then the pullback and the pushforward describe the transformation properties of a tensor on the manifold. In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.