be a smooth map between smooth manifolds. Then there is an associated linear map from the space of 1-forms (the linear space of sections of the cotangent bundle. This linear map is known as the pullback. More generally, any covariant tensor field in particular any differential form .

is a diffeomorphism, then the pullback, together with the pushforward, viewed as a change of coordinates (perhaps between different charts), 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.

Pullback of smooth functions and smooth maps

. Then the pullback is a smooth function on an open set. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions to the direct image.)

.

Pullback of bundles and sections

is a vector bundle (or indeed any fiber bundle is a smooth map, then the pullback bundle is a vector bundle (or fiber bundle whose fiber.

is a section, then the pullback section.

Pullback of multilinear forms

Let be a linear map between vector spaces V and W (i.e., Φ is an element of , also denoted ), and let

be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank , where s is the number of factors of W in the product). 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 (linear) operator from multilinear forms on W to multilinear forms on V. As a special case, note that if F is a linear form (or (0,1)-tensor) 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

be a smooth map between smooth manifolds. Then the differential, is a vector bundle morphism) from the tangent bundle to the pullback bundle. The transpose, the cotangent bundle.

is a section (a 1-form to obtain a pullback section. Applying the above bundle map (pointwise) to this section yields the pullback.

Pullback of (covariant) tensor fields

The construction of the previous section generalizes immediately to tensor bundles tensor field, then the pullback.

Pullback of differential forms

A particular important case of the pullback of covariant tensor fields is the pullback of differential forms-form, i.e., a section of the exterior bundle.

The pullback of differential forms has two properties which make it extremely useful.

1. It is compatible with the wedge product

2. It is compatible with the exterior derivative

Pullback by diffeomorphisms

between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields

according to the tensor product, then the pullback and the pushforward describe the transformation properties of a tensor. 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.

Pullback by automorphisms

. This induces a pullback action on sections of any bundle associated to the frame bundle by a representation of the general linear group).

Pullback and Lie derivative

See Lie derivative, and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.

Pullback of connections (covariant derivatives)

is a connection (or covariant derivative, then there is a pullback connection

See also

• Pushforward (differential)

• Pullback bundle

• Pullback (category theory)

References

• See sections 1.5 and 1.6.

• See section 1.7 and 2.3.