Pullback (differential geometry)

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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.

Pullback of smooth functions and smooth maps[edit]

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 direct image 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 .

Pullback of bundles and sections[edit]

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 .

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[edit]

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

be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors of W in the product). Then the pullback ΦF 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, ΦF 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 ΦF 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., WW ⊗ ⋅⋅⋅ ⊗ W. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from VV ⊗ ⋅⋅⋅ ⊗ V to WW ⊗ ⋅⋅⋅ ⊗ W 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 (r, s).

Pullback of cotangent vectors and 1-forms[edit]

Let be a smooth map between smooth manifolds. Then the differential of , written , , or , is a vector bundle morphism (over ) 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 (a 1-form on ), and precompose with to obtain a pullback section of . Applying the above bundle map (pointwise) 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[edit]

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 -forms

By taking equal to the (pointwise) 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[edit]

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 (fiberwise) 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.

  1. It is compatible with the wedge product in the sense that for differential forms and on ,
  2. It is compatible with the exterior derivative : if is a differential form on then

Pullback by diffeomorphisms[edit]

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 map

can 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.

Pullback by automorphisms[edit]

The construction of the previous section has a representation-theoretic interpretation when is a diffeomorphism from a manifold to itself. In this case the derivative is a section of . This induces a pullback action on sections of any bundle associated to the frame bundle of by a representation of the general linear group (where ).

Pullback and Lie derivative[edit]

See Lie derivative. By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on , and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.

Pullback of connections (covariant derivatives)[edit]

If is a connection (or covariant derivative) on a vector bundle over and is a smooth map from to , then there is a pullback connection on over , determined uniquely by the condition that

See also[edit]


  • Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-42627-2. See sections 1.5 and 1.6.
  • Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN 0-8053-0102-X. See section 1.7 and 2.3.