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.

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

References 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.