More generally, any covariant tensor field – in particular any differential form – on
is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from
, 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.
is a smooth function on an open set
, then the same formula defines a smooth function on the open set
(In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on
is a smooth map, then the pullback bundle
In this situation, precomposition defines a pullback operation on sections of
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).
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., W ⊗ W ⊗ ⋅⋅⋅ ⊗ W. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from V ⊗ V ⊗ ⋅⋅⋅ ⊗ V to W ⊗ W ⊗ ⋅⋅⋅ ⊗ 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).
Applying the above bundle map (pointwise) to this section yields the pullback of
The construction of the previous section generalizes immediately to tensor bundles of rank
equal to the (pointwise) differential of a smooth map
The pullback of differential forms has two properties which make it extremely useful.
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.
A general mixed tensor field will then transform using
, 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.
The construction of the previous section has a representation-theoretic interpretation when
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.
is a connection (or covariant derivative) on a vector bundle