Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*. More generally, any covariant tensor field - in particular any differential form - on N may be pulled back to M using φ.
When the map φ is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice-versa. In particular, if φ is a diffeomorphism between open subsets of Rn and Rn, viewed as a change of coordinates (perhaps between different charts on a manifold M), 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 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.
Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ*f on M defined by (φ*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)
More generally, if f:N→A is a smooth map from N to any other manifold A, then φ*f(x)=f(φ(x)) is a smooth map from M to A.
If E is a vector bundle (or indeed any fiber bundle) over N and φ:M→N is a smooth map, then the pullback bundle φ*E is a vector bundle (or fiber bundle) over M whose fiber over x in M is given by (φ*E)x = Eφ(x).
Let Φ:V→ W 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
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. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from to given by
Let φ : M → N be a smooth map between smooth manifolds. Then the differential of φ, φ* = dφ (or Dφ), is a vector bundle morphism (over M) from the tangent bundle TM of M to the pullback bundle φ*TN. The transpose of φ* is therefore a bundle map from φ*T*N to T*M, the cotangent bundle of M.
Now suppose that α is a section of T*N (a 1-form on N), and precompose α with φ to obtain a pullback section of φ*T*N. Applying the above bundle map (pointwise) to this section yields the pullback of α by φ, which is the 1-form φ*α on M defined by
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 α and β on N,
A general mixed tensor field will then transform using Φ and Φ-1 according to the tensor product decomposition of the tensor bundle into copies of TN and T*N. When M = N, then the pullback and the pushforward describe the transformation properties of a tensor on the manifold M. 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 φ is a diffeomorphism from a manifold M to itself. In this case the derivative dφ is a section of GL(TM,φ*TM). This induces a pullback action on sections of any bundle associated to the frame bundle GL(M) of M by a representation of the general linear group GL(m) (m = dim M).
See Lie derivative. By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on M, and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.
If is a connection (or covariant derivative) on a vector bundle E over N and φ is a smooth map from M to N, then there is a pullback connection on φ*E over M, determined uniquely by the condition that