Suppose that is a smooth map between smooth manifoldsM and N. Then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback, and is frequently denoted by φ∗. More generally, any covarianttensor 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, then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional 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 turns several constructions in differential geometry into contravariant functors.
Let be a smooth map between manifolds M and N, and suppose is a smooth function on N. Then the pullback of f by φ is the smooth function φ∗f on M defined by. Similarly, if f is a smooth function on an open setU in N, then the same formula defines a smooth function on the open set φ−1 in M. More generally, if is a smooth map from N to any other manifold A, then is a smooth map from M to A.
If E is a vector bundle over N and φ:M→N is a smooth map, then the pullback bundleφ∗E is a vector bundle over M whose fiber over x in M is given by x = Eφ. In this situation, precomposition defines a pullback operation on sections of E: if s is a section of E over N, then the pullback section is a section of φ∗E over M.
Pullback of multilinear forms
Let be a linear map between vector spacesV and W, and let be a multilinear form on W. 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 operator from multilinear forms on W to multilinear forms on V. As a special case, note that if F is a linear form on W, so that F is an element ofW∗, 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
Let φ : M → N be a smooth map between smooth manifolds. Then the differential of φ, written φ*, dφ, or Dφ, is a vector bundle morphism from the tangent bundleTM 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, and precompose α with φ to obtain a pullback section of φ*T*N. Applying the above bundle map to this section yields the pullback of α by φ, which is the 1-formφ*α on M defined by for x in M and X in TxM.
The construction of the previous section generalizes immediately to tensor bundles of rank for any natural number s: a tensor field on a manifold N is a section of the tensor bundle on N whose fiber at y in N is the space of multilinear s-forms By taking Φ equal to the differential of a smooth mapφ from M to N, the pullback of multilinear forms can be combined with the pullback of sections to yield a pullback tensor field on M. More precisely if S is a -tensor field on N, then the pullback of S by φ is the -tensor field φ*S on M defined by for x in M and Xj in TxM.
A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If α is a differential k-form, i.e., a section of the exterior bundle ΛkT*N of alternating k-forms on TN, then the pullback of α is the differential k-form on M defined by the same formula as in the previous section: for x in M and Xj in TxM. 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, 2. It is compatible with the exterior derivatived: if α is a differential form on N then
Pullback by diffeomorphisms
When the map φ 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 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.
Pullback by automorphisms
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. This induces a pullback action on sections of any bundle associated to the frame bundle GL of M by a representation of the general linear group GL.
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.
Pullback of connections (covariant derivatives)
If ∇ is a connection on a vector bundleE over N and φ is a smooth map from M to N, then there is a pullback connectionφ∗∇ on φ∗E over M, determined uniquely by the condition that
