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Suppose that f: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
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of the cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
) to the space of 1-forms on M. This linear map is known as the pullback (by f), and is frequently denoted by f*. More generally, any covariant tensor field - in particular any differential form - on N may be pulled back to M using f.

When the map f is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice-versa.






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Suppose that f: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
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of the cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
) to the space of 1-forms on M. This linear map is known as the pullback (by f), and is frequently denoted by f*. More generally, any covariant tensor field - in particular any differential form - on N may be pulled back to M using f.

When the map f is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
, 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 f is a diffeomorphism between open subsets of Rn and Rn, viewed as a change of coordinates (perhaps between different charts
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 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
Pullback

Suppose that f: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 to the space of 1-forms on M....
 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


Let f: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 f is the smooth function f*f on M defined by (f*f)(x) = f(f(x)). Similarly, if f is a smooth function on an open set
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 U in N, then the same formula defines a smooth function on the open set f-1(U) in M. (In the language of sheaves
Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one....
, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by f 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 f*f(x)=f(f(x)) is a smooth map from M to A.

Pullback of bundles and sections


If E is a vector bundle
Vector bundle

In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...
 (or indeed any fiber bundle
Fiber bundle

File:Roundhairbrush.JPGIn mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B ? F, but globally may have a different topological structure....
) over N and f:M?N is a smooth map, then the pullback bundle
Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
 f*E is a vector bundle (or fiber bundle
Fiber bundle

File:Roundhairbrush.JPGIn mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B ? F, but globally may have a different topological structure....
) over M whose fiber
Vector bundle

In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...
 over x in M is given by (f*E)x = Ef(x).

In this situation, precomposition defines a pullback operation on sections of E: if s is a section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of E over N, then the pullback section
Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
  is a section of f*E over M.

Pullback of multilinear forms


Let F:V? W be a linear map between vector spaces V and W (i.e., F 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
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
 - 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 F*F of F by F is a multilinear form on V defined by precomposing F with F. More precisely, given vectors v1,v2,...,vs in V, F*F is defined by the formula which is a multilinear form on V. Hence F* 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
Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra....
 of W, then F*F is an element of V*, and so pullback by F defines a linear map between dual spaces which acts in the opposite direction to the linear map F 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
Tensor product

In mathematics, the tensor product, denoted by , may be applied in different contexts to vector spaces, matrix , tensors, vector spaces, algebra over a field, topological vector spaces, and module s....
  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 Nevertheless, it follows from this that if F is invertible, pullback can be defined using pushforward by the inverse function F-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


Let f : M ? N be a smooth map between smooth manifolds. Then the differential of f, f* = df (or Df), is a vector bundle morphism (over M) from the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
 TM of M to the pullback bundle
Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
 f*TN. The transpose
Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra....
 of f* is therefore a bundle map from f*T*N to T*M, the cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
 of M.

Now suppose that a is a section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of T*N (a 1-form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
 on N), and precompose a with f to obtain a pullback section
Pullback bundle

In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
 of f*T*N. Applying the above bundle map (pointwise) to this section yields the pullback of a by f, which is the 1-form f*a on M defined by for x in M and X in TxM.

Pullback of (covariant) tensor fields

The construction of the previous section generalizes immediately to tensor bundle
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
s of rank (0,s) for any natural number s: a (0,s) tensor field
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
 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 F equal to the (pointwise) differential of a smooth map f from M to N, the pullback of multilinear forms can be combined with the pullback of sections to yield a pullback (0,s) tensor field on M. More precisely if S is a (0,s)-tensor field on N, then the pullback of S by f is the (0,s)-tensor field f*S on M defined by for x in M and Xj in TxM.

Pullback of differential forms

A particular important case of the pullback of covariant tensor fields is the pullback of differential form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
s. If a is a differential k-form, i.e., a section of the exterior bundle
Exterior bundle

In mathematics, the exterior bundle of a manifold M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors....
 ?kT*N of (fiberwise) alternating k-forms on TN, then the pullback of a 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 a and ß on N, 2. It is compatible with the exterior derivative
Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree zero, to differential forms of higher degree....
 d: if a is a differential form on N then

Pullback by diffeomorphisms

When the map f between manifolds is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
, that is, it has a smooth inverse, then pullback can be defined for the vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s 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 F and F-1 according to the tensor product
Tensor product

In mathematics, the tensor product, denoted by , may be applied in different contexts to vector spaces, matrix , tensors, vector spaces, algebra over a field, topological vector spaces, and module s....
 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
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
 on the manifold M. In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
; 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 f is a diffeomorphism from a manifold M to itself. In this case the derivative df is a section of GL(TM,f*TM). This induces a pullback action on sections of any bundle associated to the frame bundle
Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered basis, or frames, for Ex....
 GL(M) of M by a representation of the general linear group
General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrix, together with the operation of ordinary matrix multiplication....
 GL(m) (m = dim M).

Pullback and Lie derivative


See Lie derivative
Lie derivative

In mathematics, the Lie derivative, named after Sophus Lie by Wladyslaw Slebodzinski, evaluates the change of one vector field along the flow of another vector field....
. 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
Connection (vector bundle)

In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points....
 (or covariant derivative
Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a connection on the frame bundle &mdas...
) on a vector bundle E over N and f is a smooth map from M to N, then there is a pullback connection on f*E over M, determined uniquely by the condition that

See also

  • Pushforward (differential)
  • Pullback bundle
    Pullback bundle

    In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
  • Pullback (category theory)
    Pullback (category theory)

    In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X ? Z and g : Y ? Z with a common codomain; it is the limit of the cospan ....