Pullback
Encyclopedia
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 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.
(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.
(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).
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.
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 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 (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. 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 (r,s).
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
for x in M and X in TxM.
s of rank (0,s) for any natural number s: a (0,s) 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 (pointwise) differential of a smooth map φ 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 φ is the (0,s)-tensor field φ*S on M defined by
for x in M and Xj in TxM.
s. If α is a differential k-form, i.e., a section of the exterior bundle
ΛkT*N of (fiberwise) 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 derivative
d: if α is a differential form on N then
, that is, it has a smooth inverse, then pullback can be defined for the vector field
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 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.
GL(M) of M by a representation of the general linear group
GL(m) (m = dim M).
. 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.
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
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
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 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
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, 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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
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 the pullback is essentially the notion of precomposition
Pullback
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 to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
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 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
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" 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 φ-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...
, 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.
Pullback of bundles and sections
If E is a vector bundleVector bundle
In mathematics, a vector bundle is a topological construction that 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...
(or indeed any fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
) over N and φ: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 map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
φ*E is a vector bundle (or fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
) over M whose fiber
Vector bundle
In mathematics, a vector bundle is a topological construction that 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...
over x in M is given by (φ*E)x = Eφ(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 is a continuous right inverse of the function π...
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 map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
is a section of φ*E over M.Pullback of multilinear forms
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 letbe a multilinear form on W (also known as a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
- 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 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 (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 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
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
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 byNevertheless, 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
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 bundleTangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M 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 map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
φ*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 φ* is therefore a bundle map from φ*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 α is a section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of T*N (a 1-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
on N), and precompose α with φ 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 map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
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
for x in M and X in TxM.
Pullback of (covariant) tensor fields
The construction of the previous section generalizes immediately to tensor bundleTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
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 assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
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 (pointwise) differential of a smooth map φ 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 φ is the (0,s)-tensor field φ*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 formDifferential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s. If α 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. It has special significance, because one can define a connection-independent derivative on it, namely the exterior derivative.Sections of the exterior bundle...
ΛkT*N of (fiberwise) 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 derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
d: if α is a differential form on N then
Pullback by diffeomorphisms
When the map φ between manifolds is a diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, that is, it has a smooth inverse, then pullback can be defined for the vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
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 Φ and Φ-1 according to the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
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
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
on the manifold M. In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
; 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(TM,φ*TM). This induces a pullback action on sections of any bundle associated to the frame bundleFrame 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 bases, 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 matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(m) (m = dim M).
Pullback and Lie derivative
See Lie derivativeLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor 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 connectionConnection (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. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
(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...
) 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