Suppose that is a
smooth map between
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s ''M'' and ''N''. Then there is an associated
linear map from the space of
1-forms
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to eac ...
on ''N'' (the
linear space of
sections of the
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. It may be described also as the dual bundle to the tangent bundle. This ...
) 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
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
– on ''N'' may be pulled back to ''M'' using ''φ''.
When the map ''φ'' is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
, 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 R
''n'' and R
''n'', 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
In music, precompositional decisions are those decisions which a composer decides upon before or while beginning to create a composition. These limits may be given to the composer, such as the length or style needed, or entirely decided by the com ...
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 functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
s.
Pullback of smooth functions and smooth maps
Let be a smooth map between (smooth) 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 set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
''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 is a smooth map from ''N'' to any other manifold ''A'', then is a smooth map from ''M'' to ''A''.
Pullback of bundles and sections
If ''E'' is a
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 (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
(or indeed any
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
) over ''N'' and is a smooth map, then the
pullback bundle ''φ''
∗''E'' is a vector bundle (or
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
) over ''M'' whose
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
over ''x'' in ''M'' is given by .
In this situation, precomposition defines a pullback operation on sections of ''E'': if ''s'' is a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
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 spaces ''V'' and ''W'' (i.e., Φ is an element of , also denoted ), and let
:
be a multilinear form on ''W'' (also known as a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
– not to be confused with a tensor field – of rank , 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 ''v''
1, ''v''
2, ..., ''v''
''s'' 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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
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
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
(over ''M'') from the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
''TM'' of ''M'' to the
pullback bundle ''φ''
*''TN''. The
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of ''φ''
* is therefore a bundle map from ''φ''
*''T''
*''N'' to ''T''
*''M'', the
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. It may be described also as the dual bundle to the tangent bundle. This ...
of ''M''.
Now suppose that ''α'' is a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of ''T''
*''N'' (a
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction ...
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 ''T''
''x''''M''.
Pullback of (covariant) tensor fields
The construction of the previous section generalizes immediately to
tensor bundles of rank (0,''s'') for any natural number ''s'': a (0,''s'')
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analys ...
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 ''X''
''j'' in ''T''
''x''''M''.
Pullback of differential forms
A particular important case of the pullback of covariant tensor fields is the pullback of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. If ''α'' is a differential ''k''-form, i.e., a section of the
exterior bundle In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor prod ...
Λ
''k''''T''*''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 ''X''
''j'' in ''T''
''x''''M''.
The pullback of differential forms has two properties which make it extremely useful.
# It is compatible with the
wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
in the sense that for differential forms ''α'' and ''β'' on ''N'',
#:
# It is compatible with the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
''d'': if ''α'' is a differential form on ''N'' then
#:
Pullback by diffeomorphisms
When the map ''φ'' between manifolds is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
, 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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
on the manifold ''M''. In traditional terms, the pullback describes the transformation properties of the covariant indices of a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
; 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 bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts na ...
GL(''M'') of ''M'' by a representation of the
general linear group GL(''m'') (where ''m'' = dim ''M'').
Pullback and Lie derivative
See
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vecto ...
. 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 (or
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 differ ...
) on a vector bundle ''E'' over ''N'' and ''φ'' is a smooth map from ''M'' to ''N'', then there is a pullback connection ''φ''
∗∇ on ''φ''
∗''E'' over ''M'', determined uniquely by the condition that
:
See also
*
Pushforward (differential)
*
Pullback bundle
*
Pullback (category theory)
References
* ''See sections 1.5 and 1.6''.
* ''See section 1.7 and 2.3''.
{{Manifolds
Tensors
Differential geometry