Pullback (differential Geometry)
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Suppose that is a
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
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 ma ...
s ''M'' and ''N''. Then there is an associated
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
from the space of 1-forms on ''N'' (the
linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of
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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 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 applications, ...
– 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 two m ...
, then the pullback, together with the
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
, 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 In mathematics, an ordered basis of a vector space of finite dimension (vector space), dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a finite sequence, sequence of scalar (mathematics), ...
(perhaps between different
charts A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tab ...
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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
into contravariant
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 are suf ...
''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 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 ...
on ''N'' to the
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
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 every po ...
(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 p ...
) over ''N'' and is a smooth map, then the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
''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 p ...
) 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 incorporate ...
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 sig ...
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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between vector spaces ''V'' and ''W'' (i.e., Φ is an element of , also denoted ), and let :F:W \times W \times \cdots \times W \rightarrow \mathbf 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 tenso ...
– 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 :(\Phi^*F)(v_1,v_2,\ldots,v_s) = F(\Phi(v_1), \Phi(v_2), \ldots ,\Phi(v_s)), 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: :\Phi\colon V\rightarrow W, \qquad \Phi^*\colon W^*\rightarrow V^*. 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) 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 element of V \otimes W ...
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 :\Phi_*(v_1\otimes v_2\otimes\cdots\otimes v_r)=\Phi(v_1)\otimes \Phi(v_2)\otimes\cdots\otimes \Phi(v_r). 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 In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
between
smooth manifolds 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 ma ...
. Then the differential of ''φ'', written ''φ''*, ''dφ'', or ''Dφ'', is a vector bundle morphism (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 of ...
''TM'' of ''M'' to the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
''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 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 sig ...
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 : (\varphi^*\alpha)_x(X) = \alpha_(d\varphi_x(X)) 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 bundle In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of t ...
s 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 analysis ...
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 : F\colon T_y N\times\cdots \times T_y N\to \mathbf. 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 : (\varphi^*S)_x(X_1,\ldots, X_s) = S_(d\varphi_x(X_1),\ldots, d\varphi_x(X_s)) 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 applications, ...
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 produ ...
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: : (\varphi^*\alpha)_x(X_1,\ldots, X_k) = \alpha_(d\varphi_x(X_1),\ldots, d\varphi_x(X_k)) 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'', #: \varphi^*(\alpha \wedge \beta)=\varphi^*\alpha \wedge \varphi^*\beta. # 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 #: \varphi^*(d\alpha) = d(\varphi^*\alpha).


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 two m ...
, 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 :\Phi = d\varphi_x \in \operatorname\left(T_x M, T_N\right) can be inverted to give :\Phi^ = \left(\right)^ \in \operatorname\left(T_N, T_x M\right). 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) 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 element of V \otimes W ...
decomposition of the tensor bundle into copies of ''TN'' and ''T*N''. When ''M'' = ''N'', then the pullback and the
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
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 tenso ...
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 tenso ...
; by contrast, the transformation of the contravariant indices is given by a
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
.


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 nat ...
GL(''M'') of ''M'' by a representation of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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 vector fi ...
. 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) 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 :\left(\varphi^*\nabla\right)_X\left(\varphi^*s\right) = \varphi^*\left(\nabla_ s\right).


See also

*
Pushforward (differential) In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the best ...
*
Pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
*
Pullback (category theory) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...


References

* ''See sections 1.5 and 1.6''. * ''See section 1.7 and 2.3''. {{Manifolds Tensors Differential geometry