In mathematics, a pullback can be defined in several different contexts. This article focuses primarily on the pullback of tensors on differentiable manifolds. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

Given a continuously differentiable map from one differentiable manifold to another, there is an associated mapping from the cotangent bundle of Y to that of X, known as the pullback, and frequently denoted by f*. More generally, any covariant tensor pulls back under f*. Jump to: navigation, search A differentiability class in mathematics is a class of functions which share differentiability features. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In category theory, see covariant functor. ...

When the map f is a diffeomorphism of a manifold to itself, then the pullback, together with the pushforward, describe the transformation properties of the manifold under a change of coordinates. Using traditional language, these describe the transformation properties of contravariant and covariant tensors. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ...

Pullback on tensors

Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f ^* T on V can be defined. That is, given a tensor In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Jump to: navigation, search In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ...

and a set of vectors

one then defines the pullback as

.

The result f ^* T is again a tensor, so that f ^* is in fact a mapping from tensors on W to tensors on V. As a special case, note that if T is a (0,1)-tensor, so that , the dual space of W, then , and so the pullback acts in the reversed direction: Jump to: navigation, search In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

.

For a general f, a pullback can only be defined on tensors of rank (0,n). This is precisely because a pullback on mixed tensors would need to be "going in the opposed direction" for the contravariant indeces. If f is invertible, then this can be done, and one can define the pullback of an arbitrary mixed-rank tensor, as shown next. Let f be a linear isomorphism, so that is invertible. The pullback of a tensor of rank (n,0) can be defined by employing ; we make use of the identity (f ^{− 1}) ^* = (f ^* ) ^{− 1}. One then defines Contravariant is a mathematical term with a precise definition in tensor analysis. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ...

Thus we've shown that if a linear transformation is invertible, it can be used to define the pullback on general tensors of mixed rank (m,n). Perhaps the easiest way to visualize and understand the above is to keep firmly in mind that f is nothing more than a matrix, so that f(v) is just the multiplication of a vector by a matrix. Similarly, the dual space should be visualized as nothing more than a dot product. Jump to: navigation, search For the square matrix section, see square matrix. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

Pullback of (co)tangent bundles

The pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : T*N → T*M, for which the following diagram commutes: In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Here T*M and T*N are the cotangent bundles of M and N respectively, and π_M and π_N are the natural projections. Perhaps the easiest way to understand the pullback is in terms of the pushforward of f. Picking a point , the pushforward at p is a linear map between the tangent spaces Jump to: navigation, search Image File history File links SmoothPullback-01. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

where L(V,W) being the set of linear mappings from the vector space V = T_pM to the vector space W = T_f(p)N. The cotangent space is dual to the tangent space, and maps on the dual space act as the transpose. That is, consider two ordinary vectors v and w, and a matrix A. The dot product obeys the identity . Thus, if we take In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... Jump to: navigation, search In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... Jump to: navigation, search For the square matrix section, see square matrix. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

then the transpose is going to act on the 1-forms: (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

.

We use the transpose to map the cotangent spaces. For each point in the manifold, the pullback is defined as the matrix transpose of the pushforward; that is,

f ^* (p) = [f _* (p)]^T.

Note that this mapping is in a certain sense going in the "backwards" direction, that is,

.

Pullback on tensor bundles

More generally, one can construct the pullback map between tensor bundles of rank (0,n); the construction proceeds entirely analogously to that for a tensor. That is, by considering the cotangent space at point p in M, one defines the tensor space at point p as the n-fold tensor product

.

The pullback then proceeds analogously to the tensor space defined through an n-fold tensor product of . This definition applies as well to the exterior bundles Λ^kT*N and Λ^kT*M, are strict subspaces of the general tensor bundles, closed under the exterior algebra. The pullback operation commutes with the exterior algebra, and so the pullback of an alternating form is again an alternating form. That is, the pullback of a differential form on N is a differential form on M. Symbolically, we write In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

for α and β in Λ(M). Similarly, the pullback is natural with respect to derivations:

f ^* (dω) = d(f ^* ω)

for ω in Λ(M).

Pullback of diffeomorphisms

When the map f between manifolds is a diffeomorphism, that is, it is both smooth and invertible, then the pullback can be defined for the tangent space as well as for the cotangent space, and thus, by extension, for an arbitrary mixed tensor bundle on the manifold. The matrix In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ...

can be inverted to define

and thus one has, at each point p, that the pushforward is the inverse of the pullback, now acting on the tangent space (instead of the cotangent space):

f ^* (p) = [f _* (p)] ^{− 1}

so that

.

A general mixed tensor will then transform as a mixture of transposes and inverses, depending on whether the indices are contra- or co-variant. 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. In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... Jump to: navigation, search In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... In category theory, see covariant functor. ... Jump to: navigation, search In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

See also

• pullback bundle
• pullback (category theory)

Jump to: navigation, search In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See sections 1.5 and 1.6.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.7 and 2.3.