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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... 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... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... 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 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 R^m 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. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

The idea behind 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. This article needs to be cleaned up to conform to a higher standard of quality. ...

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 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.) In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

More generally, if f:N→A 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 (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 = E_φ(x). 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, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... 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 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. In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

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 let 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...

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,...,v_s in V, Φ^*F is defined by the formula In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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: 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). ...

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 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. ...

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).

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 bundle 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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a manifold M is a type of space, characterized 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, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ... 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 differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

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 In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

for x in M and X in T_xM.

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 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 In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...

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_xM.

Pullback of differential forms

A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If α is a differential k-form, i.e., a section of the exterior bundle Λ^kT*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: A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

for x in M and X_j in T_xM.

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, 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. ...

2. It is compatible with the exterior derivative d: if α is a differential form on N then In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

Pullback by diffeomorphisms

When the map φ between manifolds is a diffeomorphism, 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 In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

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. 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. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In category theory, see covariant functor. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

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 an pullback action on sections of any bundle associated to the frame bundle GL(M) of M by a representation of the general linear group GL(m) (m = dim M). In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... 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. ...

Pullback and Lie derivative

See Lie derivative. 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. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

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 on φ^*E over M, determined uniquely by the condition that In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

See also

• Pushforward (differential)
• Pullback bundle
• Pullback (category theory)

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 3-540-42627-2 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.