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*.

- Pullback of smooth functions and smooth maps
- Pullback of bundles and sections
- Pullback of multilinear forms
- Pullback of cotangent vectors and 1-forms
- Pullback of (covariant) tensor fields
- Pullback of differential forms
- Pullback by diffeomorphisms
- Pullback by automorphisms
- Pullback and Lie derivative
- Pullback of connections (covariant derivatives)
- See also
- References

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**^{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 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.

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*.)

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*.

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 this situation, precomposition defines a pullback operation on sections of *E*: if *s* is a section of *E* over *N*, then the ** pullback section ***φ*^{∗}*s* = *s* ∘ *φ* is a section of *φ*^{∗}*E* over *M*.

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

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 *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 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*, i.e., *W* ⊗ *W* ⊗ ⋅⋅⋅ ⊗ *W*. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from *V* ⊗ *V* ⊗ ⋅⋅⋅ ⊗ *V* to *W* ⊗ *W* ⊗ ⋅⋅⋅ ⊗ *W* 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*).

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 (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*.

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 *T*_{x}*M*.

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

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*.

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^{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 in the sense that for differential forms
*α*and*β*on*N*, - It is compatible with the exterior derivative
*d*: if*α*is a differential form on*N*then

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

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

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 GL(*M*) of *M* by a representation of the general linear group GL(*m*) (where *m* = dim *M*).

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.

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

In differential geometry, one can attach to every point of a smooth manifold, , a vector space called the **cotangent space** at *. Typically, the cotangent space, is defined as the dual space of the tangent space at **, , although there are more direct definitions. The elements of the cotangent space are called ***cotangent vectors** or **tangent covectors**.

In mathematics, the **tensor product***V* ⊗ *W* of two vector spaces *V* and *W* is a vector space, endowed with a bilinear map from the Cartesian product *V* × *W* to *V* ⊗ *W*. This bilinear map is universal in the sense that, for every vector space X, the bilinear maps from *V* × *W* to X are in one to one correspondence with the linear maps from *V* ⊗ *W* to X.

In the mathematical field of differential geometry, one definition of a **metric tensor** is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar *g*(*v*, *w*) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

In differential geometry, the **tangent bundle** of a differentiable manifold is a manifold which assembles all the tangent vectors in *. As a set, it is given by the disjoint union of the tangent spaces of **. That is,*

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 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, especially in geometry, topology and physics.

In mathematics, the modern component-free approach to the theory of a **tensor** views a tensor as an abstract object, expressing some definite type of multilinear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.

In mathematics and physics, 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 sciences. As a tensor is a generalization of a scalar and a vector, a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.

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*(*x*) in such a way that these vector spaces fit together to form another space of the same kind as *X*, which is then called a **vector bundle over X**.

In differential geometry, the **Lie derivative**, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) ↦*x*.

In mathematics, and especially differential geometry and gauge theory, 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. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In abstract algebra and multilinear algebra, a **multilinear form** on a vector space over a field is a map

In mathematical analysis, the **smoothness** of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be **infinitely differentiable** and referred to as a **C-infinity function**.

In differential geometry, **pushforward** is a linear approximation of smooth maps on tangent spaces. Suppose that *φ* : *M* → *N* is a smooth map between smooth manifolds; then the **differential** of *φ* at a point *x* is, in some sense, the best linear approximation of *φ* near *x*. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of *M* at *x* to the tangent space of *N* at *φ*(*x*). Hence it can be used to *push* tangent vectors on *M**forward* to tangent vectors on *N*. The differential of a map *φ* is also called, by various authors, the **derivative** or **total derivative** of *φ*.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In differential geometry, a **Lie algebra-valued form** is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

In mathematics, a **vector-valued differential form** on a manifold *M* is a differential form on *M* with values in a vector space *V*. More generally, it is a differential form with values in some vector bundle *E* over *M*. Ordinary differential forms can be viewed as **R**-valued differential forms.

In mathematics and theoretical physics, an **invariant differential operator** is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

In geometry, a **valuation** is a finitely additive function on a collection of admissible subsets of a fixed set with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies of Euclidean space . Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic.

- Jost, Jürgen (2002).
*Riemannian Geometry and Geometric Analysis*. Berlin: Springer-Verlag. ISBN 3-540-42627-2.*See sections 1.5 and 1.6*. - Abraham, Ralph; Marsden, Jerrold E. (1978).
*Foundations of Mechanics*. London: Benjamin-Cummings. ISBN 0-8053-0102-X.*See section 1.7 and 2.3*.

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