Pullback (differential geometry)

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

${\displaystyle F:W\times W\times \cdots \times W\rightarrow \mathbf {R} }$

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

${\displaystyle (\Phi ^{*}F)(v_{1},v_{2},\ldots ,v_{s})=F(\Phi (v_{1}),\Phi (v_{2}),\ldots ,\Phi (v_{s})),}$

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:

${\displaystyle \Phi \colon V\rightarrow W,\qquad \Phi ^{*}\colon W^{*}\rightarrow V^{*}.}$

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 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

${\displaystyle \Phi _{*}(v_{1}\otimes v_{2}\otimes \cdots \otimes v_{r})=\Phi (v_{1})\otimes \Phi (v_{2})\otimes \cdots \otimes \Phi (v_{r}).}$

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 (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

${\displaystyle (\varphi ^{*}\alpha )_{x}(X)=\alpha _{\varphi (x)}(d\varphi _{x}(X))}$

for x in M and X in T_xM.

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

${\displaystyle F\colon T_{y}N\times \cdots \times T_{y}N\to \mathbf {R} .}$

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

${\displaystyle (\varphi ^{*}S)_{x}(X_{1},\ldots ,X_{s})=S_{\varphi (x)}(d\varphi _{x}(X_{1}),\ldots ,d\varphi _{x}(X_{s}))}$

for x in M and X_j in T_xM.

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:

${\displaystyle (\varphi ^{*}\alpha )_{x}(X_{1},\ldots ,X_{k})=\alpha _{\varphi (x)}(d\varphi _{x}(X_{1}),\ldots ,d\varphi _{x}(X_{k}))}$

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,

   ${\displaystyle \varphi ^{*}(\alpha \wedge \beta )=\varphi ^{*}\alpha \wedge \varphi ^{*}\beta .}$

2. It is compatible with the exterior derivative d: if α is a differential form on N then

   ${\displaystyle \varphi ^{*}(d\alpha )=d(\varphi ^{*}\alpha ).}$

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

${\displaystyle \Phi =d\varphi _{x}\in \operatorname {GL} \left(T_{x}M,T_{\varphi (x)}N\right)}$

can be inverted to give

${\displaystyle \Phi ^{-1}=\left({d\varphi _{x}}\right)^{-1}\in \operatorname {GL} \left(T_{\varphi (x)}N,T_{x}M\right).}$

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.

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

${\displaystyle \left(\varphi ^{*}\nabla \right)_{X}\left(\varphi ^{*}s\right)=\varphi ^{*}\left(\nabla _{d\varphi (X)}s\right).}$

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

• 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.