Pullback (differential geometry)

Let ${\displaystyle \phi :M\to N}$  be a smooth map between (smooth) manifolds ${\displaystyle M}$  and ${\displaystyle N}$ , and suppose ${\displaystyle f:N\to \mathbb {R} }$  is a smooth function on ${\displaystyle N}$ . Then the pullbackof${\displaystyle f}$ by${\displaystyle \phi }$  is the smooth function ${\displaystyle \phi ^{*}f}$ on${\displaystyle M}$  defined by ${\displaystyle (\phi ^{*}f)($x)=f(\phi (x))}  . Similarly, if ${\displaystyle f}$  is a smooth function on an open set ${\displaystyle U}$ in${\displaystyle N}$ , then the same formula defines a smooth function on the open set ${\displaystyle f}$ in${\displaystyle \phi ^{-1}($U)}  . (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functionson${\displaystyle N}$  to the direct imageby${\displaystyle \phi }$  of the sheaf of smooth functions on ${\displaystyle M}$ .)

More generally, if ${\displaystyle f:N\to A}$  is a smooth map from ${\displaystyle N}$  to any other manifold ${\displaystyle A}$ , then ${\displaystyle (\phi ^{*}f)($x)=f(\phi (x))}   is a smooth map from ${\displaystyle M}$ to${\displaystyle A}$ .

If${\displaystyle E}$  is a vector bundle (or indeed any fiber bundle) over ${\displaystyle N}$  and ${\displaystyle \phi :M\to N}$  is a smooth map, then the pullback bundle ${\displaystyle \phi ^{*}E}$  is a vector bundle (orfiber bundle) over ${\displaystyle M}$  whose fiber over ${\displaystyle x}$ in${\displaystyle M}$  is given by ${\displaystyle (\phi ^{*}E)_{x}=E_{\phi ($x)}}  .

In this situation, precomposition defines a pullback operation on sections of ${\displaystyle E}$ : if ${\displaystyle s}$  is a sectionof${\displaystyle E}$  over ${\displaystyle N}$ , then the pullback section ${\displaystyle \phi ^{*}s=s\circ \phi }$  is a section of ${\displaystyle \phi ^{*}E}$  over ${\displaystyle M}$ .

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^∗FofF by Φ is a multilinear form on V defined by precomposing F with Φ. More precisely, given vectors v₁, v₂, ..., v_sinV, 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 spaceofW, 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 productofr 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 ⊗ ⋅⋅⋅ ⊗ VtoW ⊗ 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 ${\displaystyle \phi :M\to N}$  be a smooth map between smooth manifolds. Then the differentialof${\displaystyle \phi }$ , written ${\displaystyle \phi _{*}}$ , ${\displaystyle d\phi }$ , or ${\displaystyle D\phi }$ , is a vector bundle morphism (over ${\displaystyle M}$ ) from the tangent bundle ${\displaystyle TM}$ of${\displaystyle M}$  to the pullback bundle ${\displaystyle \phi ^{*}TN}$ . The transposeof${\displaystyle \phi _{*}}$  is therefore a bundle map from ${\displaystyle \phi ^{*}T^{*}N}$ to${\displaystyle T^{*}M}$ , the cotangent bundleof${\displaystyle M}$ .

Now suppose that ${\displaystyle \alpha }$  is a sectionof${\displaystyle T^{*}N}$  (a1-formon${\displaystyle N}$ ), and precompose ${\displaystyle \alpha }$  with ${\displaystyle \phi }$  to obtain a pullback sectionof${\displaystyle \phi ^{*}T^{*}N}$ . Applying the above bundle map (pointwise) to this section yields the pullbackof${\displaystyle \alpha }$ by${\displaystyle \phi }$ , which is the 1-form ${\displaystyle \phi ^{*}\alpha }$ on${\displaystyle M}$  defined by

The construction of the previous section generalizes immediately to tensor bundles of rank ${\displaystyle (0,s)}$  for any natural number ${\displaystyle s}$ : a ${\displaystyle (0,s)}$  tensor field on a manifold ${\displaystyle N}$  is a section of the tensor bundle on ${\displaystyle N}$  whose fiber at ${\displaystyle y}$ in${\displaystyle N}$  is the space of multilinear ${\displaystyle s}$ -forms

A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If ${\displaystyle \alpha }$  is a differential ${\displaystyle k}$ -form, i.e., a section of the exterior bundle ${\displaystyle \Lambda ^{k}(T^{*}N)}$  of (fiberwise) alternating ${\displaystyle k}$ -forms on ${\displaystyle TN}$ , then the pullback of ${\displaystyle \alpha }$  is the differential ${\displaystyle k}$ -form on ${\displaystyle M}$  defined by the same formula as in the previous section:

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 ${\displaystyle \alpha }$  and ${\displaystyle \beta }$ on${\displaystyle N}$ ,
   $${\displaystyle \phi ^{*}(\alpha \wedge \beta )=\phi ^{*}\alpha \wedge \phi ^{*}\beta .}$$
    
2. It is compatible with the exterior derivative ${\displaystyle d}$ : if ${\displaystyle \alpha }$  is a differential form on ${\displaystyle N}$  then
   $${\displaystyle \phi ^{*}(d\alpha )=d(\phi ^{*}\alpha ).}$$
    

When the map ${\displaystyle \phi }$  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 ${\displaystyle \Phi }$  and ${\displaystyle \Phi ^{-1}}$  according to the tensor product decomposition of the tensor bundle into copies of ${\displaystyle TN}$  and ${\displaystyle T^{*}N}$ . When ${\displaystyle M=N}$ , then the pullback and the pushforward describe the transformation properties of a tensor on the manifold ${\displaystyle M}$ . 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 ${\displaystyle \phi }$  is a diffeomorphism from a manifold ${\displaystyle M}$  to itself. In this case the derivative ${\displaystyle d\phi }$  is a section of ${\displaystyle \operatorname {GM} (TM,\phi ^{*}TM)}$ . This induces a pullback action on sections of any bundle associated to the frame bundle ${\displaystyle \operatorname {GM} ($m)}  of${\displaystyle M}$  by a representation of the general linear group ${\displaystyle \operatorname {GM} ($m)}   (where ${\displaystyle m=\dim M}$ ).

See Lie derivative. By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on ${\displaystyle M}$ , and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.

If${\displaystyle \nabla }$  is a connection (orcovariant derivative) on a vector bundle ${\displaystyle E}$  over ${\displaystyle N}$  and ${\displaystyle \phi }$  is a smooth map from ${\displaystyle M}$ to${\displaystyle N}$ , then there is a pullback connection ${\displaystyle \phi ^{*}\nabla }$ on${\displaystyle \phi ^{*}E}$  over ${\displaystyle M}$ , determined uniquely by the condition that

• 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.