- Pullback (differential geometry)
Suppose that "φ":"M"→ "N" is a
smooth map between smooth manifolds "M" and "N"; then there is an associatedlinear map from the space of 1-forms on "N" (the linear space of sections of thecotangent 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, anycovariant tensor field - in particular any differential form - on "N" may be pulled back to "M" using "φ".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 Rn and Rn, viewed as achange of coordinates (perhaps between different charts on a manifold "M"), then the pullback and pushforward describe the transformation properties ofcovariant andcontravariant tensors used in more traditional (coordinate dependent) approaches to the subject.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. Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors.
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 thesheaf 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".
Pullback of bundles and sections
If "E" is a
vector bundle (or indeed anyfiber bundle ) over "N" and "φ":"M"→"N" is a smooth map, then thepullback bundle "φ"*"E" is a vector bundle (orfiber 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 varphi^*s=scircvarphi is a section of "φ"*"E" over "M".
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:F:W imes W imes cdots imes W ightarrow mathbb{R}be a multilinear form on "W" (also known as atensor - 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: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"*, thedual 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::Phicolon V ightarrow W, qquad Phi^*colon W^* ightarrow 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 Wotimes Wotimescdotsotimes W of "r" copies of "W". However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from Votimes Votimescdotsotimes V to Wotimes Wotimescdotsotimes W given by:Phi_*(v_1otimes v_2otimescdotsotimes v_r)=Phi(v_1)otimes Phi(v_2)otimescdotsotimes 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").Pullback of cotangent vectors and 1-forms
Let "φ" : "M" → "N" be a
smooth map betweensmooth manifolds . Then the differential of "φ", "φ"* = d"φ" (or "Dφ"), is avector bundle morphism (over "M") from thetangent bundle "TM" of "M" to thepullback bundle "φ"*"TN". The transpose of "φ"* is therefore a bundle map from "φ"*"T"*"N" to "T"*"M", thecotangent 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:varphi^*alpha)_x(X) = alpha_{varphi(x)}(mathrm dvarphi_x(X))for "x" in "M" and "X" in "T""x""M".
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:Fcolon T_y N imescdots imes T_y N o 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:varphi^*S)_x(X_1,ldots, X_s) = S_{varphi(x)}(mathrm dvarphi_x(X_1),ldots mathrm dvarphi_x(X_s))for "x" in "M" and "X""j" in "T""x""M".Pullback of differential forms
A particular important case of the pullback of covariant tensor fields is the pullback of
differential form s. If "α" is a differential "k"-form, i.e., a section of theexterior 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::varphi^*alpha)_x(X_1,ldots, X_k) = alpha_{varphi(x)}(mathrm dvarphi_x(X_1),ldots mathrm dvarphi_x(X_k))for "x" in "M" and "X""j" in "T""x""M".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", :varphi^*(alpha wedge eta)=varphi^*alpha wedge varphi^*eta.2. It is compatible with theexterior derivative d: if "α" is a differential form on "N" then:varphi^*(mathrm dalpha) = mathrm d(varphi^*alpha).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 thevector field s as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold. The linear
Phi=mathrm dvarphi_xin GL(T_xM,T_{varphi(x)}N)can be inverted to give:Phi^{-1}={mathrm dvarphi_x}^{-1} in GL(T_{varphi(x)}N, T_xM).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 atensor on the manifold "M". In traditional terms, the pullback describes the transformation properties of thecovariant indices of atensor ; by contrast, the transformation of thecontravariant indices is given by a pushforward.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 d"φ" 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 thegeneral linear group GL("m") ("m" = dim "M").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.Pullback of connections (covariant derivatives)
If abla 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 varphi^* abla on "φ"*E over "M", determined uniquely by the condition that:varphi^* abla)_X(varphi^*s) = varphi^*( abla_{mathrm dvarphi(X)} s).ee also
*
Pushforward (differential)
*Pullback bundle
*Pullback (category theory) 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".
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