View source for Weil bundle

Consider a [[Weil algebra|Weil algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200601.png" /> and a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200602.png" />. A quite simple argument shows that the kernel of each such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200603.png" /> contains the ideal of all functions vanishing up to some order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200604.png" />, at a unique point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200605.png" />, cf. [[#References|[a3]]], 35.8. Then evaluation at the Taylor series of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200606.png" /> reveals that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200607.png" /> is completely determined by its values on the coordinate functions centred at this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w1200608.png" />. Thus,

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In this picture, the functorial action of the Weil functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006010.png" /> on mappings between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006011.png" /> (cf. [[Weil algebra|Weil algebra]]) extends to all smooth manifolds and smooth mappings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006012.png" /></td> </tr></table>

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006016.png" />. At the same time, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006017.png" /> are local charts defining the smooth structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006018.png" />, then the functorial behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006019.png" /> allows one to glue together the trivial bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006020.png" /> along the cocycle of the chart transitions, and this introduces the structure of the [[Locally trivial fibre bundle|locally trivial fibre bundle]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006021.png" />. Altogether, one obtains a [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006022.png" /> on the [[Category|category]] of smooth manifolds and mappings, the so-called Weil bundle. This construction was first presented by A. Weil, [[#References|[a11]]], under the name spaces of infinitely near points. The Weil bundles are particular examples of the bundle functors on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006023.png" /> in the sense of [[#References|[a3]]] and their restrictions to manifolds of some fixed dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006024.png" />, and locally invertible mappings provide specific examples of the natural bundles of A. Nijenhuis, [[#References|[a10]]].

Another global construction of the Weil bundles on all manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006025.png" /> is due to A. Morimoto [[#References|[a9]]], see also [[#References|[a3]]], 35.15. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006026.png" /> be a Weil algebra and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006027.png" /> be the corresponding ideal of finite codimension in the algebra of germs of functions at the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006028.png" />. Two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006030.png" /> are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006033.png" />-equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006034.png" /> for all germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006035.png" /> of functions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006036.png" />. The equivalence classes are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006038.png" />-velocities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006039.png" />. Each such class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006040.png" /> determines an algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006041.png" /> on smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006042.png" />. Thus, one obtain the Weil bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006043.png" /> again and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006044.png" /> is defined by composition in this picture. Clearly, this construction generalizes Ehresmann's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006046.png" />-velocities, i.e. the jet bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006047.png" /> (cf. [[#References|[a2]]] and [[Jet|Jet]]). See [[Weil algebra|Weil algebra]] for the corresponding Weil algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006048.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006049.png" />-velocities provide the standard tangent vectors (cf. also [[Tangent vector|Tangent vector]]) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006050.png" />-jets of smooth curves (compare with the above definition via homomorphisms, which treats them as derivations on functions).

The construction of the Weil bundles is also functorial in the Weil algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006051.png" />. Moreover, the natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006052.png" /> between two such functors are in bijective correspondence with the algebra homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006053.png" />. The smooth mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006054.png" /> are simply given by composition of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006055.png" /> with the algebra homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006056.png" />. In particular, the bundle projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006057.png" /> correspond to the unique algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006058.png" />.

The Weil bundles have many striking categorical properties. First of all, they preserve products (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006059.png" /> is canonically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006060.png" />). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006061.png" /> is completely determined by its value on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006062.png" />, which is the Weil algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006063.png" /> itself. Even its algebra structure is reconstructed by the evaluation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006064.png" /> on the addition and multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006065.png" />. As a consequence, one can also see that the composite of two Weil bundles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006066.png" />, is naturally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006067.png" />. Since there is a canonical flip isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006068.png" />, there is a corresponding natural equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006069.png" /> of the Weil bundles. The simplest example is the well-known flip in the second iterated tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006070.png" />.

The Weil bundles map many classes of mappings into itself, e.g. immersions, embeddings, submersions, surjective submersions, etc. As a consequence, many structures which are defined by diagrams involving products are transfered to the values canonically. In particular, the values of Weil bundles on Lie groups are again Lie groups (cf. also [[Lie group|Lie group]]) and all structural mappings, like the exponential or adjoint ones, are defined simply by the functorial action. Similarly, they behave nicely on vector bundles, principal bundles, etc.

A nice geometrical impact of these properties is the possibility of a unified approach to geometrical constructions and classification results for geometrical objects on the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006071.png" />. Particular attention has been paid to liftings of geometrical objects, like vector fields and connections, from the underlying manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006072.png" /> to the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006073.png" />, see e.g. [[#References|[a9]]], [[#References|[a6]]], [[#References|[a3]]]. An archetypical example is the natural lift of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006074.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006075.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006076.png" />: For all natural bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006077.png" /> there is a so-called flow prolongation of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006079.png" /> to vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006080.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006081.png" />, whose flows are given by evaluation of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006082.png" /> on the flows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006083.png" />. For each Weil bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006084.png" />, these flow prolongations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006085.png" /> are given by the formula

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All natural operators of this type are then given by composition of this operator with the natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006088.png" />. The latter transformations are parametrized by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006090.png" />, see [[#References|[a6]]] or [[#References|[a3]]], Section 42, for details. Many concepts and ideas of [[Synthetic differential geometry|synthetic differential geometry]], cf. [[#References|[a5]]], can be recovered within such a framework.

Surprisingly, much of the structure of tangent bundles generalizes to all Weil bundles. In particular, the sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006092.png" />, are exactly the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006093.png" />-linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006094.png" /> satisfying the expansion property

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which generalizes the representation of vector fields as derivations on smooth functions. There is a canonical group structure on the space of all sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006096.png" />, with associated Lie algebra and exponential mapping, see [[#References|[a4]]] or [[#References|[a3]]], Sect. 37. Of course, the latter structures are trivial in the case of tangent bundles.

For each product-preserving functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006098.png" /> is a formally real finite-dimensional commutative [[unital algebra]]; thus, a finite sum of Weil algebras is not trivial. Surprisingly enough, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w12006099.png" /> itself is, up to some subtle topological and combinatorial phenomena, the product of the corresponding Weil bundles. Moreover, if one additionally assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060100.png" /> restricts to a natural bundle in each dimension, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060101.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060103.png" />. These results were proved in [[#References|[a1]]], [[#References|[a7]]], and most completely in [[#References|[a4]]], see also [[#References|[a3]]], Sect. 36. Note also that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060104.png" /> is a bundle functor on manifolds satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060106.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060107.png" /> is a Weil algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060108.png" /> is naturally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060109.png" />, see [[#References|[a3]]], Sect. 38.

The theory of product-preserving functors and Weil bundles has been extended to the infinite-dimensional manifolds modelled over convenient vector spaces, see [[#References|[a8]]], Sect. 31.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Eck,   "Product-preserving functors on smooth manifolds"  ''J. Pure Appl. Algebra'' , '''42'''  (1986)  pp. 133–140</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ch. Ehresmann,   "Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060110.png" /> de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060111.png" /> dans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120060/w120060112.png" />. III. Transitivité des prolongements"  ''C.R. Acad. Sci. Paris'' , '''233'''  (1951)  pp. 598–600; 777–779; 1081–1083</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Kolář,   P.W. Michor,   J. Slovák,   "Natural operations in differential geometry" , Springer  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Kainz,   P.W. Michor,   "Natural transformations in differential geometry"  ''Czech. Math. J.'' , '''37''' :  112  (1987)  pp. 584–607</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Kock,   "Synthetic differential geometry" , ''Lecture Notes'' , '''51''' , London Math. Soc.  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Kolář,   "On the natural operators on vector fields"  ''Ann. Global Anal. Geom.'' , '''6'''  (1988)  pp. 109–117</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  O.O. Luciano,   "Categories of multiplicative functors and Weil's infinitely near points"  ''Nagoya Math. J.'' , '''109'''  (1988)  pp. 69–89</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P.W. Michor,   A. Kriegl,   "The convenient setting of global analysis" , ''Math. Surveys Monogr.'' , '''53''' , Amer. Math. Soc.  (1997)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Morimoto,   "Prolongation of connections to bundles of infinitely near points"  ''J. Diff. Geom.'' , '''11'''  (1976)  pp. 479–498</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Nijenhuis,   "Natural bundles and their general properties. Geometric objects revisited" , ''Diff. Geometry, in Honor of Kentaro Yano''  (1972)  pp. 317–334</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Weil,   "Théorie des points proches sur les variétés différentiables"  ''Colloq. Internat. Centre Nat. Rech. Sci.'' , '''52'''  (1953)  pp. 111–117</TD></TR></table>