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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
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<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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Consider a [[Weil algebra|Weil algebra]] $A$ and a homomorphism $\varphi \in \operatorname{Hom}( C ^ { \infty } ( \mathbf{R} ^ { m } , \mathbf{R} ) , A )$. A quite simple argument shows that the kernel of each such $\varphi$ contains the ideal of all functions vanishing up to some order $k\ge 0$, at a unique point $x _ { 0 } \in \mathbf{R} ^ { m }$, cf. [[#References|[a3]]], 35.8. Then evaluation at the Taylor series of order $k$ reveals that $\varphi$ is completely determined by its values on the coordinate functions centred at this point $x _ { 0 }$. Thus,
  
<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/w12006013.png" /></td> </tr></table>
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\begin{equation*} \operatorname{Hom}( C ^ { \infty } ( \mathbf{R} ^ { m } , \mathbf{R} ) , A ) \simeq A ^ { m } = T _ { A } \mathbf{R} ^ { m }. \end{equation*}
  
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]]].
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In this picture, the functorial action of the Weil functor $T _ { A }$ on mappings between the $\mathbf{R} ^ { m }$ (cf. [[Weil algebra|Weil algebra]]) extends to all smooth manifolds and smooth mappings
  
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).
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\begin{equation*} T _ { A } M = \operatorname { Hom } ( C ^ { \infty } ( M , \mathbf{R} ) , A ), \end{equation*}
  
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" />.
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\begin{equation*} T _ { A } f ( \varphi ) ( g ) = \varphi ( g \circ f ), \end{equation*}
  
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" />.
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where $\varphi \in \operatorname{Hom}( C ^ { \infty } ( M , \mathbf{R} ) , A )$, $f : M \rightarrow M ^ { \prime }$, $g : M ^ { \prime } \rightarrow \mathbf{R}$. At the same time, if $\varphi _ { i } : U _ { i } \subset \mathbf{R} ^ { m } \rightarrow M$ are local charts defining the smooth structure on $M$, then the functorial behaviour of $T _ { A }$ allows one to glue together the trivial bundles $T _ { A } U _ { i } = U _ { i } \times N ^ { m } \subset T _ { A } \mathbf{R} ^ { m }$ along the cocycle of the chart transitions, and this introduces the structure of the [[Locally trivial fibre bundle|locally trivial fibre bundle]] on $T _ { A } M$. Altogether, one obtains a [[Functor|functor]] $T _ { A } : \mathcal{M} f \rightarrow \mathcal{M} f$ 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 $\mathcal{M} f$ in the sense of [[#References|[a3]]] and their restrictions to manifolds of some fixed dimension $m$, and locally invertible mappings provide specific examples of the natural bundles of A. Nijenhuis, [[#References|[a10]]].
  
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.
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Another global construction of the Weil bundles on all manifolds $M$ is due to A. Morimoto [[#References|[a9]]], see also [[#References|[a3]]], 35.15. Let $A$ be a Weil algebra and let $\mathcal{A}$ be the corresponding ideal of finite codimension in the algebra of germs of functions at the origin in ${\bf R} ^ { n }$. Two mappings $f , g : \mathbf{R} ^ { n } \rightarrow M$ with $f ( 0 ) = g ( 0 ) = x \in M$ are said to be $\mathcal{A}$-equivalent if $h \circ f - h \circ g \in \mathcal{A}$ for all germs $h$ of functions at $x \in M$. The equivalence classes are called $A$-velocities on $M$. Each such class $[ f ]$ determines an algebra homomorphism $h \mapsto [ h \circ f ] \in C ^ { \infty } ( {\bf R }^ { n } , {\bf R} ) /{\cal A}$ on smooth functions on $M$. Thus, one obtain the Weil bundles $T _ { A }$ again and the action of $T _ { A }$ is defined by composition in this picture. Clearly, this construction generalizes Ehresmann's $( n , r )$-velocities, i.e. the jet bundles $J ^ { r_0 }  ( \mathbf{R} ^ { n } , M )$ (cf. [[#References|[a2]]] and [[Jet|Jet]]). See [[Weil algebra|Weil algebra]] for the corresponding Weil algebras $\mathcal{D} _ { n } ^ { r }$. The $( 1,1 )$-velocities provide the standard tangent vectors (cf. also [[Tangent vector|Tangent vector]]) as $1$-jets of smooth curves (compare with the above definition via homomorphisms, which treats them as derivations on functions).
  
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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The construction of the Weil bundles is also functorial in the Weil algebra $A$. Moreover, the natural transformations $\eta : T _ { A } \rightarrow T _ { B }$ between two such functors are in bijective correspondence with the algebra homomorphisms $A \rightarrow B$. The smooth mappings $\eta_M : T_A M \to T_B M$ are simply given by composition of the homomorphism $A \rightarrow B$ with the algebra homomorphisms $\varphi \in T _ { A } M$. In particular, the bundle projections $T _ { A } M \rightarrow M$ correspond to the unique algebra homomorphism $A \rightarrow \mathbf{R}$.
  
<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/w12006086.png" /></td> </tr></table>
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The Weil bundles have many striking categorical properties. First of all, they preserve products (i.e. $T _ { A } ( M \times M ^ { \prime } )$ is canonically isomorphic to $T _ { A } M \times T _ { A } M ^ { \prime }$). In particular, $T _ { A }$ is completely determined by its value on $\mathbf{R}$, which is the Weil algebra $A$ itself. Even its algebra structure is reconstructed by the evaluation of $T _ { A }$ on the addition and multiplication on $\mathbf{R}$. As a consequence, one can also see that the composite of two Weil bundles, $T _ { B } \circ T _ { A }$, is naturally equivalent to $T _ { B  \otimes A}$. Since there is a canonical flip isomorphism $B \otimes A \rightarrow A \otimes B$, there is a corresponding natural equivalence $T _ { B } \circ T _ { A } = T _ { A } \circ T _ { B }$ of the Weil bundles. The simplest example is the well-known flip in the second iterated tangent bundle $\kappa _ { M } : T T M \rightarrow T T M$.
  
<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/w12006087.png" /></td> </tr></table>
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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 transferred 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.
  
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.
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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 $T _ { A } M$. Particular attention has been paid to liftings of geometrical objects, like vector fields and connections, from the underlying manifolds $M$ to the bundles $T _ { A } M$, see e.g. [[#References|[a9]]], [[#References|[a6]]], [[#References|[a3]]]. An archetypical example is the natural lift of vector fields $\xi $ on $M$ to $T _ { A } M$: For all natural bundles $F$ there is a so-called flow prolongation of vector fields $\xi $ on $M$ to vector fields $F \xi$ on $F M$, whose flows are given by evaluation of the functor $F$ on the flows of $\xi $. For each Weil bundle $T _ { A }$, these flow prolongations ${\cal T} _ { A } \xi$ are given by the formula
  
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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\begin{equation*} {\cal T} _ { A } \xi = \kappa _ { M } \circ T _ { A } \xi, \end{equation*}
  
<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/w12006095.png" /></td> </tr></table>
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\begin{equation*} T _ { A } M \rightarrow T _ { A } T M \rightarrow T T _ { A } M. \end{equation*}
  
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.
+
All natural operators of this type are then given by composition of this operator with the natural transformations $T T _ { A } \rightarrow T T _ { A }$. The latter transformations are parametrized by elements $a \in A$ and $D \in \operatorname { Der } A$, 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.
  
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.
+
Surprisingly, much of the structure of tangent bundles generalizes to all Weil bundles. In particular, the sections of $T _ { A } M$, $A = \mathbf{R} \cdot1 \oplus N$, are exactly the $\mathbf{R}$-linear mappings $\xi : C ^ { \infty } ( M , \mathbf{R} ) \rightarrow C ^ { \infty } ( M , N )$ satisfying the expansion property
 +
 
 +
\begin{equation*} \xi ( f . g ) = \xi ( f ) g + f . \xi ( g ) + \xi ( f ) . \xi ( g ), \end{equation*}
 +
 
 +
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 $T _ { A } M$, 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 $F : \mathcal{M} f \rightarrow \mathcal{M} f$, $F \mathbf{R}$ is a formally real finite-dimensional commutative [[unital algebra]]; thus, a finite sum of Weil algebras is not trivial. Surprisingly enough, $F$ itself is, up to some subtle topological and combinatorial phenomena, the product of the corresponding Weil bundles. Moreover, if one additionally assumes that $F$ restricts to a natural bundle in each dimension, then $F$ is of the form $T _ { A }$ with $A = F \mathbf{R}$. 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 $F : \mathcal{M} f \rightarrow \mathcal{M} f$ is a bundle functor on manifolds satisfying $\operatorname { dim }( F \mathbf{R} ^ { m } ) = m \operatorname { dim } ( F \mathbf{R} )$ for all $m = 0,1 , \ldots$, then $F \mathbf{R}$ is a Weil algebra and $F$ is naturally equivalent to $T _ { F \mathbf{R} }$, 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.
 
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====
 
====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>
+
<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 $r$ de $V _ { n }$ dans $V _ { m }$. 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>

Latest revision as of 06:30, 15 February 2024

Consider a Weil algebra $A$ and a homomorphism $\varphi \in \operatorname{Hom}( C ^ { \infty } ( \mathbf{R} ^ { m } , \mathbf{R} ) , A )$. A quite simple argument shows that the kernel of each such $\varphi$ contains the ideal of all functions vanishing up to some order $k\ge 0$, at a unique point $x _ { 0 } \in \mathbf{R} ^ { m }$, cf. [a3], 35.8. Then evaluation at the Taylor series of order $k$ reveals that $\varphi$ is completely determined by its values on the coordinate functions centred at this point $x _ { 0 }$. Thus,

\begin{equation*} \operatorname{Hom}( C ^ { \infty } ( \mathbf{R} ^ { m } , \mathbf{R} ) , A ) \simeq A ^ { m } = T _ { A } \mathbf{R} ^ { m }. \end{equation*}

In this picture, the functorial action of the Weil functor $T _ { A }$ on mappings between the $\mathbf{R} ^ { m }$ (cf. Weil algebra) extends to all smooth manifolds and smooth mappings

\begin{equation*} T _ { A } M = \operatorname { Hom } ( C ^ { \infty } ( M , \mathbf{R} ) , A ), \end{equation*}

\begin{equation*} T _ { A } f ( \varphi ) ( g ) = \varphi ( g \circ f ), \end{equation*}

where $\varphi \in \operatorname{Hom}( C ^ { \infty } ( M , \mathbf{R} ) , A )$, $f : M \rightarrow M ^ { \prime }$, $g : M ^ { \prime } \rightarrow \mathbf{R}$. At the same time, if $\varphi _ { i } : U _ { i } \subset \mathbf{R} ^ { m } \rightarrow M$ are local charts defining the smooth structure on $M$, then the functorial behaviour of $T _ { A }$ allows one to glue together the trivial bundles $T _ { A } U _ { i } = U _ { i } \times N ^ { m } \subset T _ { A } \mathbf{R} ^ { m }$ along the cocycle of the chart transitions, and this introduces the structure of the locally trivial fibre bundle on $T _ { A } M$. Altogether, one obtains a functor $T _ { A } : \mathcal{M} f \rightarrow \mathcal{M} f$ on the category of smooth manifolds and mappings, the so-called Weil bundle. This construction was first presented by A. Weil, [a11], under the name spaces of infinitely near points. The Weil bundles are particular examples of the bundle functors on the category $\mathcal{M} f$ in the sense of [a3] and their restrictions to manifolds of some fixed dimension $m$, and locally invertible mappings provide specific examples of the natural bundles of A. Nijenhuis, [a10].

Another global construction of the Weil bundles on all manifolds $M$ is due to A. Morimoto [a9], see also [a3], 35.15. Let $A$ be a Weil algebra and let $\mathcal{A}$ be the corresponding ideal of finite codimension in the algebra of germs of functions at the origin in ${\bf R} ^ { n }$. Two mappings $f , g : \mathbf{R} ^ { n } \rightarrow M$ with $f ( 0 ) = g ( 0 ) = x \in M$ are said to be $\mathcal{A}$-equivalent if $h \circ f - h \circ g \in \mathcal{A}$ for all germs $h$ of functions at $x \in M$. The equivalence classes are called $A$-velocities on $M$. Each such class $[ f ]$ determines an algebra homomorphism $h \mapsto [ h \circ f ] \in C ^ { \infty } ( {\bf R }^ { n } , {\bf R} ) /{\cal A}$ on smooth functions on $M$. Thus, one obtain the Weil bundles $T _ { A }$ again and the action of $T _ { A }$ is defined by composition in this picture. Clearly, this construction generalizes Ehresmann's $( n , r )$-velocities, i.e. the jet bundles $J ^ { r_0 } ( \mathbf{R} ^ { n } , M )$ (cf. [a2] and Jet). See Weil algebra for the corresponding Weil algebras $\mathcal{D} _ { n } ^ { r }$. The $( 1,1 )$-velocities provide the standard tangent vectors (cf. also Tangent vector) as $1$-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 $A$. Moreover, the natural transformations $\eta : T _ { A } \rightarrow T _ { B }$ between two such functors are in bijective correspondence with the algebra homomorphisms $A \rightarrow B$. The smooth mappings $\eta_M : T_A M \to T_B M$ are simply given by composition of the homomorphism $A \rightarrow B$ with the algebra homomorphisms $\varphi \in T _ { A } M$. In particular, the bundle projections $T _ { A } M \rightarrow M$ correspond to the unique algebra homomorphism $A \rightarrow \mathbf{R}$.

The Weil bundles have many striking categorical properties. First of all, they preserve products (i.e. $T _ { A } ( M \times M ^ { \prime } )$ is canonically isomorphic to $T _ { A } M \times T _ { A } M ^ { \prime }$). In particular, $T _ { A }$ is completely determined by its value on $\mathbf{R}$, which is the Weil algebra $A$ itself. Even its algebra structure is reconstructed by the evaluation of $T _ { A }$ on the addition and multiplication on $\mathbf{R}$. As a consequence, one can also see that the composite of two Weil bundles, $T _ { B } \circ T _ { A }$, is naturally equivalent to $T _ { B \otimes A}$. Since there is a canonical flip isomorphism $B \otimes A \rightarrow A \otimes B$, there is a corresponding natural equivalence $T _ { B } \circ T _ { A } = T _ { A } \circ T _ { B }$ of the Weil bundles. The simplest example is the well-known flip in the second iterated tangent bundle $\kappa _ { M } : T T M \rightarrow T T M$.

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 transferred to the values canonically. In particular, the values of Weil bundles on Lie groups are again Lie groups (cf. also 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 $T _ { A } M$. Particular attention has been paid to liftings of geometrical objects, like vector fields and connections, from the underlying manifolds $M$ to the bundles $T _ { A } M$, see e.g. [a9], [a6], [a3]. An archetypical example is the natural lift of vector fields $\xi $ on $M$ to $T _ { A } M$: For all natural bundles $F$ there is a so-called flow prolongation of vector fields $\xi $ on $M$ to vector fields $F \xi$ on $F M$, whose flows are given by evaluation of the functor $F$ on the flows of $\xi $. For each Weil bundle $T _ { A }$, these flow prolongations ${\cal T} _ { A } \xi$ are given by the formula

\begin{equation*} {\cal T} _ { A } \xi = \kappa _ { M } \circ T _ { A } \xi, \end{equation*}

\begin{equation*} T _ { A } M \rightarrow T _ { A } T M \rightarrow T T _ { A } M. \end{equation*}

All natural operators of this type are then given by composition of this operator with the natural transformations $T T _ { A } \rightarrow T T _ { A }$. The latter transformations are parametrized by elements $a \in A$ and $D \in \operatorname { Der } A$, see [a6] or [a3], Section 42, for details. Many concepts and ideas of synthetic differential geometry, cf. [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 $T _ { A } M$, $A = \mathbf{R} \cdot1 \oplus N$, are exactly the $\mathbf{R}$-linear mappings $\xi : C ^ { \infty } ( M , \mathbf{R} ) \rightarrow C ^ { \infty } ( M , N )$ satisfying the expansion property

\begin{equation*} \xi ( f . g ) = \xi ( f ) g + f . \xi ( g ) + \xi ( f ) . \xi ( g ), \end{equation*}

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 $T _ { A } M$, with associated Lie algebra and exponential mapping, see [a4] or [a3], Sect. 37. Of course, the latter structures are trivial in the case of tangent bundles.

For each product-preserving functor $F : \mathcal{M} f \rightarrow \mathcal{M} f$, $F \mathbf{R}$ is a formally real finite-dimensional commutative unital algebra; thus, a finite sum of Weil algebras is not trivial. Surprisingly enough, $F$ itself is, up to some subtle topological and combinatorial phenomena, the product of the corresponding Weil bundles. Moreover, if one additionally assumes that $F$ restricts to a natural bundle in each dimension, then $F$ is of the form $T _ { A }$ with $A = F \mathbf{R}$. These results were proved in [a1], [a7], and most completely in [a4], see also [a3], Sect. 36. Note also that if $F : \mathcal{M} f \rightarrow \mathcal{M} f$ is a bundle functor on manifolds satisfying $\operatorname { dim }( F \mathbf{R} ^ { m } ) = m \operatorname { dim } ( F \mathbf{R} )$ for all $m = 0,1 , \ldots$, then $F \mathbf{R}$ is a Weil algebra and $F$ is naturally equivalent to $T _ { F \mathbf{R} }$, see [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 [a8], Sect. 31.

References

[a1] D.J. Eck, "Product-preserving functors on smooth manifolds" J. Pure Appl. Algebra , 42 (1986) pp. 133–140
[a2] Ch. Ehresmann, "Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre $r$ de $V _ { n }$ dans $V _ { m }$. III. Transitivité des prolongements" C.R. Acad. Sci. Paris , 233 (1951) pp. 598–600; 777–779; 1081–1083
[a3] I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a4] G. Kainz, P.W. Michor, "Natural transformations in differential geometry" Czech. Math. J. , 37 : 112 (1987) pp. 584–607
[a5] A. Kock, "Synthetic differential geometry" , Lecture Notes , 51 , London Math. Soc. (1981)
[a6] I. Kolář, "On the natural operators on vector fields" Ann. Global Anal. Geom. , 6 (1988) pp. 109–117
[a7] O.O. Luciano, "Categories of multiplicative functors and Weil's infinitely near points" Nagoya Math. J. , 109 (1988) pp. 69–89
[a8] P.W. Michor, A. Kriegl, "The convenient setting of global analysis" , Math. Surveys Monogr. , 53 , Amer. Math. Soc. (1997)
[a9] A. Morimoto, "Prolongation of connections to bundles of infinitely near points" J. Diff. Geom. , 11 (1976) pp. 479–498
[a10] A. Nijenhuis, "Natural bundles and their general properties. Geometric objects revisited" , Diff. Geometry, in Honor of Kentaro Yano (1972) pp. 317–334
[a11] A. Weil, "Théorie des points proches sur les variétés différentiables" Colloq. Internat. Centre Nat. Rech. Sci. , 52 (1953) pp. 111–117
How to Cite This Entry:
Weil bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil_bundle&oldid=40948
This article was adapted from an original article by Jan Slovak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article