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The classical theory of differential-geometric objects was revisited from the functorial point of view by A. Nijenhuis, [[#References|[a3]]]. He defined a natural bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200601.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200602.png" />-dimensional manifolds as a [[Functor|functor]] transforming every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200603.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200604.png" /> into a fibred manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200605.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200606.png" /> (cf. [[Fibred space|Fibred space]]) and every local [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200607.png" /> into a fibred manifold morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200608.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200609.png" />. Later it was taken into consideration that certain geometric objects can be constructed on certain special types of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [[#References|[a1]]].
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From this point of view, a geometric construction on the elements of one bundle of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006010.png" /> with values in the bundle of another functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006011.png" /> over the same base has the form of a natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006012.png" />. Moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006014.png" />th order natural operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006016.png" /> (cf. [[Natural operator in differential geometry|Natural operator in differential geometry]]) are in bijection with the natural transformations of the bundle functor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006017.png" />th [[Jet|jet]] prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006019.png" />.
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In the simplest case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006021.png" /> are two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006022.png" />th order natural bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006023.png" />-dimensional manifolds, the natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006024.png" /> are in bijection with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006025.png" />-equivariant mappings between their standard fibres, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006026.png" /> is the jet group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006027.png" /> in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006028.png" />. Several methods for finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006029.png" />-equivariant mappings in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006030.png" />-case are collected in [[#References|[a1]]]. If manifolds with an additional structure are studied, one has to consider the corresponding subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006031.png" />.
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The classical theory of differential-geometric objects was revisited from the functorial point of view by A. Nijenhuis, [[#References|[a3]]]. He defined a natural bundle $F$ over $m$-dimensional manifolds as a [[Functor|functor]] transforming every $m$-dimensional [[Manifold|manifold]] $M$ into a fibred manifold $F M \rightarrow M$ over $M$ (cf. [[Fibred space|Fibred space]]) and every local [[Diffeomorphism|diffeomorphism]] $f : M \rightarrow N$ into a fibred manifold morphism $F f : F M \rightarrow F N$ over $f$. Later it was taken into consideration that certain geometric objects can be constructed on certain special types of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [[#References|[a1]]].
  
Many problems on finding natural transformations between geometrically interesting pairs of bundle functors are solved in [[#References|[a1]]]. Even a negative answer can be of geometric interest. For example, in [[#References|[a1]]] it is deduced that there is no natural equivalence between the iterated tangent functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006032.png" /> and the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006033.png" /> of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006034.png" />, there is no natural [[Symplectic structure|symplectic structure]] on the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006035.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006036.png" />.
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From this point of view, a geometric construction on the elements of one bundle of a functor $F$ with values in the bundle of another functor $G$ over the same base has the form of a natural transformation $F \rightarrow G$. Moreover, the $k$th order natural operators of $F$ into $G$ (cf. [[Natural operator in differential geometry|Natural operator in differential geometry]]) are in bijection with the natural transformations of the bundle functor of the $k$th [[Jet|jet]] prolongation $J ^ { k } F$ into $G$.
  
The complete description of all natural transformations between two product-preserving bundle functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006038.png" /> on the category of all manifolds and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006039.png" />-mappings was deduced in the framework of the theory of bundle functors determined by local algebras, which was established by A. Weil, [[#References|[a4]]] (cf. also [[Weil algebra|Weil algebra]]). Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006041.png" /> corresponds to a local algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006042.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006043.png" />, respectively, and all natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006044.png" /> are in bijection with the algebra homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006045.png" />, see [[#References|[a1]]] for a survey. An analogous characterization of all natural transformations between two product preserving bundle functors on the category of fibred manifolds was deduced by W. Mikulski, [[#References|[a2]]].
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In the simplest case, if $F$ and $G$ are two $r$th order natural bundles over $m$-dimensional manifolds, the natural transformations $F \rightarrow G$ are in bijection with the $G_m ^ { r }$-equivariant mappings between their standard fibres, where $G_m ^ { r }$ is the jet group of order $r$ in dimension $m$. Several methods for finding $G_m ^ { r }$-equivariant mappings in the $C ^ { \infty }$-case are collected in [[#References|[a1]]]. If manifolds with an additional structure are studied, one has to consider the corresponding subgroup of $G_m ^ { r }$.
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Many problems on finding natural transformations between geometrically interesting pairs of bundle functors are solved in [[#References|[a1]]]. Even a negative answer can be of geometric interest. For example, in [[#References|[a1]]] it is deduced that there is no natural equivalence between the iterated tangent functor $T T$ and the composition $T ^ { * } T$ of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle $T ^ { * } M$, there is no natural [[Symplectic structure|symplectic structure]] on the [[Tangent bundle|tangent bundle]] $T M$ of a manifold $M$.
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The complete description of all natural transformations between two product-preserving bundle functors $F$ and $G$ on the category of all manifolds and all $C ^ { \infty }$-mappings was deduced in the framework of the theory of bundle functors determined by local algebras, which was established by A. Weil, [[#References|[a4]]] (cf. also [[Weil algebra|Weil algebra]]). Each $F$ or $G$ corresponds to a local algebra $A$ or $B$, respectively, and all natural transformations $F \rightarrow G$ are in bijection with the algebra homomorphisms $A \rightarrow B$, see [[#References|[a1]]] for a survey. An analogous characterization of all natural transformations between two product preserving bundle functors on the category of fibred manifolds was deduced by W. Mikulski, [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kolář,  P.W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Mikulski,  "Product preserving bundle functors on fibered manifolds"  ''Archivum Math. (Brno)'' , '''32'''  (1996)  pp. 307–316</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Nijenhuis,  "Natural bundles and their general properties" , ''Diff. Geom. in Honor of K. Yano'' , Kinokuniya  (1972)  pp. 317–334</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Weil,  "Théorie des points proches sur les variétés différentiables"  ''Colloq. C.N.R.S., Strasbourg''  (1953)  pp. 111–117</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Kolář,  P.W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  W. Mikulski,  "Product preserving bundle functors on fibered manifolds"  ''Archivum Math. (Brno)'' , '''32'''  (1996)  pp. 307–316</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Nijenhuis,  "Natural bundles and their general properties" , ''Diff. Geom. in Honor of K. Yano'' , Kinokuniya  (1972)  pp. 317–334</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A. Weil,  "Théorie des points proches sur les variétés différentiables"  ''Colloq. C.N.R.S., Strasbourg''  (1953)  pp. 111–117</td></tr></table>

Latest revision as of 17:02, 1 July 2020

The classical theory of differential-geometric objects was revisited from the functorial point of view by A. Nijenhuis, [a3]. He defined a natural bundle $F$ over $m$-dimensional manifolds as a functor transforming every $m$-dimensional manifold $M$ into a fibred manifold $F M \rightarrow M$ over $M$ (cf. Fibred space) and every local diffeomorphism $f : M \rightarrow N$ into a fibred manifold morphism $F f : F M \rightarrow F N$ over $f$. Later it was taken into consideration that certain geometric objects can be constructed on certain special types of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [a1].

From this point of view, a geometric construction on the elements of one bundle of a functor $F$ with values in the bundle of another functor $G$ over the same base has the form of a natural transformation $F \rightarrow G$. Moreover, the $k$th order natural operators of $F$ into $G$ (cf. Natural operator in differential geometry) are in bijection with the natural transformations of the bundle functor of the $k$th jet prolongation $J ^ { k } F$ into $G$.

In the simplest case, if $F$ and $G$ are two $r$th order natural bundles over $m$-dimensional manifolds, the natural transformations $F \rightarrow G$ are in bijection with the $G_m ^ { r }$-equivariant mappings between their standard fibres, where $G_m ^ { r }$ is the jet group of order $r$ in dimension $m$. Several methods for finding $G_m ^ { r }$-equivariant mappings in the $C ^ { \infty }$-case are collected in [a1]. If manifolds with an additional structure are studied, one has to consider the corresponding subgroup of $G_m ^ { r }$.

Many problems on finding natural transformations between geometrically interesting pairs of bundle functors are solved in [a1]. Even a negative answer can be of geometric interest. For example, in [a1] it is deduced that there is no natural equivalence between the iterated tangent functor $T T$ and the composition $T ^ { * } T$ of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle $T ^ { * } M$, there is no natural symplectic structure on the tangent bundle $T M$ of a manifold $M$.

The complete description of all natural transformations between two product-preserving bundle functors $F$ and $G$ on the category of all manifolds and all $C ^ { \infty }$-mappings was deduced in the framework of the theory of bundle functors determined by local algebras, which was established by A. Weil, [a4] (cf. also Weil algebra). Each $F$ or $G$ corresponds to a local algebra $A$ or $B$, respectively, and all natural transformations $F \rightarrow G$ are in bijection with the algebra homomorphisms $A \rightarrow B$, see [a1] for a survey. An analogous characterization of all natural transformations between two product preserving bundle functors on the category of fibred manifolds was deduced by W. Mikulski, [a2].

References

[a1] I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a2] W. Mikulski, "Product preserving bundle functors on fibered manifolds" Archivum Math. (Brno) , 32 (1996) pp. 307–316
[a3] A. Nijenhuis, "Natural bundles and their general properties" , Diff. Geom. in Honor of K. Yano , Kinokuniya (1972) pp. 317–334
[a4] A. Weil, "Théorie des points proches sur les variétés différentiables" Colloq. C.N.R.S., Strasbourg (1953) pp. 111–117
How to Cite This Entry:
Natural transformation in differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_transformation_in_differential_geometry&oldid=50471
This article was adapted from an original article by Ivan Kolář (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article