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In the simplest case, one considers two natural bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200401.png" />-dimensional manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200403.png" />, cf. [[Natural transformation in differential geometry|Natural transformation in differential geometry]]. A natural operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200404.png" /> is a system of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200405.png" /> transforming every section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200406.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200407.png" /> into a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200408.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n1200409.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004010.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004011.png" /> with the following properties:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004012.png" /> commutes with the action of diffeomorphisms, i.e.
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In the simplest case, one considers two natural bundles over $m$-dimensional manifolds $F$ and $G$, cf. [[Natural transformation in differential geometry|Natural transformation in differential geometry]]. A natural operator $A : F \rightarrow G$ is a system of operators $A _ {M}$ transforming every section $s$ of $F M$ into a section $A _ { M } ( s )$ of $G M$ for every $m$-dimensional [[Manifold|manifold]] $M$ with the following properties:
  
for every [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004014.png" />;
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1) $A$ commutes with the action of diffeomorphisms, i.e.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004015.png" /> has the localization property, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004016.png" /> for every open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004017.png" />;
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\begin{equation*} A _ { N } ( F f \circ s \circ f ^ { - 1 } ) = ( G f ) \circ A _ { M } ( s ) \circ f ^ { - 1 } \end{equation*}
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004018.png" /> is regular, i.e. every smoothly parametrized family of sections is transformed into a smoothly parametrized family.
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for every [[Diffeomorphism|diffeomorphism]] $f : M \rightarrow N$;
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2) $A$ has the localization property, i.e. $A _ { U } ( s | _ { U } ) = A _ { M } ( s ) | _ { U }$ for every open subset $U \subset M$;
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3) $A$ is regular, i.e. every smoothly parametrized family of sections is transformed into a smoothly parametrized family.
  
 
This idea has been generalized to other categories over manifolds and to operators defined on certain distinguished classes of sections in [[#References|[a2]]].
 
This idea has been generalized to other categories over manifolds and to operators defined on certain distinguished classes of sections in [[#References|[a2]]].
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004020.png" />th order natural operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004021.png" /> are in bijection with the natural transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004022.png" />th [[Jet|jet]] prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004023.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004024.png" />. In this case the methods from [[#References|[a2]]] can be applied for finding natural operators. So it is important to have some criteria guaranteeing that all natural operators of a prescribed type have finite order. Fundamental results in this direction were deduced by J. Slovák, who developed a far-reaching generalization of the Peetre theorem to non-linear problems, [[#References|[a2]]]. However, in certain situations there exist natural operators of infinite order.
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The $k$th order natural operators $F \rightarrow G$ are in bijection with the natural transformations of the $k$th [[Jet|jet]] prolongation $J ^ { k } F$ into $G$. In this case the methods from [[#References|[a2]]] can be applied for finding natural operators. So it is important to have some criteria guaranteeing that all natural operators of a prescribed type have finite order. Fundamental results in this direction were deduced by J. Slovák, who developed a far-reaching generalization of the Peetre theorem to non-linear problems, [[#References|[a2]]]. However, in certain situations there exist natural operators of infinite order.
  
The first result about natural operators was deduced by R. Palais, [[#References|[a3]]], who proved that all linear natural operators transforming exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004025.png" />-forms into exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004026.png" />-forms are constant multiples of the exterior differential (cf. also [[Exterior form|Exterior form]]). In [[#References|[a2]]] new methods are used to prove that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004027.png" /> linearity even follows from naturality.
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The first result about natural operators was deduced by R. Palais, [[#References|[a3]]], who proved that all linear natural operators transforming exterior $p$-forms into exterior $( p + 1 )$-forms are constant multiples of the exterior differential (cf. also [[Exterior form|Exterior form]]). In [[#References|[a2]]] new methods are used to prove that for $p \geq 1$ linearity even follows from naturality.
  
 
Many concrete problems on finding all natural operators are solved in [[#References|[a2]]].
 
Many concrete problems on finding all natural operators are solved in [[#References|[a2]]].
  
The following result on the natural operators on morphisms of fibred manifolds is closely related to the geometry of the calculus of variations. On a fibred manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004028.png" />-dimensional base, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004029.png" />, there is no natural operator transforming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004030.png" />th order Lagrangeans into Poincaré–Cartan morphisms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004031.png" />, see [[#References|[a1]]]. In this case, one has to use an additional structure to distinguish a single Poincaré-Cartan form determined by a Lagrangean.
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The following result on the natural operators on morphisms of fibred manifolds is closely related to the geometry of the calculus of variations. On a fibred manifold with $m$-dimensional base, $m \geq 2$, there is no natural operator transforming $r$th order Lagrangeans into Poincaré–Cartan morphisms for $r \geq 3$, see [[#References|[a1]]]. In this case, one has to use an additional structure to distinguish a single Poincaré-Cartan form determined by a Lagrangean.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kolář,  "Natural operators related with the variational calculus" , ''Proc. Conf. Diff. Geom. Appl., Silesian Univ. Opava''  (1993)  pp. 461–472</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Kolář,  P.W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Palais,  "Natural operations on differential forms"  ''Trans. Amer. Math. Soc.'' , '''92'''  (1959)  pp. 125–141</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Kolář,  "Natural operators related with the variational calculus" , ''Proc. Conf. Diff. Geom. Appl., Silesian Univ. Opava''  (1993)  pp. 461–472</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Kolář,  P.W. Michor,  J. Slovák,  "Natural operations in differential geometry" , Springer  (1993)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R. Palais,  "Natural operations on differential forms"  ''Trans. Amer. Math. Soc.'' , '''92'''  (1959)  pp. 125–141</td></tr></table>

Latest revision as of 15:30, 1 July 2020

In the simplest case, one considers two natural bundles over $m$-dimensional manifolds $F$ and $G$, cf. Natural transformation in differential geometry. A natural operator $A : F \rightarrow G$ is a system of operators $A _ {M}$ transforming every section $s$ of $F M$ into a section $A _ { M } ( s )$ of $G M$ for every $m$-dimensional manifold $M$ with the following properties:

1) $A$ commutes with the action of diffeomorphisms, i.e.

\begin{equation*} A _ { N } ( F f \circ s \circ f ^ { - 1 } ) = ( G f ) \circ A _ { M } ( s ) \circ f ^ { - 1 } \end{equation*}

for every diffeomorphism $f : M \rightarrow N$;

2) $A$ has the localization property, i.e. $A _ { U } ( s | _ { U } ) = A _ { M } ( s ) | _ { U }$ for every open subset $U \subset M$;

3) $A$ is regular, i.e. every smoothly parametrized family of sections is transformed into a smoothly parametrized family.

This idea has been generalized to other categories over manifolds and to operators defined on certain distinguished classes of sections in [a2].

The $k$th order natural operators $F \rightarrow G$ are in bijection with the natural transformations of the $k$th jet prolongation $J ^ { k } F$ into $G$. In this case the methods from [a2] can be applied for finding natural operators. So it is important to have some criteria guaranteeing that all natural operators of a prescribed type have finite order. Fundamental results in this direction were deduced by J. Slovák, who developed a far-reaching generalization of the Peetre theorem to non-linear problems, [a2]. However, in certain situations there exist natural operators of infinite order.

The first result about natural operators was deduced by R. Palais, [a3], who proved that all linear natural operators transforming exterior $p$-forms into exterior $( p + 1 )$-forms are constant multiples of the exterior differential (cf. also Exterior form). In [a2] new methods are used to prove that for $p \geq 1$ linearity even follows from naturality.

Many concrete problems on finding all natural operators are solved in [a2].

The following result on the natural operators on morphisms of fibred manifolds is closely related to the geometry of the calculus of variations. On a fibred manifold with $m$-dimensional base, $m \geq 2$, there is no natural operator transforming $r$th order Lagrangeans into Poincaré–Cartan morphisms for $r \geq 3$, see [a1]. In this case, one has to use an additional structure to distinguish a single Poincaré-Cartan form determined by a Lagrangean.

References

[a1] I. Kolář, "Natural operators related with the variational calculus" , Proc. Conf. Diff. Geom. Appl., Silesian Univ. Opava (1993) pp. 461–472
[a2] I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a3] R. Palais, "Natural operations on differential forms" Trans. Amer. Math. Soc. , 92 (1959) pp. 125–141
How to Cite This Entry:
Natural operator in differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_operator_in_differential_geometry&oldid=49928
This article was adapted from an original article by Ivan Kolář (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article