View source for Natural operator in differential geometry

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:

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.

<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/n/n120/n120040/n12004013.png" /></td> </tr></table>

for every [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120040/n12004014.png" />;

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" />;

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.

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.

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.

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.

====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>