Natural operator in differential geometry
In the simplest case, one considers two natural bundles over -dimensional manifolds
and
, cf. Natural transformation in differential geometry. A natural operator
is a system of operators
transforming every section
of
into a section
of
for every
-dimensional manifold
with the following properties:
1) commutes with the action of diffeomorphisms, i.e.
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for every diffeomorphism ;
2) has the localization property, i.e.
for every open subset
;
3) 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 th order natural operators
are in bijection with the natural transformations of the
th jet prolongation
into
. 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 -forms into exterior
-forms are constant multiples of the exterior differential (cf. also Exterior form). In [a2] new methods are used to prove that for
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 -dimensional base,
, there is no natural operator transforming
th order Lagrangeans into Poincaré–Cartan morphisms for
, 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 |
Natural operator in differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_operator_in_differential_geometry&oldid=13188