Natural operator in differential geometry

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