##
**Some natural operators in differential geometry.**
*(English)*
Zbl 0653.58003

Differential geometry and its applications, Proc. Conf., Brno/Czech. 1986, Math. Appl., East Eur. Ser. 27, 91-110 (1987).

[For the entire collection see Zbl 0624.00014.]

Denote by \b{M}\({}_ n\) the category of n-dimensional smooth manifolds in which the morphisms are local diffeomorphisms and by the category of smooth fibre bundles. A natural bundle is defined as a functor F of \b{M}\({}_ n\) into such that for any \(M\in \underline M_ n\), FM is a fibre bundle over M and for any morphism f of \b{M}\({}_ n\), Ff is a fibre bundle morphism over f. Natural operators A: \(F\to G\) between two natural bundles are defined as families \(\{A_ M: C^{\infty}FM\to C^{\infty}GM| \quad M\in \underline M_ n\}\) of mappings \(A_ M\) between the corresponding spaces of smooth sections such that \(A_ M\) maps any family of smooth sections of FM, which is smooth in a certain sense, into a smooth family of smooth sections of GM. Moreover, some compatibility conditions with respect to the action of diffeomorphisms on the space of sections and to restrictions of bundles and sections to open submanifolds are claimed.

The author generalizes a result of R. Palais to the following: For \(p>0\) all natural operators between the p-th and \((p+1)\)-st exterior power of the cotangent bundle are the constant multiples of the exterior differential. In a similar way all natural operators of the connection bundle \(QH^ 1\) of the natural bundle \(H^ 1\) of 1-st order frames (i.e., the sections of \(QH^ 1\) are simply the principal connections of \(H^ 1)\) into itself are classified. Replacing in these definitions the category \b{M}\({}_ n\) by the category \(P_ n(G)\) of all principal G- bundles (where G is some fixed Lie group) over n-dimensional manifolds, one gets in a similar way the definitions of gauge-natural bundles and gauge-natural operators introduced by D. Eck. The remaining part of the paper is devoted to some properties of gauge-natural operators related to the connection bundle.

Denote by \b{M}\({}_ n\) the category of n-dimensional smooth manifolds in which the morphisms are local diffeomorphisms and by the category of smooth fibre bundles. A natural bundle is defined as a functor F of \b{M}\({}_ n\) into such that for any \(M\in \underline M_ n\), FM is a fibre bundle over M and for any morphism f of \b{M}\({}_ n\), Ff is a fibre bundle morphism over f. Natural operators A: \(F\to G\) between two natural bundles are defined as families \(\{A_ M: C^{\infty}FM\to C^{\infty}GM| \quad M\in \underline M_ n\}\) of mappings \(A_ M\) between the corresponding spaces of smooth sections such that \(A_ M\) maps any family of smooth sections of FM, which is smooth in a certain sense, into a smooth family of smooth sections of GM. Moreover, some compatibility conditions with respect to the action of diffeomorphisms on the space of sections and to restrictions of bundles and sections to open submanifolds are claimed.

The author generalizes a result of R. Palais to the following: For \(p>0\) all natural operators between the p-th and \((p+1)\)-st exterior power of the cotangent bundle are the constant multiples of the exterior differential. In a similar way all natural operators of the connection bundle \(QH^ 1\) of the natural bundle \(H^ 1\) of 1-st order frames (i.e., the sections of \(QH^ 1\) are simply the principal connections of \(H^ 1)\) into itself are classified. Replacing in these definitions the category \b{M}\({}_ n\) by the category \(P_ n(G)\) of all principal G- bundles (where G is some fixed Lie group) over n-dimensional manifolds, one gets in a similar way the definitions of gauge-natural bundles and gauge-natural operators introduced by D. Eck. The remaining part of the paper is devoted to some properties of gauge-natural operators related to the connection bundle.

Reviewer: H.Gollek

### MSC:

58A20 | Jets in global analysis |

53C05 | Connections (general theory) |

58A10 | Differential forms in global analysis |