# zbMATH — the first resource for mathematics

On natural operators on sectorform fields. (English) Zbl 0706.58003
If M is a smooth manifold consider the tangent bundles $p_ M: TM\to M,\quad p_{TM}: T(T(M))\quad \to \quad TM,\quad...,\quad p_{T_{k- 1}M}: T(T_{k-1}M)\quad \to \quad T_{k-1}M$ where $$T_ kM=T(T(...TM))$$, k-times. There exist k vector bundle structures on $$T_ kM$$ over $$T_{k-1}M$$, $$T_ rp_{T_{k-r-1}M}: T_ kM\to T_{k-1}M$$, $$r=0,1,...,k-1$$. A map $$\Omega_{k,x}: (T_ kM)_ x\to {\mathbb{R}}$$ linear with respect to all the above k vector bundle structures is called a k- sectorform on M at the point x.
Using a general method for finding all natural operators developed by I. Kolář [Differential geometry and its applications, Proc. Conf., Brno, Czech. 1986, Math. Appl., East Eur. Ser. 27, 91-110 (1987; Zbl 0653.58003)] in this paper the author determines all natural operators of the following types: from 1-sectorform bundle to 2- sectorform bundle, from 1-sectorform bundle to 3-sectorform bundle and from 2-sectorform bundle to 3-sectorform bundle. Taking into account these results the author shows that the fundamental operator is the differential of sectorform fields introduced by J. E. White [The methods of iterated tangents with applications to local Riemannian geometry (1982; Zbl 0478.58002)] and I. Kolář [Lie derivatives of sectorform fields, Colloq. Math. 55, No.1, 71-78 (1988; Zbl 0691.58008)].
Reviewer: D.Andrica

##### MSC:
 58A20 Jets in global analysis
##### Keywords:
sectorform fields
##### Citations:
Zbl 0653.58003; Zbl 0478.58002; Zbl 0691.58008
Full Text: