zbMATH — the first resource for mathematics

Natural operators lifting vector fields on manifolds to the bundles of covelocities. (English) Zbl 0854.58006
Bureš, J. (ed.) et al., Proceedings of the Winter School on geometry and physics, Srní, Czech Republic, January 1994. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 39, 105-116 (1996).
The author proves that for a manifold $$M$$ of dimension greater than 2 the sets of all natural operators $$TM \to (T^{r*}_k M, T^{q*}_\ell M)$$ and $$TM \to TT^{r*}_k M$$, respectively, are free finitely generated $$C^\infty ((\mathbb{R}^k)^r)$$-modules. The space $$T^{r*}_k M = J^r(M, \mathbb{R}^k)_0$$, this is, jets with target 0 of maps from $$M$$ to $$\mathbb{R}^k$$, is called the space of all $$(k,r)$$-covelocities on $$M$$. Examples of such operators are shown and the bases of the modules are explicitly constructed. The definitions and methods are those of the book of I. Kolář, P. W. Michor and J. Slovák [Natural operations in differential geometry, Springer-Verlag, Berlin (1993; Zbl 0782.53013)].
For the entire collection see [Zbl 0840.00036].
MSC:
 58A20 Jets in global analysis 53A55 Differential invariants (local theory), geometric objects
Keywords:
bundle functors; natural operators