Non-existence of some canonical constructions on connections. (English) Zbl 1099.58004

Mikulski, W. M.

Commentat. Math. Univ. Carol. 44, No. 4, 691-695 (2003).

Let \(H\) be a vector bundle valued bundle functor on manifolds with the point property.
The main result of the paper reads that \(H\) preserves products if and only if for every \(m\) and \(n\) there exists a natural operator transforming connections on every \((m,n)\)-dimensional fibered manifold \(p\: Y \to M\) into connections on \(Hp \: HY \to HM\). In a similar way the author deduces that for every natural bundle over fibered \((m,n)\)-manifolds with the property that the induced natural bundle \(\widetilde E\) over \(m\)-manifolds \(\widetilde {E}M = E (M \times \mathbb{R}^n)\), \(\widetilde {E} \phi = E (\phi \times \text{id} \mathbb{R}_n)\) is not of order zero there is no natural operator transforming connections on \(Y \to M\) into connections on \(EY \to M\).

Reviewer: Ivan Kolář (Brno)

MSC:

58A20	Jets in global analysis
53C05	Connections (general theory)

Keywords:

natural operator