On the Kolář connection. (English) Zbl 1299.53071

Mikulski, Włodzimierz M.

Arch. Math., Brno 49, No. 4, 223-240 (2013).

Let \(Y\to M\) be a fibred manifold and \(E\to M\) be a vector bundle, \(\operatorname {dim} Y= \operatorname {dim} E\). In [I. Kolář, Int. J. Geom. Methods Mod. Phys. 7, No. 4, 705–711 (2010; Zbl 1197.53029)] a classical connection \((\Gamma, \Lambda, \Phi, \Delta)\) is constructed on \(Y\), determined by a general connection \(\Gamma \) on \(Y\to M\), a torsion free classical linear connection \(\Lambda \) on \(M\), a vertical parallelism \(\Phi \colon Y\times_ME\to VY\) on \(Y\) and a linear connection \(\Delta \) on \(E\to M\). The author deduces that all natural operators of this type form a 12-parameter family, and he characterizes this family geometrically.

Reviewer: Ivan Kolář (Brno)

MSC:

53C05	Connections (general theory)
58A32	Natural bundles

Keywords:

general connection; linear connection; classical linear connection; vertical parallelism; natural operators

Citations:

Zbl 1197.53029