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Some gauge-natural operators on connections. (English) Zbl 0806.53025
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 435-445 (1992).
The author considers gauge-natural bundles over $$m$$-dimensional manifolds and gauge-natural operators, introduced by D. J. Eck, solving two problems. He determines all gauge-natural operators of the curvature type. More precisely, let $$P$$ be a principal $$G$$-bundle, $$QP \to BP$$ the connection bundle of $$P$$ and $$LP$$ the fibre bundle associated to $$P$$, with standard fibre the Lie algebra $${\mathfrak g}$$, with respect to the adjoint action. The author proves that all gauge-natural operators $$Q \to LP \otimes \overset {2} \otimes T^* B$$ are the modified curvature operators, i.e. they are obtained combining the curvature operator and the vector bundle morphism from LP into LP induced by the linear maps of $${\mathfrak g}$$ commuting with the adjoint action of $$G$$. Furthermore, he studies the gauge-natural operators transforming a connection $$\Gamma$$ on a principal fibre bundle $$\pi : P \to BP$$ and a linear connection $$\Lambda$$ on $$BP$$ into a linear connection on $$P$$. He constructs an operator $$N$$ which transforms $$(\Gamma,\Lambda)$$ into a linear connection $$N(\Gamma,\Lambda)$$ on $$P$$. Supposing that $$\Lambda$$ is without torsion and $$\dim G = n$$, he proves that the gauge-operators transforming $$(\Gamma,\Lambda)$$ into a linear connection on $$P$$ form a family depending on $$n^ 3 + n^ 2 + 2n$$ parameters, generated by $$N$$ and three families of gauge-natural difference tensors.
For the entire collection see [Zbl 0764.00002].
Reviewer: A.M.Pastore (Bari)

##### MSC:
 53C05 Connections (general theory)