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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)
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