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Some functorial prolongations of general connections. (English) Zbl 1424.53054

The present paper is devoted to the prolongation of general connections with respect to a product preserving bundle functor. A general connection on a fibered manifold \(p:Y\to M\) can be defined as a lifting map \(\Gamma:Y\times_M TM\to TY\), which can be also equivalently interpreted as a section \(Y\to J^1Y\) of the first jet prolongation of \(Y\).
Using the flow prolongation of vector fields, the author constructed the vertical prolongation \(\mathcal V\Gamma\) of \(\Gamma\) on the vertical bundle \(VY\to M\), \(\mathcal V\Gamma:VY\times_MTM\to VY\) and studied some geometric properties of \(\mathcal V\Gamma\). Then he proved the formula for the curvature \(C_\Gamma=\frac{1}{2}[\omega_\Gamma,\omega_\Gamma]\) and \([\omega_\Gamma,[\omega_\Gamma,\omega_\Gamma]]=0\), where \(\omega_\Gamma\) is some tangent valued \(1\)-form corresponding to \(\Gamma\). In the rest of the paper the author discussed some properties of Weil functors and constructed the induced connection \(\mathcal T^A\Gamma\) on \(T^AY\to T^AM\), where \(T^A\) is the Weil functor corresponding to the Weil algebra \(A\). Finally he clarified that the curvature of \(\mathcal T^A\Gamma\) is the \(\mathcal T^A\)-prolongation of the curvature of \(\Gamma\).

MSC:

53C05 Connections (general theory)
58A20 Jets in global analysis
58A32 Natural bundles
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