## 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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### References:

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