## Lifts of foliated linear connections to the second order transverse bundles.(English)Zbl 1365.58003

Summary: The second order transverse bundle $$T^2_{\mathrm{tr}}M$$ of a foliated manifold $$M$$ carries a natural structure of a smooth manifold over the algebra $$\mathbb{D}^2$$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $$\mathbb{D}^2$$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $$\mathbb{D}^2$$-smooth foliated diffeomorphism between two second order transverse bundles maps the lift of a foliated linear connection into the lift of a foliated linear connection.

### MSC:

 58A20 Jets in global analysis 53C12 Foliations (differential geometric aspects) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58A32 Natural bundles
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### References:

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