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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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