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Summary:
Given a fibered manifold $Y \to X$, a 2-connection on $Y$ means a section $J^1 Y \to J^2 Y$. The authors determine all first order natural operators transforming a 2-connection on $Y$ and a classical linear connection on $X$ into a connection on $J^1 Y \to Y$. (The proof implies that there is no first order natural operator transforming 2-connections on $Y$ into connections on $J^1Y \to Y$.) Using this result, the authors deduce several properties of characterizable connections on $J^1 Y \to X$.
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