Geodesic
Total connections in Lie groupoids
Archivum mathematicum, Tome 31 (1995) no. 3, pp. 183-200
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A total connection of order $r$ in a Lie groupoid $\Phi $ over $M$ is defined as a first order connections in the $(r-1)$-st jet prolongations of $\Phi $. A connection in the groupoid $\Phi $ together with a linear connection on its base, ie. in the groupoid $\Pi (M)$, give rise to a total connection of order $r$, which is called simple. It is shown that this simple connection is curvature-free iff the generating connections are. Also, an $r$-th order total connection in $\Phi $ defines a total reduction of the $r$-th prolongation of $\Phi $ to $\Phi \times \Pi (M)$. It is shown that when $r>2$ then this total reduction of a simple connection is holonomic iff the generating connections are curvature free and the one on $M$ also torsion-free.
A total connection of order $r$ in a Lie groupoid $\Phi $ over $M$ is defined as a first order connections in the $(r-1)$-st jet prolongations of $\Phi $. A connection in the groupoid $\Phi $ together with a linear connection on its base, ie. in the groupoid $\Pi (M)$, give rise to a total connection of order $r$, which is called simple. It is shown that this simple connection is curvature-free iff the generating connections are. Also, an $r$-th order total connection in $\Phi $ defines a total reduction of the $r$-th prolongation of $\Phi $ to $\Phi \times \Pi (M)$. It is shown that when $r>2$ then this total reduction of a simple connection is holonomic iff the generating connections are curvature free and the one on $M$ also torsion-free.
Classification : 53C05, 58A20
Keywords: Lie groupoids; semi-holonomic jets; higher order connections; total connections; simple connections 
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     title = {Total connections in {Lie} groupoids},
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     language = {en},
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Virsik, George. Total connections in Lie groupoids. Archivum mathematicum, Tome 31 (1995) no. 3, pp. 183-200. http://geodesic.mathdoc.fr/item/ARM_1995_31_3_a2/

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