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Zbl 0841.53024
Virsik, George
Total connections in Lie groupoids. (English)
[J] Arch. Math., Brno 31, No.3, 183-200 (1995). ISSN 0044-8753; ISSN 1212-5059/e

First of all, the author introduces the concept of iterated $r$-jet, which generalizes classical non-holonomic jets. Then he studies the higher-order connections in a Lie groupoid $\Phi$ (which is equivalent to a principal fiber bundle). A total connection of order $r$ in $\Phi$ is defined to be a first-order connection in the $(r - 1)$st non-holonomic prolongation of $\Phi$. A connection in $\Phi$ together with a linear connection on its base $M$ give rise to a total connection of order $r$, which is called simple. Moreover, an $r$-th order total connection in $\Phi$ defines a total reduction of the $r$-th prolongation of $\Phi$ to $\Phi \times \Pi(M)$, where $\Pi(M)$ denotes the Lie groupoid of all invertible one-jets on $M$. It is shown that when $r > 2$ then the total reduction of a simple connection is holonomic if and only if the generating connections are curvature-free and the one on $M$ is also torsion-free.
[I.Kolář (Brno)]

MSC 2000:: *53C05 Connections, general theory
58A20 Jets

Keywords: simple connection; higher-order connections; Lie groupoid; total connection

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