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**Total connections in Lie groupoids.**
*(English)*
Zbl 0841.53024

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.

Reviewer: I.Kolář (Brno)