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Higher order Cartan connections

Juraj Virsik — 1996

Archivum Mathematicum

A Cartan connection associated with a pair P ( M , G ' ) P ( M , G ) is defined in the usual manner except that only the injectivity of ω : T ( P ' ) T ( G ) e is required. For an r -th order connection associated with a bundle morphism Φ : P ' P the concept of Cartan order q r is defined, which for q = r = 1 , Φ : P ' P , and dim M = dim G / G ' coincides with the classical definition. Results are obtained concerning the Cartan order of r -th order connections that are the product of r first order (Cartan) connections.

Total connections in Lie groupoids

Juraj Virsik — 1995

Archivum Mathematicum

A total connection of order r in a Lie groupoid Φ over M is defined as a first order connections in the ( r - 1 ) -st jet prolongations of Φ . A connection in the groupoid Φ together with a linear connection on its base, ie. in the groupoid Π ( 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 Φ defines a total reduction of the r -th prolongation of Φ to Φ × Π ( M ) ....

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