Abstract
Synectic extensions of complete lifts of linear connections in tangent bundles were introduced by A. P. Shirokov in the seventies of the last century [1; 2]. He established that these connections are linear and are real realizations of linear connections on first-order tangent bundles endowed with a smooth structure over the algebra of dual numbers. He also proved the existence of a smooth structure on tangent bundles of arbitrary order on a smooth manifold M over the algebra of plural numbers. Studying holomorphic linear connections on over an algebra , A. P. Shirokov obtained real realizations of these connections, which he called Synectic extensions of a linear connection defined on M. A natural generalization of the algebra of plural numbers is the A. Weyl algebra, and a generalization of the tangent bundle is the A. Weyl bundle. It was shown in [3] that a synectic extension of linear connections defined on M a smooth manifold can also be constructed on A. Weyl bundles , where is the A. Weyl algebra. The geometry of these bundles has been studied by many authors — A. Morimoto, V. V. Shurygin and others. A detailed analysis of these works can be found in [3]. In this paper, we study synectic lifts of linear connections defined on A. Weyl bundles.
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