On the local representation of synectic connections on Weil bundles
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.
Reference
1. Shirokov, A. P.: A note on structures in tangent bundles. Tr. Geom. Sem., 5, 311—318 (1974).
2. Vishnevskiy, V. V., Shirokov, А. P., Shurygin, V. V.: Spaces over algebras. Kazan (1984).
3. Sultanov, A. Ya.: Extensions of tensor fields and connections to Weil bundles. Izvestia Vuzov. Math., 9, 64—72 (1999).
4. Sultanov, А. Ya.: On the real realization of a holomorphic path connection over an algebra. DGMF. Kaliningrad. 38, 136—139 (2007).
5. Shurygin, V. V.: Smooth varieties over local algebras and Weil bundles. Itogi nauki i tekhn. Sovrem. math. and its app. Theme reviews, 73, 162—236 (2002).