Differential Geometry of Manifolds

2022 №53

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On the local representation of synectic connections on Weil bundles



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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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).