### On the local representation of synectic connections on Weil bundles

- DOI
- 10.5922/0321-4796-2022-53-11
- Pages
- 118-126

#### 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).