Smooth manifolds over local algebras and Weil bundles
type: Item
instance of: scholarly article
title: Smooth manifolds over local algebras and Weil bundles
zbMATH Open document ID: 1007.58001
DOI: 10.1023/A:1012848404391
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-07-18T00:00:00Z
review text: The theory of smooth manifolds over local algebras belongs to the fields of geometry and topology of manifolds that have multiaffine structures defined by associative commutative algebras. This field of study is closely related to the geometry of jet bundles, the theory of differential-geometric objects, and the topology of foliations. The main idea in constructing the theory of smooth manifolds over algebras is the replacement of the field of real numbers \(\mathbb R\) and linear spaces over \(\mathbb R\) by a certain associative commutative finite-dimensional algebra \(\mathbb A\) over \(\mathbb R\) and \(\mathbb A\)-modules of finite dimension over \(\mathbb R\), respectively. The general theory of manifolds over algebras was studied by A. P. Shirokov, J. Vanzura, V. V. Vishnevskij, G. I. Kruchkovich, and others. \textit{A. P. Shirokov} [Tr. Geom. Semin. 1, 425-456 (1966; Zbl 0198.54601)] proved that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of \(\mathbb A\)-closed points in the sense of A.~Weil. In this survey, the author presents the review of new trends in studies of the differential geometry and topology of smooth manifolds over local algebras. The paper consists of seven sections. After presenting the introductory information in the first two sections, in section 3 the author considers the structure of an \(n\)-dimensional \(\mathbb A\)-smooth manifold on the Weil \(\mathbb A\)-jet bundle \(J^{\mathbb A}M_n\) over a real smooth manifold \(M_n\). The existence of the structure of an \(\mathbb A\)-smooth manifold on \(J^{\mathbb A}M_n\) leads to the appearance of the structural \(\mathbb A\)-affine differential group \(D_n(\mathbb A)\) of the bundle \(J^{\mathbb A}M_n\), and the principal bundle \(B({\mathbb A})M_n\), called the \(\mathbb A\)-affine frame bundle on \(M_n\) associated with \(J^{\mathbb A}M_n\), with the structural group \(D_n(\mathbb A)\). The author studies connections in bundles with t…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 58B05
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58-02
zbMATH DE Number: 1768745
zbMATH Keywords: local Weil algebra
zbMATH Keywords: mapping over algebra
zbMATH Keywords: A-smooth map
zbMATH Keywords: manifold over algebra
zbMATH Keywords: A-manifold
zbMATH Keywords: A-connection
author: Vadim V. jun. Shurygin
MaRDI profile type: Publication
full work available at URL: a:1012848404391
OpenAlex ID: W191163166
description: scientific article; zbMATH DE number 1768745