About differential equations of the curvature tensors of a fundamental group and affine connections
The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvatu...
Published in: | Differential Geometry of Manifolds of Figures |
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Immanuel Kant Baltic Federal University
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ftdoajarticles:oai:doaj.org/article:1f31faf8ca8a4b1f84e8179cd12516ec 2023-05-15T17:07:17+02:00 About differential equations of the curvature tensors of a fundamental group and affine connections N. Ryazanov 2019-08-01T00:00:00Z https://doi.org/10.5922/0321-4796-2019-50-15 https://doaj.org/article/1f31faf8ca8a4b1f84e8179cd12516ec EN RU eng rus Immanuel Kant Baltic Federal University https://journals.kantiana.ru/geometry/4279/12676/ https://doaj.org/toc/0321-4796 https://doaj.org/toc/2782-3229 doi:10.5922/0321-4796-2019-50-15 0321-4796 2782-3229 https://doaj.org/article/1f31faf8ca8a4b1f84e8179cd12516ec Дифференциальная геометрия многообразий фигур, Iss 50, Pp 133-140 (2019) structure equations of laptev fundamental-group connection affine connection connection object curvature tensor Mathematics QA1-939 article 2019 ftdoajarticles https://doi.org/10.5922/0321-4796-2019-50-15 2022-12-31T15:05:11Z The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamental-group connection, as well as for the components of the curvature tensor of the affine connection. Article in Journal/Newspaper laptev Directory of Open Access Journals: DOAJ Articles Lemma ENVELOPE(19.530,19.530,69.873,69.873) Differential Geometry of Manifolds of Figures 50 133 140 |
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Directory of Open Access Journals: DOAJ Articles |
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language |
English Russian |
topic |
structure equations of laptev fundamental-group connection affine connection connection object curvature tensor Mathematics QA1-939 |
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structure equations of laptev fundamental-group connection affine connection connection object curvature tensor Mathematics QA1-939 N. Ryazanov About differential equations of the curvature tensors of a fundamental group and affine connections |
topic_facet |
structure equations of laptev fundamental-group connection affine connection connection object curvature tensor Mathematics QA1-939 |
description |
The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamental-group connection, as well as for the components of the curvature tensor of the affine connection. |
format |
Article in Journal/Newspaper |
author |
N. Ryazanov |
author_facet |
N. Ryazanov |
author_sort |
N. Ryazanov |
title |
About differential equations of the curvature tensors of a fundamental group and affine connections |
title_short |
About differential equations of the curvature tensors of a fundamental group and affine connections |
title_full |
About differential equations of the curvature tensors of a fundamental group and affine connections |
title_fullStr |
About differential equations of the curvature tensors of a fundamental group and affine connections |
title_full_unstemmed |
About differential equations of the curvature tensors of a fundamental group and affine connections |
title_sort |
about differential equations of the curvature tensors of a fundamental group and affine connections |
publisher |
Immanuel Kant Baltic Federal University |
publishDate |
2019 |
url |
https://doi.org/10.5922/0321-4796-2019-50-15 https://doaj.org/article/1f31faf8ca8a4b1f84e8179cd12516ec |
long_lat |
ENVELOPE(19.530,19.530,69.873,69.873) |
geographic |
Lemma |
geographic_facet |
Lemma |
genre |
laptev |
genre_facet |
laptev |
op_source |
Дифференциальная геометрия многообразий фигур, Iss 50, Pp 133-140 (2019) |
op_relation |
https://journals.kantiana.ru/geometry/4279/12676/ https://doaj.org/toc/0321-4796 https://doaj.org/toc/2782-3229 doi:10.5922/0321-4796-2019-50-15 0321-4796 2782-3229 https://doaj.org/article/1f31faf8ca8a4b1f84e8179cd12516ec |
op_doi |
https://doi.org/10.5922/0321-4796-2019-50-15 |
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Differential Geometry of Manifolds of Figures |
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50 |
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133 |
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140 |
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1766062657248428032 |