Hierarchy of spaces of projective connection
We consider the bundle of projective frames over a smooth manifold, i. e. the principal bundle whose typical fiber is the projective group. The giving fundamental-group connection in this bundle transforms it into a space of general projective connection. Differential equations of components of the...
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| Format: | Article |
| Language: | English |
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Immanuel Kant Baltic Federal University
2018-08-01
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| Series: | Дифференциальная геометрия многообразий фигур |
| Subjects: | |
| Online Access: | https://journals.kantiana.ru/geometry/4117/12355/ |
| _version_ | 1819279037184344064 |
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| author | Yu. Shevchenko |
| author_facet | Yu. Shevchenko |
| author_sort | Yu. Shevchenko |
| collection | DOAJ |
| description | We consider the bundle of projective frames over a smooth manifold, i. e. the principal bundle whose typical fiber is the projective group. The giving fundamental-group connection in this bundle transforms it into a space of general projective connection. Differential equations of components of the curvature tensor, their covariant derivatives and analogues of Bianchi identities are obtained.
When the basic and fiber indices coincide, a special case is identified, called the space of a pre-Cartan projective connection. By means of gluing the fibers to the base of the pre-Cartan connection space we obtain the space of the Cartan projective connection. In this case the curvature tensor is transformed into a curvature-torsion tensor, which contains the tensor of affine curvature-torsion with a torsion.
A hierarchy of the considered spaces is constructed, which there are a bundle of projective frames is at the beginning of the hierarchy, the space of projective connection (general, pre-Cartan and Cartan) is in the middle, a projective space is at the end of the hierarchy. |
| first_indexed | 2024-12-24T00:21:32Z |
| format | Article |
| id | doaj.art-1047fa03cd704812baa423a56fd0d085 |
| institution | Directory Open Access Journal |
| issn | 0321-4796 2782-3229 |
| language | English |
| last_indexed | 2024-12-24T00:21:32Z |
| publishDate | 2018-08-01 |
| publisher | Immanuel Kant Baltic Federal University |
| record_format | Article |
| series | Дифференциальная геометрия многообразий фигур |
| spelling | doaj.art-1047fa03cd704812baa423a56fd0d0852022-12-21T17:24:34ZengImmanuel Kant Baltic Federal UniversityДифференциальная геометрия многообразий фигур0321-47962782-32292018-08-0149178192Hierarchy of spaces of projective connectionYu. Shevchenko0Immanuel Kant Baltic Federal UniversityWe consider the bundle of projective frames over a smooth manifold, i. e. the principal bundle whose typical fiber is the projective group. The giving fundamental-group connection in this bundle transforms it into a space of general projective connection. Differential equations of components of the curvature tensor, their covariant derivatives and analogues of Bianchi identities are obtained. When the basic and fiber indices coincide, a special case is identified, called the space of a pre-Cartan projective connection. By means of gluing the fibers to the base of the pre-Cartan connection space we obtain the space of the Cartan projective connection. In this case the curvature tensor is transformed into a curvature-torsion tensor, which contains the tensor of affine curvature-torsion with a torsion. A hierarchy of the considered spaces is constructed, which there are a bundle of projective frames is at the beginning of the hierarchy, the space of projective connection (general, pre-Cartan and Cartan) is in the middle, a projective space is at the end of the hierarchy.https://journals.kantiana.ru/geometry/4117/12355/bundle of projective framesprojective connectionanalogues of bianchi identitiespre-cartan projective connectiongluingprojective cartan connectioncurvature-torsion tensortensor of affine curvature-torsion |
| spellingShingle | Yu. Shevchenko Hierarchy of spaces of projective connection Дифференциальная геометрия многообразий фигур bundle of projective frames projective connection analogues of bianchi identities pre-cartan projective connection gluing projective cartan connection curvature-torsion tensor tensor of affine curvature-torsion |
| title | Hierarchy of spaces of projective connection |
| title_full | Hierarchy of spaces of projective connection |
| title_fullStr | Hierarchy of spaces of projective connection |
| title_full_unstemmed | Hierarchy of spaces of projective connection |
| title_short | Hierarchy of spaces of projective connection |
| title_sort | hierarchy of spaces of projective connection |
| topic | bundle of projective frames projective connection analogues of bianchi identities pre-cartan projective connection gluing projective cartan connection curvature-torsion tensor tensor of affine curvature-torsion |
| url | https://journals.kantiana.ru/geometry/4117/12355/ |
| work_keys_str_mv | AT yushevchenko hierarchyofspacesofprojectiveconnection |