Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization
The space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> of centered <i>m</i>-planes is considered in projective space <inline-formula> <math display="...
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doaj-56adf3c717cb45c1ab3b80c881255b702020-11-25T00:39:07ZengMDPI AGMathematics2227-73902019-09-0171090110.3390/math7100901math7100901Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at NormalizationOlga Belova0Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016 Kaliningrad, RussiaThe space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> of centered <i>m</i>-planes is considered in projective space <inline-formula> <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula>. A principal bundle is associated with the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula>. Semi-normalized spaces <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>1</mn> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>2</mn> </msup> </semantics> </math> </inline-formula> and normalized space <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> </inline-formula> are investigated. By virtue of the Cartan−Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> to the normalized space <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/7/10/901differentiable manifoldcartan–laptev methodspace of centered planesnormalizationreductionconnection |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Olga Belova |
| spellingShingle |
Olga Belova Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization Mathematics differentiable manifold cartan–laptev method space of centered planes normalization reduction connection |
| author_facet |
Olga Belova |
| author_sort |
Olga Belova |
| title |
Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization |
| title_short |
Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization |
| title_full |
Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization |
| title_fullStr |
Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization |
| title_full_unstemmed |
Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization |
| title_sort |
reduction of bundles, connection, curvature, and torsion of the centered planes space at normalization |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2019-09-01 |
| description |
The space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> of centered <i>m</i>-planes is considered in projective space <inline-formula> <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula>. A principal bundle is associated with the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula>. Semi-normalized spaces <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>1</mn> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>2</mn> </msup> </semantics> </math> </inline-formula> and normalized space <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> </inline-formula> are investigated. By virtue of the Cartan−Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space <inline-formula> <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> </inline-formula> to the normalized space <inline-formula> <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> </inline-formula>. |
| topic |
differentiable manifold cartan–laptev method space of centered planes normalization reduction connection |
| url |
https://www.mdpi.com/2227-7390/7/10/901 |
| work_keys_str_mv |
AT olgabelova reductionofbundlesconnectioncurvatureandtorsionofthecenteredplanesspaceatnormalization |
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1725295005947396096 |