Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization
The space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> of centered m -planes is considered in projective space <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>n</mi> <...
Published in: | Mathematics |
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Main Author: | |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
MDPI AG
2019
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Subjects: | |
Online Access: | https://doi.org/10.3390/math7100901 https://doaj.org/article/56adf3c717cb45c1ab3b80c881255b70 |
Summary: | The space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> of centered m -planes is considered in projective space <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>n</mi> </msub> </semantics> </math> . A principal bundle is associated with the space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> . Semi-normalized spaces <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>1</mn> </msup> </semantics> </math> , <math display="inline"> <semantics> <msup> <mo>Π</mo> <mn>2</mn> </msup> </semantics> </math> and normalized space <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> 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 <math display="inline"> <semantics> <mo>Π</mo> </semantics> </math> to the normalized space <math display="inline"> <semantics> <msup> <mo>Π</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </semantics> </math> . |
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