| Title:
|
Singer-Thorpe bases for special Einstein curvature tensors in dimension 4 (English) |
| Author:
|
Dušek, Zdeněk |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
1101-1115 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(M,g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$. In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group ${\rm O}(4)$ acts as a transformation group between ST bases at $T_pM$ and for the so-called 2-stein curvature tensors, the group ${\rm Sp}(1)\subset {\rm SO}(4)$ acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of ${\rm SO}(4)$ which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups ${\rm SO}(2)$, $T^2$, ${\rm Sp}(1)$ or ${\rm U}(2)$ are obtained and the classes of curvature tensors whose transformation group into new ST bases is one of the mentioned groups are determined. (English) |
| Keyword:
|
Einstein manifold |
| Keyword:
|
$2$-stein manifold |
| Keyword:
|
Singer-Thorpe basis |
| MSC:
|
53C25 |
| idZBL:
|
Zbl 06537713 |
| idMR:
|
MR3441338 |
| DOI:
|
10.1007/s10587-015-0230-1 |
| . |
| Date available:
|
2016-01-13T09:27:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144795 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
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