Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1101-1115
  • ISSN: 0011-4642

Abstract

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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 O ( 4 ) acts as a transformation group between ST bases at T p M and for the so-called 2-stein curvature tensors, the group Sp ( 1 ) ⊂ SO ( 4 ) acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of SO ( 4 ) which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups SO ( 2 ) , T 2 , Sp ( 1 ) or 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.

How to cite

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Dušek, Zdeněk. "Singer-Thorpe bases for special Einstein curvature tensors in dimension 4." Czechoslovak Mathematical Journal 65.4 (2015): 1101-1115. <http://eudml.org/doc/276262>.

@article{Dušek2015,
abstract = {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.},
author = {Dušek, Zdeněk},
journal = {Czechoslovak Mathematical Journal},
keywords = {Einstein manifold; $2$-stein manifold; Singer-Thorpe basis},
language = {eng},
number = {4},
pages = {1101-1115},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Singer-Thorpe bases for special Einstein curvature tensors in dimension 4},
url = {http://eudml.org/doc/276262},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Dušek, Zdeněk
TI - Singer-Thorpe bases for special Einstein curvature tensors in dimension 4
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1101
EP - 1115
AB - 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.
LA - eng
KW - Einstein manifold; $2$-stein manifold; Singer-Thorpe basis
UR - http://eudml.org/doc/276262
ER -

References

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  8. Sekigawa, K., Vanhecke, L., 10.1007/BFb0076638, Differential Geometry, Proc. Second Int. Symp., Peñí scola, Spain, 1985 Lecture Notes in Math. 1209 Springer, Berlin (1986), 275-291 A. M. Naveira et al. (1986) Zbl0605.53031MR0863763DOI10.1007/BFb0076638
  9. Singer, I. M., Thorpe, J. A., The curvature of 4-dimensional Einstein spaces, Global Analysis, Papers in Honor of K. Kodaira Univ. Tokyo Press, Tokyo (1969), 355-365. (1969) Zbl0199.25401MR0256303

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