Previous |  Up |  Next


Einstein manifold; $2$-stein manifold; Singer-Thorpe basis
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.
[1] Carpenter, P., Gray, A., Willmore, T. J.: The curvature of Einstein symmetric spaces. Q. J. Math., Oxf. II. Ser. 33 (1982), 45-64. DOI 10.1093/qmath/33.1.45 | MR 0689850 | Zbl 0509.53045
[2] Dušek, Z., Kowalski, O.: Transformations between Singer-Thorpe bases in 4-dimensional Einstein manifolds. Hokkaido Math. J. 44 (2015), 441-458.
[3] Euh, Y., Park, J. H., Sekigawa, K.: A generalization of a 4-dimensional Einstein manifold. Math. Slovaca 63 (2013), 595-610. MR 3071978
[4] Euh, Y., Park, J., Sekigawa, K.: Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds. Differ. Geom. Appl. 29 (2011), 642-646. DOI 10.1016/j.difgeo.2011.07.001 | MR 2831820 | Zbl 1228.58010
[5] Gilkey, P. B.: The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds. ICP Advanced Texts in Mathematics 2 Imperial College, London (2007). MR 2351705 | Zbl 1128.53041
[6] Kowalski, O., Vanhecke, L.: Ball-homogeneous and disk-homogeneous Riemannian manifolds. Math. Z. 180 (1982), 429-444. DOI 10.1007/BF01214716 | MR 0666999 | Zbl 0476.53023
[7] Sekigawa, K., Vanhecke, L.: Volume-preserving geodesic symmetries on four-dimensional 2-stein spaces. Kodai Math. J. 9 (1986), 215-224. DOI 10.2996/kmj/1138037204 | MR 0842869 | Zbl 0613.53010
[8] Sekigawa, K., Vanhecke, L.: Volume-preserving geodesic symmetries on four-dimensional Kähler manifolds. 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. DOI 10.1007/BFb0076638 | MR 0863763 | Zbl 0605.53031
[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. MR 0256303 | Zbl 0199.25401
Partner of
EuDML logo