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

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.