Singer-Thorpe bases for special Einstein curvature tensors in dimension 4. (English) Zbl 1374.53073

A Riemannian manifold \((M,g)\) is Einstein if \(\operatorname{Ric}_{ab} = R_a{}^r{}_{br}\) is a constant multiple of \(g_{ab}\). Here \(R_{abcd}\) is the curvature tensor of the Levi-Civita connection \(\nabla\) of \(g\), \(\operatorname{Ric}_{ab}\) is the Ricci tensor and we raise and lower abstract indices using the metric and its inverse. There are many generalizations of the Einstein property. One possibility is to consider so called 2-stein manifolds, i.e., to require \[ R_{r(ab}{}^s R^r{}_{cd)s} = \Lambda g_{(ab} g_{cd)} \quad \text{for some } \Lambda \in \mathbb{R}. \] In the Einstein case in dimension \(n=4\), there exist orthonormal bases of tangent spaces of \(M\) which “simplify” the curvature tensor in the sense that \(R_{ijkl}=0\) (these are not abstract indices!) for \(|\{i,j,k,l,\}|=3\) and remaining components of \(R\) are given by only five functions. These are known as Singer-Thorpe (ST) bases. If \((M,g)\) is in addition 2-stein, all components of the curvature tensor are determined by just two functions.
ST bases are not unique. Considering \(x\in M\) (or working rather with algebraic curvature tensors), the author together with O. Kowalski determined in [Hokkaido Math. J. 44, No. 3, 441–458 (2015; Zbl 1339.53048)] the subgroup in \(\mathrm{SO}(4)\) of all transformations of \(T_xM\) which map ST bases to ST bases for all 4-dimensional Einstein structures. This subgroup is discrete. On the other hand, it is known that the case of 2-stein manifolds yields the Lie group \(\mathrm{Sp}(1) \subseteq \mathrm{SO}(4)\) of transformations of ST bases. Thus one can ask: for which subclasses of Einstein 4-dimensional structures is there a Lie group of transformations of ST-bases? The author first classifies 1-parameter subgroups of \(\mathrm{SO}(4)\) which preserve ST bases of certain subclasses of Einstein 4-dimensional metrics. These are characterized by explicit linear relations between five components (functions) of the curvature tensor. Further discussion reveals that beside the corresponding subgroups \(\mathrm{SO}(2) \subseteq \mathrm{SO}(4)\), there are also subgroups \(\mathrm{Sp}(1), \mathrm{SO}(2) \times \mathrm{SO}(2), \mathrm{SO}(2) \times \mathrm{Sp}(1) \subseteq \mathrm{SO}(4)\) of transformations of ST bases. Finally, the author notes that the full classification of all possible groups of transformations of ST bases requires to study the discrete subgroups of \(\mathrm{SO}(4)\) in detail.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)


Zbl 1339.53048
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