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Three- and four-dimensional Einstein-like manifolds and homogeneity. (English) Zbl 0944.53026
This paper is a contribution to the problem of classifying three- and four-dimensional Ricci-curvature homogeneous Einstein-like Riemannian manifolds. We recall that a Ricci-curvature homogeneous space is characterized by the constancy of the eigenvalues of the Ricci-operator and that $$(M,g)$$ is called Einstein-like if its Ricci-curvature tensor is either cyclic parallel (type $$\mathcal A$$) or a Codazzi-tensor (type $$\mathcal B$$). For the three-dimensional case, the authors gave a complete classification: those of type $$\mathcal A$$ are naturally reductive spaces, while those of type $$\mathcal B$$ are locally symmetric. The four-dimensional case is more involved. In [Geom. Dedicata 54, 225-243 (1995; Zbl 0835.53056)], F. Podestà and A. Spiro provided the full classification of (non-Einstein) four-dimensional Ricci-curvature homogeneous manifolds of type $$\mathcal B$$. Again these are all locally symmetric. For type $$\mathcal A$$, they gave a classification under the additional assumption that not all Ricci eigenvalues are distinct, obtaining specific locally homogeneous manifolds which need not be naturally reductive. They also stated that, in their opinion, no examples of locally homogeneous manifolds of type $$\mathcal A$$ exist with four distinct Ricci eigenvalues.
The present authors refute this expectation by an explicit classification of four-dimensional locally homogeneous spaces of type $$\mathcal A$$ having four distinct Ricci eigenvalues. For this, they use the computer algebra package Maple V. The original problem of classifying four-dimensional Ricci-curvature homogeneous spaces of type $$\mathcal A$$ is still open in the generic case. The classification results are then used to clarify the relation between three- and four-dimensional manifolds of type $$\mathcal A$$ and D’Atri-spaces, i.e., Riemannian manifolds with volume-preserving (up to sign) geodesic symmetries.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds
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