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Four-dimensional Einstein-like manifolds and curvature homogeneity. (English) Zbl 0835.53056
A Riemannian manifold $$(M,g)$$ is said to be curvature homogeneous if for any two points $$p, q \in M$$, there exists a linear isometry $$F : T_p M \to T_q M$$ such that $$F^* R = R_p$$, where $$R$$ denotes the Riemannian curvature tensor. In dimension two and three this is equivalent to the constancy of the eigenvalues of the Ricci operator $$Q$$. Three-dimensional curvature homogeneous spaces have been studied in detail by O. Kowalski and F. Prüfer [see J. Anal. Anwend. 14, 43-58 (1995; Zbl 0821.53036) for a short survey]. In dimension four a result of Derdzinkski yields that a curvature homogeneous Einstein space is locally symmetric.
The main purpose of the authors is to consider four-dimensional spaces $$(M,g)$$ which have constant Ricci eigenvalues (a condition which is satisfied in the curvature homogeneous case) and such that the Ricci tensor $$\rho$$ of type (0,2) is a Codazzi tensor or is cyclic-parallel. For the first class they prove that when $$(M,g)$$ is not an Einstein space then it is locally symmetric. Together with Derdzinski’s result, this implies that any four-dimensional curvature homogeneous manifold with Codazzi tensor $$\rho$$ is locally symmetric. Further, they also consider the second class and provide a classification when not all the Ricci eigenvalues are distinct. This restriction is due to the complexity of the system of equations involved in the determination of these manifolds by means of their method of attack.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C30 Differential geometry of homogeneous manifolds
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##### References:
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