Four-dimensional Einstein-like manifolds and curvature homogeneity.

*(English)*Zbl 0835.53056A 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.

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.

Reviewer: L.Vanhecke (Leuven)

##### 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 |

##### Keywords:

curvature homogeneous Riemannian manifold; eigenvalues of the Ricci operator; locally symmetric
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\textit{F. Podestá} and \textit{A. Spiro}, Geom. Dedicata 54, No. 3, 225--243 (1995; Zbl 0835.53056)

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##### References:

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[2] | Gray, A.: Einstein-like manifolds which are not Einstein,Geom. Dedicata 7, (1978), 259-280. · Zbl 0378.53018 |

[3] | Jensen, G.R.: Homogeneous Einstein spaces of dimension 4,J. Differential Geom. 3 (1969), 309-349. · Zbl 0194.53203 |

[4] | Kowalski, O.: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ?1=?2 ? ?3, Preprint, 1992. |

[5] | Kowalski, O. and Vanhecke, L.: Four-dimensional naturally reductive homogeneous spaces,Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale (1983), 223-232. · Zbl 0631.53039 |

[6] | O’Neill, B.: The fundamental equations of a submersion,Michigan Math. J. 13 (1966), 459-469. · Zbl 0145.18602 |

[7] | Tricerri, F. and Vanhecke, L.: Curvature homogeneous Riemannian manifolds,Ann. Sci. École Norm. Sup. (4)22 (1989), 525-554. · Zbl 0698.53033 |

[8] | Yamato, K.: A characterization of locally homogeneous Riemann manifolds of dimension 3,Nagoya Math. J. 123 (1991), 77-90. · Zbl 0738.53032 |

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