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Three-dimensional conformally flat homogeneous Lorentzian manifolds. (English) Zbl 1116.53042

The reviewed paper is devoted to the classification problem of conformally flat homogeneous semi-Riemannian manifolds. Let \(M^n_q\) be an \(n\)-dimensional, \(n\geqslant 3\), semi-Riemannian manifold of index \(q\). Let \(A\) be a tensor field of type (1,1) defined by \(A=[Q-S/(2(n-2))Id]/(n-2)\), where \(Q\) and \(S\) denote the Ricci operator and scalar curvature of \(M^n_q\), respectively. Assuming that \(M^n_q\) is homogeneous and conformally flat, the authors show the identity of the eigenvalues of the tensor field \(A\). Basing on this identity they give a local classification of conformally flat homogeneous semi-Riemannian manifolds with real diagonalizable Ricci operators and next a complete classification of possible candidates for the linear operators \(A\) of conformally flat homogeneous Lorentzian manifolds. Finally, the authors show that a 3-dimensional simply connected conformally flat homogeneous Lorentzian manifold is isometric to one of the six kinds of manifolds described by them.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
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