## 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
Full Text: