Conformally Flat Homogeneous Lorentzian Manifolds

Abstract

Conformally flat homogeneous Riemannian manifolds were classified by Takagi. They are all symmetric spaces. We consider the problem to classify conformally flat homogeneous Lorentzian manifolds. Our classification depends on the form of the modified Ricci operators $A = \frac{1} {n-2}\left (Q - \frac{S} {2(n-1)}Id\right )$, where Q is the Ricci operator and S is the scalar curvature. We classified the case when A is diagonalizable with real eigenvalues in the previous paper [4]. If A is not diagonalizable with real eigenvalues, then three cases for its form may occur. For two cases, we show complete local classifications. They are not locally symmetric except one example. For the last case, we can show examples but cannot solve the classification problem at the present.