On the Einstein standard stationary space-times

In this paper, we study Einstein standard stationary space-times (M, g) = (R x Sigma, -beta dt(3) + dt circle times g Sigma(xi,.) + g Sigma(xi,.) circle times dt + g Sigma) of dimension n + 1 >= 3 with xi is conformal on the space-like hypersurface (Sigma, g Sigma). When Sigma is assumed to be connected and complete, we give a criterion so that (M, g) is Einstein. In particular, we show that if (M, g) is Einstein and Sigma is complete and connected, then the scalar curvature of M and that of Sigma are non-positive and xi is Killing with respect to g Sigma.