Einstein metrics on principal circle bundles

Let M be a compact Kahler manifold and P a principal circle bundle over M with a connection. We can define a metric on P by horizontal lifting of the metric from M. We will prove that M is a product of compact Kahler-Einstein manifolds with positive first Chern classes and the Euler class of P is determined by the first Chern classes of the Kahler-Einstein manifolds in the base M as described by M.Y. Wang and W. Ziller in 1990 if P is Einstein and the curvature form of the connection is of type (1, 1).