ABSTRACT
This chapter explains the real 4-dimensional Kahler-Einstein and scalarflat Kahler metrics with a 3-dimensional symmetry group which is transitive on 3-surfaces. Such symmetries have been classified, and the resulting metrics are often referred to as Bianchi metrics. The field equations for Kahler-Einstein or scalar-flat Kahler metrics reduce to systems of first order ordinary differential equations for the metric coefficients for the Bianchi class A metrics, and to algebraic equations for the Bianchi class B metrics. One main assumption of Kahler form is that the metric is diagonal. Another assumption is that the metric is nonhyperKahler and non-reducible. This allows a result of Lichnerowicz to be used which implies under these conditions that the Kahler form must be group invariant. In order to ensure that a metric is not hyperKahler it is sufficient to assume that the Ricci curvature is not identically zero.
