connected Einstein manifold must be constant. For this reason, the Einstein condition is often given as for a real number Kähler manifolds When the Riemannian manifold is also a complex manifold, that is it comes with an integrable almost-complex structure , it is possible to ask for a compatibility between the metric structure and the complex structure . There are many equivalent ways of formulating this compatibility condition, and one succinct interpretation is to ask that is orthogonal with respect to , so that for all vector fields , and that is preserved by the parallel transport of the Levi-Civita connection , captured by the condition . Such a triple is called a Kähler manifold. Kähler–Einstein metrics A Kähler–Einstein manifold is one which combines the above properties of being Kähler and admitting an Einstein metric. The combination of these properties implies a simplification of the Einstein equation in terms of the complex structure. Namely, on a Kähler manifold one can define the Ricci form, a real -form, by the expression where are any tangent vector fields to . The almost-complex structure forces to be antisymmetric, and the compatibility condition combined with the Bianchi identity implies that is a closed differential form. Associated to the Riemannian metric is the Kähler form defined by a similar expression . Therefore the Einstein equations for can be rewritten as the Kähler–Einstein equation. Since this is an equality of closed differential forms, it implies an equality of the associated de Rham cohomology classes and . The former class is the first Chern class of , . Therefore a necessary condition for the existence of a solution to the Kähler–Einstein equation is that , for some . This is a topological necessary condition on the Kähler manifold . Note that since the Ricci curvature is invariant under scaling , if there is a metric such that , one can always normalise to a new metric with , that is . Thus the Kähler–Einstein equation is often written depending on the sign of the topological constant . Transformation to a complex Monge–Ampere equation The situation of compact Kähler manifolds is special, because the Kähler–Einstein equation can be reformulated as a complex Monge–Ampere equation for a smooth Kähler potential on . By the topological assumption on the Kähler manifold, we may always assume that there exists some Kähler metric . The Ricci form of is given in local