Homogeneous Para - Kähler Einstein

A para-Kähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. a parallel field of skew-symmetric endomorphisms with K = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie group G admits an invariant para-Kähler structure (g, K) if and only if it is a covering of the adjoint orbit AdGh of a semisimple element h. We give a description of all invariant para-Kähler structures (g,K) on a such homogeneous manifold. Using a para-complex analogue of basic formulas of Kähler geometry, we prove that any invariant para-complex structure K on M = G/H defines a unique para-Kähler Einstein structure (g,K) with given non-zero scalar curvature. An explicit formula for the Einstein metric g is given. A survey of recent results on para-complex geometry is included.