Homogeneous para-Kähler Einstein manifolds

A para-K¨ahler manifold can be defined as a pseudo-Riemannian manifold (M, g) with a parallel skew-symmetric para-complex structure K, that is, a parallel field of skew-symmetric endomorphisms with K^2 = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L^±, that is, two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie group G admits an invariant para-K¨ahler structure (g,K) if and only if it is a covering of the adjoint orbit AdG h of a semisimple element h. A description is given of all invariant para-K¨ahler structures (g,K) on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of K¨ahler geometry it is proved that any invariant para-complex structure K on M = G/H defines a unique para-K¨ahler 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.