Homogeneous para-Kähler Einstein manifolds
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Abstract
A para-Kähler manifold can be defined as a pseudo-Riemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. 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 ± , 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.