Homogeneous para-Kähler Einstein manifolds

Abstract

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $K^2=\mathrm{Id}$ or, equivalently, as a symplectic manifold $(M,\omega )$ with a bi-Lagrangian structure $L^{\pm }$, 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 ${\mathrm{Ad}}_{G}h$ 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.

44 pages

References in corpus (1)

• Holonomy representations which are a diagonal direct sum of two faithful representations

Cited by in corpus (1)

• Geometric structures on Lie groups with flat bi-invariant metric