Homogeneous para-K\"ahler Einstein manifolds

TL;DR

This paper classifies invariant para-K"ahler Einstein structures on homogeneous manifolds related to semisimple Lie groups, providing explicit formulas and a comprehensive survey of recent developments in para-complex geometry.

Contribution

It characterizes all invariant para-K"ahler structures on certain homogeneous spaces and proves the uniqueness of Einstein metrics with non-zero scalar curvature for each para-complex structure.

Findings

Complete classification of invariant para-K"ahler structures on homogeneous manifolds.

Proof of uniqueness of para-K"ahler Einstein structures with given scalar curvature.

Explicit formula for the Einstein metric g.

Abstract

A para-K\"ahler 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-K\"ahler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\mathrm{Ad}_Gh$ of a semisimple element $h.$ We give a description of all invariant para-K\"ahler structures $(g,K)$ on a such homogeneous manifold. Using a para-complex analogue of basic formulas of K\"ahler geometry, we prove 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.