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Homogeneous para-Kähler Einstein manifolds

D. V. Alekseevsky^a, C. Medori^b, A. Tomassini<sup>b

a</sup> University of Edinburgh
^b Università degli Studi di Parma

Abstract: A para-Kähler 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 $K^2=\operatorname{Id}$ or, equivalently, as a symplectic manifold $(M,\omega)$ with a bi-Lagrangian structure $L^\pm$, that is, 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 $\operatorname{Ad}_Gh$ of a semisimple element $h$. A description is given of all invariant para-Kähler structures $(g,K)$ on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kähler geometry it is proved 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.

UDC: 514.747+514.76

MSC: Primary 53C25, 53C26; Secondary 53B35, 53C55, 53C15

Received: 09.06.2008

DOI: 10.4213/rm9262

 English version: Russian Mathematical Surveys, 2009, 64:1, 1–43

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