Homogeneous para-Kähler Einstein manifolds

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dc.creatorAlekseevsky, Dmitri V.
dc.creatorMedori, Costantino
dc.creatorTomassini, Adriano
dc.date2008-06-13
dc.date2008-12-23
dc.date.accessioned2026-07-07T12:21:01Z
dc.date.available2026-07-07T12:21:01Z
dc.descriptionA 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 $(M,ω)$ 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 $\mathrm{Ad}_Gh$ 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.
dc.description44 pages
dc.identifierhttps://arxiv.org/abs/0806.2272
dc.identifierhttp://arxiv.org/abs/0806.2272
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/213238
dc.subjectDifferential Geometry
dc.subject53C15, 53C56
dc.titleHomogeneous para-Kähler Einstein manifolds
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