Some properties of para-Kähler–Walker metrics
Volume 112 / 2014
Annales Polonici Mathematici 112 (2014), 115-125
MSC: Primary 53C50; Secondary 53B30.
DOI: 10.4064/ap112-2-2
Abstract
A Walker $4$-manifold is a pseudo-Riemannian manifold $(M_{4} ,g)$ of neutral signature, which admits a field of parallel null $2$-planes. We study almost paracomplex structures on $4$-dimensional para-Kähler–Walker manifolds. In particular, we obtain conditions under which these almost paracomplex structures are integrable, and the corresponding para-Kähler forms are symplectic. We also show that Petean's example of a nonflat indefinite Kähler-Einstein $4$-manifold is a special case of our constructions.
