A Walker 4-manifold is a pseudo-Riemannian manifold (M₄,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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap112-2-2, author = {Mustafa \"Ozkan and Murat \.I\c scan}, title = {Some properties of para-K\"ahler-Walker metrics}, journal = {Annales Polonici Mathematici}, volume = {111}, year = {2014}, pages = {115-125}, zbl = {1309.53032}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap112-2-2} }
Mustafa Özkan; Murat İşcan. Some properties of para-Kähler-Walker metrics. Annales Polonici Mathematici, Tome 111 (2014) pp. 115-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap112-2-2/