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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.
The main purpose of this article is to investigate the paraholomorphy property of the Sasaki and Cheeger-Gromoll metrics by using compatible paracomplex stuctures on the tangent bundle.
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