Counterexamples to Goldberg Conjecture with Reversed Orientation on Walker 8-Manifolds of Neutral Signature

Abstract

The famous Goldberg conjecture (Goldberg, Proc Am Math Soc 21, 96–100, 1969) [8] states that the almost complex structure of a compact almost-Kähler Einstein Riemannian manifold is Kähler. It is true if the scalar curvature of the manifold is nonnegative (Sekigawa, Math Ann 271, 333–337, 1985) [20], (Sekigawa, J Math Soc Jpn 36, 677–684, 1987) [21]. If we turn our attention to indefinite metric spaces, several counterexamples to the conjecture have been reported (cf. (Matsushita, J Geom Phys 55, 385–398, 2005) [17], (Matsushita et al., Monatsh Math 150, 41–48, 2007) [18], (Matsushita, et al., Proceedings of The 19th International Workshop on Hermitian-Grassmannian Submanifolds and Its Applications and the 10th RIRCM-OCAMI Joint Differential Geometry Workshop, Institute for Mathematical Sciences (NIMS), vol 19, pp 1–14. Daejeon, South Korea, 2015) [19]). It is important to recognize that all known counterexamples to date are constructed on Walker manifolds, equipped with an almost complex structure of normal orientation. In the present paper, we focus our attention on Walker manifolds with an opposite almost complex structure, and consider if counterexamples to the Goldberg conjecture can be constructed. We succeeded in finding such a counterexample on an 8-dimensional compact Walker manifold of neutral signature, but failed in the case of 6-dimensional compact Walker manifold of signature (4, 2) with a canonically defined opposite almost complex structure.