Anti-quasi-Sasakian manifolds

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely Kähler almost contact metric manifolds \((M,\varphi , \xi ,\eta ,g)\), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the \(\varphi \)-invariance and the \(\varphi \)-anti-invariance of the 2-form \(\textrm{d}\eta \). A Boothby–Wang type theorem allows to obtain aqS structures on principal circle bundles over Kähler manifolds endowed with a closed (2, 0)-form. We characterize aqS manifolds with constant \(\xi \)-sectional curvature equal to 1: they admit an \(Sp(n)\times 1\)-reduction of the frame bundle such that the manifold is transversely hyperkähler, carrying a second aqS structure and a null Sasakian \(\eta \)-Einstein structure. We show that aqS manifolds with constant sectional curvature are necessarily flat and cokähler. Finally, by using a metric connection with torsion, we provide a sufficient condition for an aqS manifold to be locally decomposable as the Riemannian product of a Kähler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (M, g) cannot be locally symmetric.