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 Kahler almost contact metric manifolds (M, phi, xi, eta, g), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the.-invariance and the phi-anti-invariance of the 2-form d eta. A Boothby-Wang type theorem allows to obtain aqS structures on principal circle bundles over Kahler 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) x 1-reduction of the frame bundle such that the manifold is transversely hyperkahler, 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 cokahler. 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 Kahler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (M, g) cannot be locally symmetric.