Anti-quasi-Sasakian manifolds

TL;DR

This paper introduces anti-quasi-Sasakian manifolds, characterizes their geometric properties, and explores their curvature, structure, and decomposition, expanding the understanding of almost contact metric manifolds.

Contribution

It defines and studies anti-quasi-Sasakian manifolds, providing characterizations, curvature conditions, and decomposition results, and relates them to principal circle bundles over K"ahler manifolds.

Findings

aqS manifolds with constant sectional curvature are flat or cok"ahler.

Characterization of aqS manifolds with constant $\xi$-sectional curvature 1.

Existence of aqS structures on principal circle bundles over K"ahler manifolds.

Abstract

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely K\"ahler 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 $d\eta$. A Boothby-Wang type theorem allows to obtain aqS structures on principal circle bundles over K\"ahler 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 hyperk\"ahler, 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 cok\"ahler. 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 K\"ahler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, $(M,g)$ cannot be locally symmetric.