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 K\"ahler almost contact metric manifolds \((M,\varphi, \xi,\eta,g)\), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectiv...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Dario Di Pinto, Dileo, Giulia
Format: Article
Language:English
Subjects:
Curvature
Invariance
Manifolds
Online Access:Get full text
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Summary: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.
ISSN:2331-8422