Remarks on a triple of \(K\)-contact structures
type: Item
instance of: scholarly article
title: Remarks on a triple of \(K\)-contact structures
zbMATH Open document ID: 0881.53040
author: Shukichi Tanno
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-02-19T00:00:00Z
review text: It is known that a complete regular 3-Sasakian manifold is a principal \(Sp(1)\) or \(SO(3)\) bundle over a quaternionic Kähler manifold of dimension \(4r\geq 8\). The converse question was considered by \textit{M. Konishi} in [Kōdai Math. Semin. Rep. 26, 194-200 (1975; Zbl 0308.53035)]. In the present paper, the author extends Konishi's construction to the case \(r= 1\); the notion of a quaternionic Kähler structure in dimension 4 is that of an anti-selfdual Einstein manifold. The author also proves, analogous to the fact that a 3-dimensional \(K\)-contact manifold is Sasakian, that a \(K\)-contact 3-structure on a 7-dimensional Riemannian manifold is a Sasakian 3-structure. In the Konishi construction, if the scalar curvature of the base manifold is negative, then the structure on the bundle space is a pseudo-Sasakian 3-structure, the signature of the metric being \((3,4r)\). To put this structure in a Riemannian setting the author introduces the notion of a triple of \(K\)-contact structures being of \(nS\)-type. The main result is as follows. Let \((M,g)\) be a quaternionic Kähler, non-hyperkähler manifold of dimension \(4r \geq 4\) with negative scalar curvature (normalized to the value \(-16r(r+2))\). Then there exists a canonically associated principal \(SO(3)\) bundle over \((M,g)\) which admits a triple of \(K\)-contact structures of \(nS\)-type and with respect to which the fibering is a Riemannian submersion.
reviewed by: David E. Blair
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1044138
zbMATH Keywords: Sasakian 3-structures
zbMATH Keywords: \(K\)-contact 3-structure
zbMATH Keywords: \(K\)-contact structures of \(nS\)-type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225296
OpenAlex ID: W2153263680
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DOI: 10.2748/TMJ/1178225296
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description: scientific article; zbMATH DE number 1044138