Remarks on a triple of \(K\)-contact structures (Q1362569)

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.