Ivanov, Stefan; Vassilev, Dimiter Quaternionic contact manifolds with a closed fundamental 4-form. (English) Zbl 1221.53080 Bull. Lond. Math. Soc. 42, No. 6, 1021-1030 (2010). A \(qc\) structure on a real \((4n+3)\)-dimensional manifold \(M\) is a codimension \(3\) distribution \(H\), called the horizontal space, locally given as the kernel of a \(1\)-form \(\eta = (\eta_{1}, \eta_{2}, \eta_{3})\) with values in \(\mathbb{R}^{3}\), such that, the three \(2\)-forms \(d\eta_{i}|_{H}\) are the fundamental \(2\)-forms of a quaternionic structure on \(H\). The \(1\)-form \(\eta\) is determined up to a conformal factor and the action of \(SO (3)\) on \(\mathbb{R}^{3}\). Thus, \(H\) is equipped with a conformal class \([g]\) of Riemannian metrics and a \(2\)-sphere bundle of almost complex structures, the quaternionic bundle \(Q\). The \(2\)-sphere bundle of \(1\)-forms determines uniquely the associated metric and a conformal change of the metric is equivalent to a conformal change of the \(1\)-forms. To every metric in a fixed conformal class, one can associate a complementary to \(H\) distribution \(V\) spanned by the Reeb vector fields \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\) and a linear connection \(\nabla\) preserving the \(qc\) structure and the splitting \(TM = H \oplus V\) provided \(n>1\). This connection is known as the Biquard connection.In the paper under review, the authors obtain the following main result. It is listed below preserving the labeling within the paper.Theorem 1.1: Let \(\left(M^{4n+3}, \eta, Q \right)\) be a \((4n+3)\)-dimensional \(qc\) manifold. For \(n > 1\), the following conditions are equivalent. 6.5mm (i) The fundamental \(4\)-form is closed, \(d \Omega =0\). (ii) The torsion endomorphism of the Biquard connection vanishes.(iii) Each Reeb vector field \(\xi_{l}\) preserves the fundamental \(4\)-form \({\mathbb{L}}_{\xi_{l}} \Omega = 0\). Any of the above conditions imply that the \(qc\)-scalar curvature is constant and the vertical distribution is integrable. Reviewer: Adrian Sandovici (Piatra Neamt) Cited in 1 ReviewCited in 9 Documents MSC: 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 53C17 Sub-Riemannian geometry 58D10 Spaces of embeddings and immersions Keywords:quaternionic contact manifold; closed fundamental 4-form; Biquard connection; Reeb vector field; qc-scalar curvature. PDFBibTeX XMLCite \textit{S. Ivanov} and \textit{D. Vassilev}, Bull. Lond. Math. Soc. 42, No. 6, 1021--1030 (2010; Zbl 1221.53080) Full Text: DOI arXiv