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Quaternionic contact manifolds with a closed fundamental 4-form. (English) Zbl 1221.53080

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.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C17 Sub-Riemannian geometry
58D10 Spaces of embeddings and immersions
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