Compact hyperhermitian-Weyl and quaternion Hermitian-Weyl manifolds (Q1264250)

A local conformally quaternionic-Kähler manifold is a Riemannian manifold \((M,g)\) with a conformal torsion-free connection \(D\) \((Dg=\omega\otimes g)\) which preserves a quaternionic structure \(H\). Under the assumption that \(M\) is compact, not quaternionic Kähler, and the integrable 1-dimensional and 4-dimensional distributions \(\mathcal B, \mathcal D \) spanned by \(B= g^{-1}\omega\) and \(HB\), respectively, have compact leaves, the authors prove that the manifold \(M\) admits a \(S^1\)-fibration over a manifold \(P\) with local 3-Sasakian structure. Moreover, some finite covering \(\widetilde M\) of \(M\) carries a local conformal hyper-Kähler structure (i.e., a torsion-free conformal connection which preserves a hypercomplex structure \((J_1,J_2,J_3))\) and the base \(\widetilde P\) of the associated \(S^1\)-fibration of \(\widetilde M\) is 3-Sasakian manifold. The manifold \(M\) admits an integrable complex structure which is a section of the bundle \(H\).