This paper is a detailed report on two classes of \(4n\)-dimensional Riemannian manifolds admitting local, but non-global, quaternion Kähler (q.K.) structures, corresponding to different viewpoints. Namely, class (1) consists of local q.K. manifolds, characterized by having reduced holonomy group contained in \(Sp(n)\cdot Sp(1)\), hence these manifolds are Einstein. Class (2) is formed by the locally conformal q.K. manifolds which are Einstein-Weyl. The first class was discussed by the author [P. Piccinni, ‘The geometry of positive locally quaternion Kähler manifolds’, Preprint 26, Dip. di Matem., Univ. “La Sapienza”, Roma (1997)], while class (2) was mainly studied by L. Ornea and P. Piccinni [Trans. Am. Math. Soc. 349, 641-655 (1997; Zbl 0865.53038) and ‘Compact hyperhermitian-Weyl and quaternion Hermitian-Weyl manifolds’, Preprint 14, Dip. di Matem., Univ. “La Sapienza”, Roma (1997)]. Class (1) includes locally hyperkähler manifolds. Class (2) contains some finite quotients of quaternionic Hopf manifolds. Moreover, when compact, manifolds in class (2) are locally conformal locally hyperkähler. But a characterization in terms of holonomy for class (2) is still lacking.
As concerns positive manifolds (i.e., with positive scalar curvature) in class (1), the author shows that they must be compact, locally symmetric and finitely covered by a q.K. Wolf space. Hence, a thorough examination of the possible groups acting on these Wolf spaces produces a complete list of positive q.K. manifolds. Apart from \(\mathbb{H} P^1\), they are all real or complex Grassmannians.
Although no connection was a priori apparent between the two classes, a link is found by the author using \(3\)-Sasakian geometry. Let \(\overline P\) be a compact \(3\)-Sasakian manifold with canonical \(3\)-dimensional foliation \(\mathcal K\) [see, e.g., Ch. P. Boyer, K. Galicki and B. M. Mann, J. Reine Angew. Math. 455, 183-220 (1994; Zbl 0889.53029)] and let \(\Gamma\) be a finite group of isometries of \(\overline P\). Compact manifolds in class (1) are flat principal circle bundles over locally \(3\)-Sasakian manifolds of the form \(\overline P/\Gamma\) with \(\Gamma\) preserving each leaf of \(\mathcal K\). But if \(\Gamma\) interchanges the leaves of \(\mathcal K\) on a homogeneous \(\overline P\), then positive class (1) manifolds are obtained by leaf spaces of the foliation induced by \(\mathcal K\) on \(\overline P/\Gamma\).
Clear and very well written, this paper throws some new light on the geometry of two classes of quaternion Hermitian manifolds whose study is only at the beginning.