Classification of positive quaternion-Kähler 12-manifolds
One says that an oriented connected irreducible Riemannian manifold of dimension \(m=4n\geq 8\) is a quaternionic Kähler manifold if its linear holonomy is contained in \(Sp(n)Sp(1)\). One says that such a manifold is \textit{positive} if the scalar curvature is positive and if the manifold is complete. It is conjectured that every positive quaternionic Kähler manifold is isometric to a (symmetric) Wolf space. This conjecture has been established by \textit{Y. S. Poon} and \textit{S. M. Salamon} in 8 dimensions [J. Differ. Geom. 33, No. 2, 363-378 (1991; Zbl 0733.53035)] and an analogous conjecture was established by \textit{N. Hitchin} in 4 dimensions [Proc. Lond. Math. Soc. (3) 43, 133-150 (1981; Zbl 0474.14024)]. The authors establish the conjecture in 12 dimensions showing: Theorem: Any positive \(12\) dimensional quaternionic Kähler manifold is isometric to one of the following symmetric spaces: Quaternionic projective space \(\mathbb{H}\mathbb{P}^3\), the complex Grassmannian \(\text{Gr}_2(\mathbb{C}^6)\), or the real Grassmannian \(\text{Gr}_4(\mathbb{R}^7)\).
scientific article; zbMATH DE number 1744398