Classification of positive quaternion-Kähler 12-manifolds
type: Item
instance of: scholarly article
title: Classification of positive quaternion-Kähler 12-manifolds
zbMATH Open document ID: 1007.53037
DOI: 10.1016/S1631-073X(02)02209-4
author: Haydeé Herrera
author: Rafael Herrera
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-08-04T00:00:00Z
review text: 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 \(\text{Gr}_4(\mathbb{R}^7)\).
Mathematics Subject Classification ID: 53C26
zbMATH DE Number: 1744398
zbMATH Keywords: symmetric space
zbMATH Keywords: quaternion Kähler manifold
zbMATH Keywords: positive scalar curvature
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1744398