Contact structures on twistor spaces
type: Item
instance of: scholarly article
title: Contact structures on twistor spaces
zbMATH Open document ID: 0597.53051
author: Masaru Takeuchi
author: Takashi Gyoshin Nitta
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: For a quaternionic Kähler manifold \((M,g,H)\) of nonzero scalar curvature, H being a quaternionic structure, the twistor space Z of \((M,g,H)\) admits a (possibly indefinite) Kähler-Einstein metric \(\bar g\) and a complex contact structure \(\gamma\) [\textit{S. Salamon}, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)]. First the authors prove: The natural lifting gives rise to an isomorphism from the group of automorphisms of (M,g,H) to the group of isometric contact automorphisms of \((Z,\gamma,\bar g)\). The method is to use the prolongation of the latter group to automorphisms of the \({\mathbb{C}}^*\)-bundle over Z associated to \(\gamma\). Secondly, by a refinement of Salamon's method, the following analogue of Matsushima's theorem is proved: The Lie algebra of infinitesimal contact automorphisms of \((Z,\gamma)\) is the complexification of the Lie algebra of infinitesimal isometric contact automorphisms of \((Z,\gamma,\bar g)\). Finally the authors prove a certain uniqueness in twistor theory for quaternionic Kähler manifolds. They prove first that a compact complex manifold with torsion-free first integral homology and \(h^{0,1}=0\) can admit at most one complex contact structure up to automorphisms. Making use of this fact, the following is proved: Let \((M,g,H)\) and \((M',g',H')\) be compact quaternionic Kähler manifolds of positive scalar curvature, and Z and Z' their twistor spaces. Suppose that either \((M,g)\) or \((M',g')\) is a Riemannian symmetric space. Then, if Z is biholomorphic to \(Z'\), \((M,g,H)\) is equivalent to \((M',g',H')\). It should be remarked that all the known examples of compact quaternionic Kähler manifolds of positive scalar curvature are Riemannian symmetric spaces.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 3962680
zbMATH Keywords: quaternionic Kähler manifold
zbMATH Keywords: twistor space
zbMATH Keywords: contact structure
zbMATH Keywords: contact automorphisms
zbMATH Keywords: symmetric space
zbMATH Keywords: positive scalar curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03910139
OpenAlex ID: W2082586265
DOI: 10.2969/JMSJ/03910139
description: scientific article; zbMATH DE number 3962680