Contact structures on twistor spaces
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.
scientific article; zbMATH DE number 3962680