THE TWISTOR SPACES OF A PARA-QUATERNIONIC KÄHLER MANIFOLD

Abstract

We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a c omplex structure. The twistor space Z of such a G-structure is endowed with a field of involutions J ∈ (End T Z) and a J -invariant distribution H Z . We study the conditions for the integrability of J and for the (para-)holomorphicity of HZ . Then we apply this theory to para-quaternionic Kahler manifolds of non-zero scalar curvature, which admit two natural twistor spaces (Z , J , HZ ), = ±1, such that J 2 = Id. We prove that in both cases J is integrable (recovering results of Blair, Davidov and Mu˘ and that HZ defines a holomorphic ( = −1) or para-holomorphic ( = +1) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on Z . In particular, we find that there is a unique Kahler-Einstei n ( = −1) or para-Kahler-Einstein ( = +1) metric. Finally, we prove that any Kahler or para-Kahler submanifold of a para-quaternionic Kahler manifold is minimal and describe all such submanifolds in terms of complex ( = −1), respectively, para-complex ( = +1) submanifolds of Z tangent to the contact distribution.