Veranstaltung

Vortrag
The twistor spaces of a para-quaternionic Kähler manifold
30.6.2006, 13:30 Uhr – 14:20 Uhr

We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space Z of such a G-structure is endowed with a field of involutions formula21 and a formula23-invariant distribution formula25. We study the conditions for the integrability of formula23 and for the (para-)holomorphicity of formula25. Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces formula31, formula33, such that formula35 Id. We prove that in both cases formula23 is integrable (recovering results of Blair, Davidov and Muskarov) and that formula25 defines a holomorphic (formula41) or para-holomorphic (formula43) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on formula45. In particular, we find that there is a unique Kähler-Einstein (formula41) or para-Kähler-Einstein (formula43) metric. Finally, we prove that any Kähler or para-Kähler submanifold of a para-quaternionic Kähler manifold is minimal and describe all such submanifolds in terms of complex (formula41), respectively, para-complex (formula43) submanifolds of formula45 tangent to the contact distribution. (This is joint work with Dmitri Alekseevsky.)