The twistor spaces of a para-quaternionic Kähler manifold

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 ${\cal J}\in\Gamma ({\rm End}\, TZ)$ and a ${\cal J}$-invariant distribution ${\cal H}_Z$. We study the conditions for the integrability of ${\cal J}$ and for the (para-)holomorphicity of ${\cal H}_Z$. Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces $(Z^\epsilon, {\cal J}, {\cal H}_Z)$, $\epsilon=\pm 1$, such that ${\cal J}^2=\epsilon$ Id. We prove that in both cases ${\cal J}$ is integrable (recovering results of Blair, Davidov and Muŝkarov) and that ${\cal H}_Z$ defines a holomorphic ($\epsilon=-1$) or para-holomorphic ($\epsilon=+1$) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on $Z^\epsilon$. In particular, we find that there is a unique Kähler-Einstein ($\epsilon=-1$) or para-Kähler-Einstein ($\epsilon=+1$) metric. Finally, we prove that any Kähler or para-Kähler submanifold of a para-quaternionic K\"ahler manifold is minimal and describe all such submanifolds in terms of complex ($\epsilon=-1$), respectively, para-complex ($\epsilon=+1$) submanifolds of $Z^\epsilon$ tangent to the contact distribution. (This is joint work with Dmitri Alekseevsky.)