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