THE TWISTOR SPACES OF A PARA-QUATERNIONIC KÄHLER MANIFOLD

We develop the twistor theory of $G$-structures for which\nthe (linear) Lie algebra of the structure group contains an\ninvolution, instead of a complex structure. The twistor space\n$Z$ of such a $G$-structure is endowed with a field of involutions\n$\\mathcal{J}\\in \\Gamma (\\End TZ)$ and a $\\mathcal{J}$-invariant\ndistribution $\\mathcal{H}_{Z}$. We study the conditions for\nthe integrability of $\\mathcal{J}$ and for the (para-)holomorphicity\nof $\\mathcal{H}_{Z}$. Then we apply this theory to para-quaternionic\nKähler manifolds of non-zero scalar curvature, which\nadmit two natural twistor spaces $(Z^{\\epsilon},\\mathcal{J},\\mathcal{H}_{Z})$,\n$\\epsilon=\\pm 1$, such that $\\mathcal{J}^{2}=\\epsilon \\Id$.\nWe prove that in both cases $\\mathcal{J}$ is integrable (recovering\nresults of Blair, Davidov and Mu\\u{s}karov) and that $\\mathcal{H}_{Z}$\ndefines a holomorphic ($\\epsilon=-1$) or para-holomorphic\n($\\epsilon=+1$) contact structure. Furthermore, we determine\nall the solutions of the Einstein equation for the canonical\none-parameter family of pseudo-Riemannian metrics on $Z^{\\epsilon}$.\nIn particular, we find that there is a unique Kähler-Einstein\n($\\epsilon=-1$) or para-Kähler-Einstein ($\\epsilon=+1$)\nmetric. Finally, we prove that any Kähler or para-Kähler\nsubmanifold of a para-quaternionic Kähler manifold\nis minimal and describe all such submanifolds in terms of\ncomplex ($\\epsilon=-1$), respectively, para-complex ($\\epsilon=+1$)\nsubmanifolds of $Z^{\\epsilon}$ tangent to the contact distribution.