## Osaka Journal of Mathematics

### 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 complex structure. The twistor space $Z$ of such a $G$-structure is endowed with a field of involutions $\mathcal{J}\in \Gamma (\End TZ)$ 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 $(Z^{\epsilon},\mathcal{J},\mathcal{H}_{Z})$, $\epsilon=\pm 1$, such that $\mathcal{J}^{2}=\epsilon \Id$. We prove that in both cases $\mathcal{J}$ is integrable (recovering results of Blair, Davidov and Mu\u{s}karov) and that $\mathcal{H}_{Z}$ 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 Kähler 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.

#### Article information

Source
Osaka J. Math., Volume 45, Number 1 (2008), 215-251.

Dates
First available in Project Euclid: 14 March 2008

https://projecteuclid.org/euclid.ojm/1205503566

Mathematical Reviews number (MathSciNet)
MR2416658

Zentralblatt MATH identifier
1177.53047

#### Citation

Alekseevsky, Dmitri; Cortés, Vicente. The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45 (2008), no. 1, 215--251. https://projecteuclid.org/euclid.ojm/1205503566

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