Osaka Journal of Mathematics

The twistor spaces of a para-quaternionic Kähler manifold

Dmitri Alekseevsky and Vicente Cortés

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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

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

First available in Project Euclid: 14 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry 53C28: Twistor methods [See also 32L25]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C50: Lorentz manifolds, manifolds with indefinite metrics


Alekseevsky, Dmitri; Cortés, Vicente. The twistor spaces of a para-quaternionic Kähler manifold. Osaka J. Math. 45 (2008), no. 1, 215--251.

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