Osaka Journal of Mathematics

The twistor spaces of a para-quaternionic Kähler manifold

Dmitri Alekseevsky and Vicente Cortés

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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

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1205503566

Mathematical Reviews number (MathSciNet)
MR2416658

Zentralblatt MATH identifier
1177.53047

Subjects
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

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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References

  • D.V. Alekseevsky and V. Cortés: Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type; in Lie Groups and Invariant Theory, Amer. Math. Soc. Transl. (2) 213, Amer. Math. Soc., Providence, RI., 2005, 33--62.
  • D.V. Alekseevsky and M.M. Graev: $G$-structures of twistor type and their twistor spaces, J. Geom. Phys. 10 (1993), 203--229.
  • D.V. Alekseevsky and S. Marchiafava: A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold, Ann. Mat. Pura Appl. (4) 184 (2005), 53--74.
  • M.F. Atiyah, N.J. Hitchin and I.M. Singer: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425--461.
  • A.L. Besse: Einstein Manifolds, Springer-Verlag, Berlin, 1987.
  • D.E. Blair, J. Davidov and O. Mu\uskarov: Hyperbolic twistor spaces, Rocky Mountain J. Math. 35 (2005), 1437--1465.
  • D.E. Blair: A product twistor space, Serdica Math. J. 28 (2002), 163--174.
  • R.L. Bryant: Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), 623--715.
  • M. Mamone Capria and S.M. Salamon: Yang-Mills fields on quaternionic spaces, Nonlinearity 1 (1988), 517--530.
  • V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig: Special geometry of Euclidean supersymmetry I: Vector multiplets, J. High Energy Phys. (2004), 028, 73.
  • V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig: Special geometry of Euclidean supersymmetry II: hypermultiplets and the c-map, J. High Energy Phys. (2005), 025, 37.
  • B. O'Neill: The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459--469.
  • R. Penrose: Twistor algebra, J. Mathematical Phys. 8 (1967), 345--366.
  • R. Penrose: The twistor programme, Rep. Mathematical Phys. 12 (1977), 65--76.
  • S. Salamon: Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143--171.