Quaternionic-Kähler manifolds and conformal geometry. (English) Zbl 0674.53036

LeBrun, Claude

Math. Ann. 284, No. 3, 353-376 (1989).

Quaternionic-Kähler manifolds are 4k-dimensional manifolds, \(k\geq 2\), with holonomy group \(Sp(k)Sp(1):=Sp(k)\times Sp(1)/{\mathbb{Z}}_ 2.\) In this paper the author considers how many different quaternionic-Kähler geometries exist in dimension 4k. He shows that for every real-analytic conformal pseudo-Riemannian metric of signature (3,k-1) one can construct a quaternionic Kähler 4k-manifold, such that the initial data (i.e. the original conformal metric) may be read off from the asymptotics of the quaternionic-Kähler geometry.
In order to realize this idea, the author shows that Salamon’s construction of a twistor space from a quaternionic manifold is invertible in an appropriate sense. Further, he proceeds to produce candidate twistor spaces. In the last section the resulting geometry is examined after the twistor-theoretic scaffolding has been wheeled away.

Reviewer: N.Bokan

MSC:

53B35	Local differential geometry of Hermitian and Kählerian structures
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

conformal geometry; Quaternionic-Kähler manifolds; holonomy group; real-analytic conformal pseudo-Riemannian metric; twistor space

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