×

zbMATH — the first resource for mathematics

Quaternionic-Kähler manifolds and conformal geometry. (English) Zbl 0674.53036
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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Berger, M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés Riemanniennes. Bull. Soc. Math. Fr.83, 279-330 (1955) · Zbl 0068.36002
[2] Berger, M.: Remarques sur le groupe d’holonomie des variétés Riemanniennes. C.R. Acad. Sci. Paris262, 1316-1318 (1966) · Zbl 0151.28301
[3] Bryant, R.: Metrics with exceptional holonomy. Ann. Math.126, 525-576 (1987) · Zbl 0637.53042
[4] Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’un espace donné. Ann. Inst. Fourier16, 1-95 (1966) · Zbl 0146.31103
[5] Eastwood, M., LeBrun, C. Fattening complex manifolds. Preprint (1988) · Zbl 0748.32017
[6] Grauert, H., Remmert, R.: Coherent analytic sheaves. Berlin Heidelberg New York: Springer 1984 · Zbl 0537.32001
[7] Hitchin, N.J., Karlhede, A., Lindström, U., Ro?ek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys.108, 535-589 (1987) · Zbl 0612.53043
[8] Kodaira, K.: On complete analytic families of compact complex submanifolds. Ann. Math.75, 146-162 (1962) · Zbl 0112.38404
[9] Kodaira, K.: Complex manifolds and deformations of complex structures. Berlin Heidelberg New York: Springer 1986 · Zbl 0581.32012
[10] Lebrun, C.: Spaces of complex null geodesics in complex-Riemannian geometry. Trans. Am. Math. Soc.278, 209-231 (1983) · Zbl 0562.53018
[11] LeBrun, C.:H-space with a cosmological constant. Proc. R. Soc. Lond. Ser. A380, 171-185 (1982) · Zbl 0549.53042
[12] Penrose, R.: Non-linear gravitons and curved twistor theory. Gen. Relativ. Gravitation7, 31-52 (1976) · Zbl 0354.53025
[13] Pedersen, H.: Einstein metrics, spinning top motions, and monopoles. Math. Ann.274, 35-59 (1986) · Zbl 0566.53058
[14] Salamon, S.M.: Quaternionic Kähler manifolds. Invent. Math.67, 143-171 (1982) · Zbl 0486.53048
[15] Salamon, S.M.: Differential geometry of quaternionic manifolds. Ann. Sc. Ec. Norm. Super., IV. Ser.19, 31-55 (1986) · Zbl 0616.53023
[16] Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech.14, 1033-1047 (1965) · Zbl 0141.38202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.