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Quaternionic reduction and quaternionic orbifolds. (English) Zbl 0628.53060
An analogue of the process of symplectic reduction is defined for quaternionic Kähler manifolds. Certain features of the process are explored. In each dimension 4n, \(n>1\), the construction yields an infinite family of compact, simply-connected Riemannian orbifolds which have \(Sp_ 1Sp_ n\) holonomy and are not locally symmetric. In dimension 4, it yields infinite family of compact, simply-connected Riemannian orbifolds (“weighted” complex projective planes) which are Einstein, self-dual and of positive scalar curvature. This contrasts interestingly with a result of N. Hitchin in the non-singular case

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:
[1] Alekseevskii, D.V.: Classification of quaternionic spaces with transitive solvable group of motions. Math. USSR-Izv.9, 297 (1975) · Zbl 0324.53038 · doi:10.1070/IM1975v009n02ABEH001479
[2] Berger, M.: Sur les groupes d’holonomie homogène des variétés à connexion affines et des variétés riemanniennes. Bull. Soc. Math. France83, 279 (1955) · Zbl 0068.36002
[3] Bourguignon, J.P., Lawson, H.B. Jr.: Stability and isolation phenomena for Yang-Mills fields. Commun. Math. Phys.79, 169 (1980)
[4] Galicki, K.: A generalization of the momentum mapping construction for quaternionic Kähler manifolds. Commun. Math. Phys.108, 117 (1987) · Zbl 0608.53058 · doi:10.1007/BF01210705
[5] Galicki, K.: New metrics withSp n Sp 1 holonomy. Nucl. Phys. B289, 573 (1987) · doi:10.1016/0550-3213(87)90394-4
[6] Hitchin, N.J.: Kählerian twistor spaces. Proc. London Math. Soc. (3)43, 133 (1981) · Zbl 0474.14024 · doi:10.1112/plms/s3-43.1.133
[7] Hitchin, N.J.: Not published
[8] Ishihara, S.: Quaternion Kählerian manifolds. J. Diff. Geom.9, 483 (1974) · Zbl 0297.53014
[9] Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121 (1974) · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4
[10] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J.13, 459 (1966) · Zbl 0145.18602 · doi:10.1307/mmj/1028999604
[11] Pedersen, H.: Einstein metrics, spinning top motions and monopoles. Math. Ann.274, 35 (1986) · Zbl 0566.53058 · doi:10.1007/BF01458016
[12] Salamon, S.: Quaternionic Kähler manifolds. Invent. Math.67, 143 (1982) · Zbl 0486.53048 · doi:10.1007/BF01393378
[13] Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech.14, 1033 (1965) · Zbl 0141.38202
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