Abstract
Following the point ofview of Gray and Hervella, we derive detailed conditions whichcharacterize each one of the classes of almostquaternion-Hermitian 4n-manifolds, n > 1. Previously, bycompleting a basic result of Swann, we give explicitdescriptions of the tensors contained in the space of covariantderivatives of the fundamental form Ω and split thecoderivative of Ω into its Sp(n)Sp(1)-components. For 4n > 8, Swann also proved that all theinformation about the intrinsic torsion ∇Ω iscontained in the exterior derivative dΩ. Thus, wegive alternative conditions, expressed in terms of dΩ, to characterize the different classes of almostquaternion-Hermitian manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevski, D. V. and Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures, Ann. Mat. Pura Appl. (4) 171 (1996), 205–273.
Berger, M.: Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330.
Bérard Bergery, L. and Ochoiai, T.: On some generalizations of the construction of twistor spaces, in: T. J. Willmore and N. J. Hitchin (eds), Global Riemannian Geometry, Ellis Horwood, Chichester, 1984.
Cordero, L. A., Fernández, M. and de LeÓn, M.: On the quaternionic Heisenberg group, Boll. Unione Mat. Ital. A (7) 1(1) (1987), 31–37.
Dotti, I. and Fino, A.: HyperKähler torsion structures invariant by nilpotent Lie groups, Classical Quantum Gravity 19(3) (2002), 551–562.
Bröcker, T. and Dieck, T.: Representations of Compacts Lie Groups, Graduate Text in Math. 98, Springer-Verlag, Berlin, 1985.
Gray, A.: Minimal varieties and almost Hermitian submanifolds, Michigan Math. J. 12 (1965), 273–279.
Gray, A. and Hervella, L. M.: The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35–58.
Ivanov, S.: Geometry of quaternionic Kähler connections with torsion, J. Geom. Phys. 41(3) (2002), 235–257.
Joyce, D.: Compact Hypercomplex and Quaternionic manifolds, J. Differential Geom. 35 (1992), 743–761.
Ornea, L. and Piccinni, P.: Locally conformal Kähler structures in quaternionic geometry, Trans. Amer. Math. Soc. 349(2) (1997), 641–655.
Salamon, S.: Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 142–171.
Salamon, S.: Differential geometry of quaternionic manifolds, Ann. Scient. Éc. Norm. Sup. 19 (1986), 31–55.
Salamon, S.: Riemannian Geometry and Holonomy Groups, Pitman Res. Notes Mat. Ser. 201, Longman Sci. Tech., Harlow, 1989.
Salamon, S.: Almost parallel structures, Global differential geometry: The mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288 (2000), 162–181
Swann, A. F.: Aspects symplectiques de la géometrie quaternionique, C. R. Acad. Sci. Paris 308 (1989), 225–228.
Swann, A. F.: Quaternionic Kähler geometry and the fundamental 4-form, in: C. T. J. Dodson (ed.), Proceedings of the Curvature Geometry Workshop, UDLM Publ., Lancaster, 1989, pp. 165–174.
Swann, A. F.: HyperKähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421–450.
Swann, A. F.: Some remarks on quaternion-Hermitian manifolds, Arch.Math. (Brno) 33 (1997), 349–354.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cabrera, F.M. Almost Quaternion-Hermitian Manifolds. Annals of Global Analysis and Geometry 25, 277–301 (2004). https://doi.org/10.1023/B:AGAG.0000023249.48228.93
Issue Date:
DOI: https://doi.org/10.1023/B:AGAG.0000023249.48228.93