×

Hyper-Hermitian metrics with symmetry. (English) Zbl 0945.53042

Authors’ abstract: “We show that any four-dimensional hyper-Hermitian manifold admitting a non-trivial tri-holomorphic Killing vector field is locally determined by the solution of a monopole-like equation on a three-dimensional Einstein-Weyl space of a special type. Conversely, any four-dimensional hyper-Hermitian manifold admitting a non-trivial tri-holomorphic Killing vector field arises in this way”.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C80 Applications of global differential geometry to the sciences
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. Apostolov and P Gauduchon, A Riemannian Goldberg-Sachs Theorem, Internat. J. Math., to appear.
[2] Atiyah, M.F.; Hitchin, N.J.; Singer, I.M., Self-duality in four dimensional geometry, (), 425-461 · Zbl 0389.53011
[3] Bonan, E., Sur LES G-structures de type quaternionien, Cahiers de topologie et Géométrie différentielle, Vol. 9, 389-461, (1967) · Zbl 0171.20802
[4] Boyer, C., Conformal duality and compact complex surfaces, Math. ann., 274, 517-526, (1986) · Zbl 0571.32017
[5] Boyer, C., A note on Hyperhermitian four-manifolds, (), 157-164 · Zbl 0642.53073
[6] Cartan, E., Sur une classe d’espaces de Weyl, Ann. sci. ec. norm. sup., 60, 1-16, (1943) · Zbl 0028.30802
[7] Chave, T.; Tod, K.P.; Valent, G., (4, 0) and (4, 4) sigma models with a triholomorphic Killing vector, Phys. lett. B, 383, 262-270, (1996)
[8] Gauduchon, P., La 1-forme de torsion d’une variété hermitienne compacte, Math. ann., 267, 495-518, (1984) · Zbl 0523.53059
[9] Gauduchon, P., Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S1 × S3, J. reine angew. math., 469, 1-50, (1995) · Zbl 0858.53039
[10] Hitchin, N.J., On compact four dimensional Einstein manifolds, J. differential geom., 9, 435-442, (1974) · Zbl 0281.53039
[11] Jones, P.E.; Tod, K.P., Minitwistor spaces and Einstein-Weyl spaces, Class. quant. grav., 2, 565-577, (1985) · Zbl 0575.53042
[12] LeBrun, C., Explicit self-dual metrics on CP^{2}#… #CP2, J. differential geom., 34, 233-253, (1991)
[13] Obata, M., Affine connections on manifolds with almost complex quaternionic or Hermitian structures, Jap. J. math., 26, 43-79, (1956)
[14] Papadopoulos, G., Elliptic monopoles and (4, 0)-supersymmetric sigma models with torsion, Phys. lett. B, 356, 249-255, (1995)
[15] Pedersen, H.; Swann, A., Riemannian submersions, four-manifolds and Einstein geometry, (), 381-399, (66) · Zbl 0742.53014
[16] Pedersen, H.; Tod, K.P., Three-dimensional Einstein-Weyl geometry, Adv. math., 97, 74-109, (1993) · Zbl 0778.53041
[17] Salamon, S., Riemannian geometry and holonomy groups, () · Zbl 0685.53001
[18] Tod, K.P., Compact three-dimensional Einstein-Weyl structures, J. London math. soc., 45, 2, 341-351, (1992) · Zbl 0761.53026
[19] Ward, R.S., Einstein-Weyl spaces and SU(∞) Toda fields, Class. quant. grav., 7, L95-L98, (1990) · Zbl 0687.53044
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.