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Symplectic 4-manifolds with Hermitian Weyl tensor


Authors: Vestislav Apostolov and John Armstrong
Journal: Trans. Amer. Math. Soc. 352 (2000), 4501-4513
MSC (2000): Primary 53B20, 53C25
DOI: https://doi.org/10.1090/S0002-9947-00-02624-6
Published electronically: June 13, 2000
MathSciNet review: 1779485
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Abstract: It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.


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  • 1. V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs Theorem, Int. J. of Math. 8 (1997), 421-439. MR 98g:53080
  • 2. V. Apostolov and T. Draghici, Almost Kähler 4-manifolds with $J$-invariant Ricci tensor and special Weyl tensor, Quart. J. Math. Oxford Ser.(2), to appear.
  • 3. J. Armstrong, On four-dimensional almost Kähler manifolds, Quart. J. Math. Oxford Ser.(2) 48 (1997), 405-415. MR 98k:53054
  • 4. J. Armstrong, An Ansatz for Almost-Kähler, Einstein 4-manifolds, preprint, 1997.
  • 5. J. Armstrong, Almost Kähler Geometry, Ph.D. Thesis, Oxford, 1998.
  • 6. T. Aubin, Non-linear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, Grund. Math. Wiss., 252, 1982. MR 85j:58002
  • 7. W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Springer-Verlag, 1984. MR 86c:32026
  • 8. J.-P. Bourguignon, Formules de Weitzenböck en dimension 4, in Géométrie riemannienne en dimension 4, Séminaire Arthur Besse, 1978/79, eds. Bérard-Bergery, M. Berger, C. Houzel, Cedic/Fernand Nathan, Paris 1981. CMP 17:05
  • 9. J.-P. Bourguignon, Les variétés de dimension 4 à signature non nule dont la courbure est harmonique sont d'Einstein, Invent. Math. 63 (1981), 263-286. MR 82g:5305
  • 10. B.-Y. Chen, Some topological obstructions to Bochner-Kähler metrics and their applications, J. Diff. Geom. 13 (1978), 574-588. MR 81f:32037
  • 11. A. Derdzinski, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), 405-433. MR 84h:53060
  • 12. T. Draghici, Almost Kähler 4-manifolds with $J$-invariant Ricci tensor, Houston J. of Math., Vol.25, no.1, (1999), 133-145. CMP 99:13
  • 13. G. Ganchev, On Bochner curvature tensors in almost Hermitian manifolds, Pliska Studia Matematica Bulgarica 9 (1987), 33-43. MR 88e:53045
  • 14. J. Gasqui, Sur la résolubilité locale des équations d'Einstein, Compositio Math. 47 (1982), 43-69. MR 84f:58115
  • 15. J. Goldberg and R. Sachs, A theorem on Petrov types, Acta Phys. Polon. 22 Suppl. (1962), 13-23. MR 27:6599
  • 16. S.I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96-100. MR 38:6514
  • 17. H. Goldschmidt, Integrability criteria for systems of non-linear partial differential equations, J. Diff. Geom. 1 (1967), 269-307. MR 37:1746
  • 18. M. Itoh, Self-duality of Kähler surfaces, Compositio Math. 51 (1984), 265-273. MR 85m:53079
  • 19. T. Oguro and K. Sekigawa Non-existence of almost Kähler structure on hyperbolic spaces of dimension $2n(\geq 4)$, Math. Ann. 300 (1994), 317-329. MR 95h:53066
  • 20. M. Przanowski and B. Broda, Locally Kähler gravitational instantons, Acta Phys. Polon. B 14 (1983), 637-661. MR 85g:83016b
  • 21. I. Robinson and A. Schild, A generalization of a theorem by Goldberg and Sachs, J. Math. Phys. 4 (1963), 484. MR 26:7421
  • 22. K. Sekigawa, On some 4-dimensional compact Einstein almost Kähler manifolds, Math. Ann. 271 (1985), 333-337. MR 86g:53054
  • 23. F. Tricerri and L.Vanhecke Curvature tensors on almost Hermitian manifods, Trans. Amer. Math. Soc. 267 (1981), 365-398. MR 82j:53071

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Additional Information

Vestislav Apostolov
Affiliation: IHÉS - EPDI, Le Bois-Marie, 35, route de Chartres, F-91440 Bures-sur-Yvette Cedex, France
Email: apostolo@ihes.fr

John Armstrong
Affiliation: 10 Alan Bullock Close, Oxford OX4 1AU, United Kingdom
Email: John.Armstrong@madge.com

DOI: https://doi.org/10.1090/S0002-9947-00-02624-6
Received by editor(s): June 21, 1999
Published electronically: June 13, 2000
Additional Notes: The first author was supported in part by NSF grant INT-9903302
Article copyright: © Copyright 2000 American Mathematical Society

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