Self-dual Kähler manifolds and Einstein manifolds of dimension four

Andrzej Derdziński

Compositio Mathematica (1983)

  • Volume: 49, Issue: 3, page 405-433
  • ISSN: 0010-437X

How to cite


Derdziński, Andrzej. "Self-dual Kähler manifolds and Einstein manifolds of dimension four." Compositio Mathematica 49.3 (1983): 405-433. <>.

author = {Derdziński, Andrzej},
journal = {Compositio Mathematica},
keywords = {self-dual metrics; 4-dimensional Einstein metrics; locally symmetric; Kaehler metrics; extremal Kaehler metrics},
language = {eng},
number = {3},
pages = {405-433},
publisher = {Martinus Nijhoff Publishers},
title = {Self-dual Kähler manifolds and Einstein manifolds of dimension four},
url = {},
volume = {49},
year = {1983},

AU - Derdziński, Andrzej
TI - Self-dual Kähler manifolds and Einstein manifolds of dimension four
JO - Compositio Mathematica
PY - 1983
PB - Martinus Nijhoff Publishers
VL - 49
IS - 3
SP - 405
EP - 433
LA - eng
KW - self-dual metrics; 4-dimensional Einstein metrics; locally symmetric; Kaehler metrics; extremal Kaehler metrics
UR -
ER -


  1. [1] T. Adati and T. Miyazawa: On a Riemannian space with recurrent conformal curvature. Tensor, N. S.18 (1967) 348-354. Zbl0152.39103MR215251
  2. [2] M.F. Atiyah, N.J. Hitchin and I.M. Singer: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. LondonA362 (1978) 425-461. Zbl0389.53011MR506229
  3. [3] R. Bach: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Zeitsch.9 (1921) 110-135. Zbl48.1035.01MR1544454JFM48.1035.01
  4. [4] L. Bérard Bergery: Sur de nouvelles variétés riemanniennes d'Einstein, preprint, Institut Elie Cartan, Université de Nancy I (1981). Zbl0544.53038MR725678
  5. [5] S. Bochner: Vector fields and Ricci curvature. Bull. Amer. Math. Soc.52 (1946) 776-797. Zbl0060.38301MR18022
  6. [6] A. Borel: Compact Clifford-Klein forms of symmetric spaces. Topology2 (1963) 111-122. Zbl0116.38603MR146301
  7. [7] J.P. Bourguignon: Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein. Invent. Math.63 (1981) 263-286. Zbl0456.53033MR610539
  8. [8] E. Calabi: Extremal Kähler metrics. Seminar on Differential Geometry (edited by Shing-Tung Yau). Princeton Univ. Press and Univ. of Tokyo Press, Princeton (1982), pp. 259-290. Zbl0487.53057MR645743
  9. [9] J. Carrell, A. Howard and C. Kosniowski: Holomorphic vector fields on complex surfaces. Math. Ann.204 (1973) 73-81. Zbl0242.14008MR372262
  10. [10] A. Derdziński: Exemples de métriques de Kähler et d'Einstein auto-duales sur le plan complexe. Géométrie riemannienne en dimension 4 (Séminaire Arthur Besse 1978/79). Cedic/Fernand Nathan, Paris (1981), pp. 334-346. Zbl0477.53025
  11. [11] A. Derdziński and A. Rigas: Unflat connections in 3-sphere bundles over S4. Trans. Amer. Math. Soc.265 (1981) 485-493. Zbl0465.53028MR610960
  12. [12] A. Derdziński and W. Roter: On conformally symmetric manifolds with metrics of indices 0 and 1. Tensor, N. S.31 (1977) 255-259. Zbl0379.53027MR467596
  13. [13] D. Deturck and J.L. Kazdan: Some regularity theorems in Riemannian geometry. Ann. Sci. Ec. Norm. Sup. (4) 14 (1981) 249-260. Zbl0486.53014MR644518
  14. [14] H. Donnelly: Topology and Einstein Kähler metrics. J. Differential Geometry11 (1976) 259-264. Zbl0332.32004
  15. [15] TH. Friedrich and H. Kurke: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature. Math. Nachrichten (to appear). Zbl0503.53035MR675762
  16. [16] A. Gray: Invariants of curvature operators of four-dimensional Riemannian manifolds. Proc. 13th Biennial Seminar Canadian Math. Congress (1971), vol. 2, 42-65. Zbl0278.53034MR405446
  17. [17] N. Hitchin: Compact four-dimensional Einstein manifolds. J. Differential Geometry9 (1974) 435-441. Zbl0281.53039MR350657
  18. [18] N. Hitchin: Kählerian twistor spaces. Proc. London Math. Soc.43 (1981133-150. Zbl0474.14024MR623721
  19. [19] G.R. Jensen: Homogeneous Einstein spaces of dimension four. J. Differential Geometry3 (1969) 309-349. Zbl0194.53203MR261487
  20. [20] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, vol. I. Interscience, New York (1963). Zbl0119.37502MR152974
  21. [21] A. Lichnerowicz: Espaces homogènes kählériens. Colloque de Géométrie Différentielle, Strasbourg (1953) 171-201. Zbl0053.11603MR66017
  22. [22] A. Lichnerowicz: Isométries et transformations analytiques d'une variété kählérienne compacte. Bull. Soc. Math. France87 (1959) 427-437. Zbl0192.28403MR114187
  23. [23] R. Mandelbaum and B. Moishezon: The topological type of algebraic surfaces: Hypersurfaces in P3(C). Proc. Symposia in Pure Math.30 (1977), Part I, 277-283. Zbl0355.57002MR478178
  24. [24] Y. Matsushima: Remarks on Kähler-Einstein manifolds. Nagoya Math. J.46 (1972) 161-173. Zbl0249.53050MR303478
  25. [25] J.W. Milnor and J.D. Stasheff: Characteristic Classes. Princeton University Press, Princeton (1974). Zbl0298.57008MR440554
  26. [26] D. Page: A compact rotating gravitational instanton. Phys. Lett.79B (1978) 235-238. 
  27. [27] A. Polombo: Nombres caractéristiques d'une surface kählérienne compacte. C. R. Acad. Sci. ParisA283 (1976) 1025-1028. Zbl0339.53041MR431066
  28. [28] W. Roter: Some existence theorems on conformally recurrent manifolds (in preparation). 
  29. [29] I.M. Singer and J.A. Thorpe: The curvature of 4-dimensional Einstein spaces. Global Analysis, Papers in Honor of K. Kodaira, Princeton (1969) 355-365. Zbl0199.25401MR256303
  30. [30] S. Tanno: Curvature tensors and covariant derivatives. Ann. Mat. Pura Appl.96 (1973) 233-241. Zbl0277.53013MR326619
  31. [31] J. Thorpe: Some remarks on the Gauss-Bonnet integral. J. of Math. Mech.18 (1969) 779-786. Zbl0183.50503MR256307
  32. [32] K. Yano and I. Mogi: On real representations of Kaehlerian manifolds. Ann. of Math.61 (1955) 170-189. Zbl0064.16302MR68291
  33. [33] S.-T. Yau: Calabi's conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. U.S.A.74 (1977) 1798-1799. Zbl0355.32028MR451180

Citations in EuDML Documents

  1. Sun-Yung A. Chang, Matthew J. Gursky, Paul C. Yang, A conformally invariant sphere theorem in four dimensions
  2. Mitsuhiro Itoh, Self-duality of Kähler surfaces
  3. Andrzej Derdziński, Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces
  4. A. Polombo, De nouvelles formules de Weitzenböck pour des endomorphismes harmoniques. Applications géométriques
  5. D. Burns, P. De Bartolomeis, Applications harmoniques stables dans P n
  6. Vestislav Apostolov, Oleg Muškarov, Weakly-Einstein hermitian surfaces
  7. Vestislav Apostolov, Paul Gauduchon, Selfdual Einstein hermitian four-manifolds
  8. Paul Gauduchon, Variétés riemanniennes autoduales
  9. Zindine Djadli, Opérateurs géométriques et géométrie conforme

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