Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Weyl tensor of a self-dual complex 4-manifold

Author: Florin Alexandru Belgun
Journal: Trans. Amer. Math. Soc. 356 (2004), 853-880
MSC (2000): Primary 53C21, 53A30, 32Qxx
Published electronically: October 21, 2003
MathSciNet review: 1984459
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the $\beta$-surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of $\mathbb{CP} ^2$.

References [Enhancements On Off] (What's this?)

  • 1. M.F. Atiyah, N.J. Hitchin, I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proceedings of the Royal Society of London, Series A 362 (1978), 425-461. MR 80d:53023
  • 2. F.A. Belgun, Null-geodesics on complex conformal mmanifolds and the LeBrun correspondence, J. Reine Angew Math. 536 (2001), 43-63. MR 2002g:53080
  • 3. A.L. Besse, Einstein manifolds, Springer Verlag (1987). MR 88f:53087
  • 4. F. Campana, On twistor spaces of the class $\mathcal{C}$, J. Diff. Geom. 33 (1991), 541-549. MR 92g:32059
  • 5. P. Gauduchon, Connexion canonique et structures de Weyl en géométrie conforme, Preprint, juin 1990.
  • 6. N.J. Hitchin, Complex manifolds and Einstein's equations, in Twistor Geometry and Non-linear Systems, Lecture Notes in Mathematics 970 (1982) Springer-Verlag, 73-99. MR 84i:32041
  • 7. B. Klingler, Structures affines et projectives sur les surfaces complexes, Ann. Inst. Fourier 48 (1998), 441-477. MR 99c:32038
  • 8. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry I, Interscience, John Wiley (1963). MR 27:2945
  • 9. S. Kobayashi, T. Ochiai, Holomorphic projective structures on compact complex surfaces, Math. Annalen, 249 (1980), 75-94. MR 81g:32021
  • 10. K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifols of complex manifolds, Ann. Math., II Ser.,75 (1962), 146-162. MR 24:A3665b
  • 11. K. Kodaira, On compact complex analytic spaces I, Ann. Math. 71 (1960), 111-152; II, ibid. 77 (1963), 563-626; III, ibid. 78 (1963), 1-40. MR 24:A2396; MR 32:1730
  • 12. C.R. LeBrun, $\mathcal{H}$-Space with a cosmological constant, Proceedings of the Royal Society of London, Series A 380 (1982), 171-185. MR 83d:83019
  • 13. C.R. LeBrun, Twistors, Ambitwistors, and Conformal Gravity, in Twistors in mathematics and physics, London Math. Soc. Lecture Note Ser., 156 (1990), Cambridge Univ. Press, Cambridge, 71-86. MR 92a:83042
  • 14. C.R. LeBrun, Foliated CR Manifolds, Journal of Differential Geometry 22 (1985), 81-96. MR 88e:32028
  • 15. C.R. LeBrun, Spaces of Complex Null Geodesics in Complex-Riemannian Geometry, Trans. Am. Math. Soc. 278 (1983), 209-231. MR 84e:32023
  • 16. R. Penrose, The Structure of Space-Time, in Battelle Rencontres (ed. C. DeWitt & J. Wheeler), New York: Benjamin (1968), 121-235. MR 38:955
  • 17. R. Penrose, Nonlinear Gravitons and Curved Twistor Theory, General Relativity and Gravitation 7 (1976), 31-52. MR 55:11905
  • 18. S. Salamon, Topics in four-dimensional Riemannian geometry, in Geometry Seminar ``Luigi Bianchi'', Lecture Notes in Mathematics 1022 (1982) Springer-Verlag, 36-124. MR 85i:53002
  • 19. I.M. Singer, J.A. Thorpe, The curvature of 4-dimensional Einstein spaces, in Global Analysis, in honor of Kodaira, Princeton Math. Series 29 (1969), Princeton Univ. Press, 355-365. MR 41:959
  • 20. T.Y. Thomas, A projective theory of affinely connected manifolds, Math. Zeit. 25 (1926), 723-733.
  • 21. J.H.C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math. 3 (1932), 33-42.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C21, 53A30, 32Qxx

Retrieve articles in all journals with MSC (2000): 53C21, 53A30, 32Qxx

Additional Information

Florin Alexandru Belgun
Affiliation: Centre de Mathématiques, UMR 7640 CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France
Address at time of publication: Mathematisches Institut, Augustusplatz 10/11, 04109 Leipzig, Germany

Received by editor(s): February 27, 2000
Published electronically: October 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society