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Complex manifolds and Einstein’s equations

  • N. J. Hitchin
Twistor Geometry
Part of the Lecture Notes in Mathematics book series (LNM, volume 970)

Abstract

We present a generalization of Penrose’s twistor theory based on the geometry of rational curves in complex manifolds. The analytical counterpart of this complex geometry consists, in the three simplest cases, of a system of differential equations closely connected with Einstein’s equations.

Keywords

Vector Bundle Normal Bundle Projective Line Conformal Structure Twistor Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    ATIYAH, M.F., HITCHIN, N.J. & SINGER, I.M. Self-duality in four dimensional Riemannian geometry, Proc. R. Soc. Lond. A362, (1978), 425–461ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    CARTAN, E. Sur une classe d’espaces de Weyl, Ann. Ec. Norm 60 (1943), 1–16MathSciNetMATHGoogle Scholar
  3. [3]
    DOUGLAS, J. The general geometry of paths, Ann. Math. 29 (1927), 143–168MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    EDDINGTON, A.S. The mathematical theory of relativity, Cambridge Univ. Press, Cambridge (1922)MATHGoogle Scholar
  5. [5]
    EISENHART, L.P. A treatise on the differential geometry of curves and surfaces, Ginn & Co., Boston (1909)MATHGoogle Scholar
  6. [6]
    HANSEN, R.O. NEWMAN, E.T. PENROSE, R. & TOD, K.P. The metric and curvature properties of H-space, Proc. R. Soc. Lond. A363 (1978), 445–468ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    HITCHIN, N.J. Polygons and gravitons, Math. Proc. Camb. Phil. Soc. 85 (1979), 465–476MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    KODAIRA, K. A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. Math. 84 (1962), 146–162MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    LANG, S. Introduction to differentiable manifolds, Wiley & Sons, New York (1962)MATHGoogle Scholar
  10. [10]
    LEBRUN, C. Spaces of complex geodesics and related structures, D.Phil. thesis, Oxford (1980)Google Scholar
  11. [11]
    NAKANO, S. On the inverse of monoidal transformation, Publ. RIMS Kyoto Univ. 6 (1970) 483–502MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    PENROSE, R. Non-linear gravitons and curved twistor theory, Gen. Relativ. Grav. 7 (1976) 31–52ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    WARD, R.S. On self-dual gauge fields, Phys. Lett. 61A (1977), 81–82ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    WARD, R.S. Self-dual spacetimes with a cosmological constant, Commun. Math. Phys. 78 (1980)Google Scholar

Copyright information

© Spring-Verlag 1982

Authors and Affiliations

  • N. J. Hitchin
    • 1
  1. 1.St. Catherine’s CollegeOxfordEngland

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