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Twistor spaces for real four-dimensional Lorentzian manifolds

  • Yoshinori Machida (a1) and Hajime Sato (a2)

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It is R. Penrose who constructed the twistor theory which gives a correspondence between complex space-times and 3-dimensional complex manifolds called twistor spaces. He and his colleagues investigated conformally invariant equations (e.g. massless field equations, self-dual Yang-Mills equations) on the space-time by transforming them into objects in complex analytical geometry. See e.g. Penrose-Ward [P-W] or Ward-Wells [W-W]. After that, Atiyah-Hitchin-Singer ([A-H-S], cf. [Fr]) constructed the twistor spaces corresponding to real 4-dimensional Riemannian manifolds. Their construction as well as that of Penrose is mainly effective under the condition of the self-duality. In this paper we will construct twistor spaces more geometrically from real 4-dimensional Lorentzian manifolds under a suitable curvature condition.

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References

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[A-H-S] Atiyah, M.F., Hitchin, NJ. and Singer, I.M., Self-duality in four-dimensional Riemannian geometry, Proc. R. S. Lond., A 362 (1978), 425461.
[BB-O] Bergery, L. Berard and Ochiai, T., On some generalization of the construction of twistor spaces, Global Riemannian geometry (Willmore, T. J. and Hitchin, NJ., eds.), Ellis Horwood, Chichester, 1984, pp. 5259.
[Be] Besse, A. L., Einstein manifolds, Springer-Verlag, Berlin-Heidelberg-New York, 1987.
[Fr] Friedrich, T., (ed.), Self-dual Riemannian geometry and instantons, Teubner, Leipzig, 1981.
[Ma] Manin, Yu. I., Mathematics and physics, Birkhäuser, Boston-Basel, 1981.
[O’N] O’Neill, B., Semi-Riemannian geometry, Academic Press, New York, 1983.
[P-W] Penrose, R. and Ward, R.S., Twistors for flat and curved space-time, General relativity and gravitation vol. 2 (Held, A., ed.), Plenum Press, New York-London, 1980, pp. 283328.
[S-Y] Sato, H. and Yamaguchi, K., Lie contact manifolds, Geometry of manifolds (Shiohama, K., ed.), Academic Press, New York, 1989, pp. 191238.
[W-W] Ward, R. S. and Wells, R.O. Jr., Twistor geometry and field theory, Cambridge Univ. Press, Cambridge, 1990.
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Twistor spaces for real four-dimensional Lorentzian manifolds

  • Yoshinori Machida (a1) and Hajime Sato (a2)

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