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The singularities of H-space

  • K. P. Tod (a1)

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The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.

In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.

Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.

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References

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(1)Douady, A.Ann. Inst. Fourier. Grenoble 16 (1966), 1.
(2)Eastwood, M. G. and Tod, K. P.Edth – A differential operator on the sphere. Proc. Cambridge Philos. Soc. 92 (1982), 317330.
(3)Eells, J.Elliptic operators on manifolds in complex analysis and its applications, vol. I (IAEA, Vienna, 1976).
(4)Hansen, R., Newman, E. T., Penrose, R. and Tod, K. P.Proc. Roy. Soc. London, Ser. A 363 (1978), 445.
(5)Kodaira, K.Ann. Math. 75 (1962), 146. Am. J. Math. 85 (1963), 79.
(6)Newman, E. T.Gen. Eel. Grav. 7 (1976), 107.
(7)Palais, R. S. (ed.). Seminar on the Atiyah-Singer index theorem, Ann. Study 57 (Princeton University Press, 1966).
(8)Penrose, R. and MacCallum, M. A. H.Phys. Rep. 6c. (1973), 241.
(9)Penrose, R.Gen. Bel. Grav. 7 (1976), 31.
(10)Sparling, G. A. J. and Tod, K. P.J. Math. Phys. 22 (1981), 331.

The singularities of H-space

  • K. P. Tod (a1)

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