Einstein-Weyl geometry, the dKP equation and twistor theory

Journal of Geometry and Physics

Volumn 37, Issue 1-2, 2001, Pages 63-93

  Dunajski, Maciej ^a   Mason, Lionel J ^a   Tod, Paul ^a  

^a Applied Mathematics Mathematical Institute   (United Kingdom)

Author keywords

58F07; 81R25; DKP equation; Dynamical systems; Einstein Weyl geometry; General relativity; Integrable systems; Twistors

Indexed keywords

EID: 0001969881     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(00)00033-4     Document Type: Article

References (32)

1. Boyer C. A note on hyper-Hermitian four-manifolds. Proc. Am. Math. Soc. 102:1988;157-164.
2. D.M. Calderbank, H. Pedersen, Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier 50 (2000), in press.
3. D.M. Calderbank, K.P. Tod, Einstein metrics, hypercomplex structures and the Toda field equation, Diff. Geom. Appl. (2000), in press.
4. Cartan E. Sur une classe d'espaces de Weyl. Ann. Sci. Ecole Norm. Supp. 60:1943;1-16.
5. M. Dunajski, The nonlinear graviton as an integrable system, D.Phil. Thesis, Oxford, 1998.
6. Dunajski M. The twisted photon associated to hyper-Hermitian four manifolds. J. Geom. Phys. 30:1999;266-281.
7. M. Dunajski, K.P. Tod, Einstein-Weyl structures from hyper-Kähler metrics with conformal Killing vectors, Diff. Geom. Appl. (1999), submitted for publication.
8. Finley J.D., Plebański J.F. The classification of all. H spaces admitting a Killing vector J. Math. Phys. 20:1979;1938.
9. Gauduchon P., Tod K.P. Hyper-Hermitian metrics with symmetry. J. Geom. Phys. 25:1998;291-304.
10. Gibbons G.W., Hawking S.W. Gravitational multi-instantons. Phys. Lett. B. 78:1978;430-432.
11. J. Gibbons, The Zabolotskaya-Khokhlov equation and the inverse scattering problem of classical mechanics, in: S. Takeno (Ed.), Dynamical Problems in Soliton Theory, 1985.
12. Gibbons J., Kodama Y. A method for solving the dispersionless KP hierarchy and its exact solutions, II. Phys. Lett. A. 135:1989;167-170.
13. Grant J.D.E., Strachan I.A.B. Hypercomplex integrable systems. Nonlinearity. 12:1999;1247-1261.
14. P. Guha, K. Takasaki, Dispersionless hierarchies, Hamilton-Jacobi theory and twistor correspondences, 1997, solv-int/9705013.
15. N. Hitchin, Complex manifolds and Einstein's equations, in: H.D. Doebner, T.D. Palev (Eds.), Twistor Geometry and Non-linear Systems, Lecture Notes in Mathematics, Vol. 970, Springer, Berlin, 1982.
16. Jones P., Tod K.P. Minitwistor spaces and Einstein-Weyl spaces. Classical Quantum Gravity. 2:1985;565-577.
17. Kodama Y. A method for solving the dispersionless KP equation and its exact solutions. Phys. Lett. A. 129:1988;223-226.
18. Kodaira K. On stability of compact submanifolds of complex manifolds. Am. J. Math. 85:1963;79-94.
19. 2 J. Diff. Geom. 34:1991;233-253.
20. L.J. Mason, Generalized twistor correspondences, d -bar problems, and the KP equations, in: S. Huggett (Ed.), Twistor Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 169, Marcel Dekker, New York.
21. Mason L.J., Newman E.T. A connection between Einstein and Yang-Mills equations. Commun. Math. Phys. 121:1989;659-668.
22. L.J. Mason, N.M.J. Woodhouse, Integrability, self-duality, and twistor theory, LMS Monographs New Series, Vol. 15, Oxford Univ. Press, Oxford, 1996.
23. Penrose R. Nonlinear gravitons and curved twistor theory. Gen. Relativity Gravitation. 7:1976;31-52.
24. R. Penrose, Rindler, Spinors and Space-Time, Vols. 1 and 2, Cambridge Univ. Press, Cambridge, 1986.
25. Pedersen H., Tod K.P. Three-dimensional Einstein-Weyl geometry. Adv. Math. 97:1993;74-109.
26. Plebański J.F. Some solutions of complex Einstein equations. J. Math. Phys. 16:1975;2395-2402.
27. Strachan I.A.B. The Moyal bracket and the dispersionless limit of the KP hierarchy. J. Phys. A. 28:1995;1967-1976.
28. Takasaki K., Takebe T. Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7:1995;743-808.
29. Tod K.P. Scalar-flat Kähler metrics from Painlevé-III. Classical Quantum Gravity. 12:1995;1535-1547.
30. Ward R.S. Self-dual space-times with cosmological constant. Commun. Math. Phys. 78:1980;1-17.
31. Ward R.S. Twistors in 2+1 dimensions. J. Math. Phys. 30:1989;2246-2251.
32. Ward R.S. Einstein-Weyl spaces and. SU(∞) Toda fields Classical Quantum Gravity. 7:1990;L95-L98.