Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 075, 21 pages      arXiv:1306.3195      https://doi.org/10.3842/SIGMA.2013.075

Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

Mikhail B. Sheftel a and Andrei A. Malykh b
a) Department of Physics, Boğaziçi University 34342 Bebek, Istanbul, Turkey
b) Department of Numerical Modelling, Russian State Hydrometeorlogical University, 98 Malookhtinsky Ave., 195196 St. Petersburg, Russia

Received June 14, 2013, in final form November 19, 2013; Published online November 27, 2013

Abstract
We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.

Key words: Monge-Ampère equation; Boyer-Finley equation; partner symmetries; symmetry reduction; non-invariant solutions; group foliation; anti-self-dual gravity; Ricci-flat metric.

pdf (439 kb)   tex (30 kb)

References

  1. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
  2. Boyer C.P., Finley III J.D., Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982), 1126-1130.
  3. Calderbank D.M.J., Tod P., Einstein metrics, hypercomplex structures and the Toda field equation, Differential Geom. Appl. 14 (2001), 199-208, math.DG/9911121.
  4. Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
  5. Dunajski M., Gutowski J., Sabra W., Enhanced Euclidean supersymmetry, 11D supergravity and SU(∞) Toda equation, J. High Energy Phys. 2013 (2013), no. 10, 089, 20 pages, arXiv:1301.1896.
  6. Dunajski M., Mason L.J., Twistor theory of hyper-Kähler metrics with hidden symmetries, J. Math. Phys. 44 (2003), 3430-3454, math.DG/0301171.
  7. Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), 213-393.
  8. Hitchin N., Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435-441.
  9. Lie S., Über Differentialinvarianten, Math. Ann. 24 (1884), 537-578.
  10. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries of the complex Monge-Ampère equation yield hyper-Kähler metrics without continuous symmetries, J. Phys. A: Math. Gen. 36 (2003), 10023-10037, math-ph/0305037.
  11. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries and non-invariant solutions of four-dimensional heavenly equations, J. Phys. A: Math. Gen. 37 (2004), 7527-7545, math-ph/0403020.
  12. Malykh A.A., Nutku Y., Sheftel M.B., Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics, J. Phys. A: Math. Theor. 40 (2007), 9371-9386, arXiv:0704.3335.
  13. Malykh A.A., Sheftel M.B., Recursions of symmetry orbits and reduction without reduction, SIGMA 7 (2011), 043, 13 pages, arXiv:1005.0153.
  14. Martina L., Sheftel M.B., Winternitz P., Group foliation and non-invariant solutions of the heavenly equation, J. Phys. A: Math. Gen. 34 (2001), 9243-9263, math-ph/0108004.
  15. Mason L.J., Newman E.T., A connection between the Einstein and Yang-Mills equations, Comm. Math. Phys. 121 (1989), 659-668.
  16. Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs. New Series, Vol. 15, The Clarendon Press, Oxford University Press, New York, 1996.
  17. Nutku Y., Sheftel M.B., A family of heavenly metrics, gr-qc/0105088.
  18. Nutku Y., Sheftel M.B., Differential invariants and group foliation for the complex Monge-Ampère equation, J. Phys. A: Math. Gen. 34 (2001), 137-156.
  19. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  20. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
  21. Plebañski J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
  22. Schrüfer E., EXCALC: A differential geometry package, in REDUCE, User's and Contributed Packages Manual, Version 3.8, Editor A.C. Hearn, Santa Monica, CA, 2003, 333-343.
  23. Sheftel M.B., Method of group foliation and non-invariant solutions of partial differential equations. Example: the heavenly equation, Eur. Phys. J. B 29 (2002), 203-206.
  24. Sheftel M.B., Malykh A.A., Lift of invariant to non-invariant solutions of complex Monge-Ampère equations, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 385-395, arXiv:0802.1463.
  25. Sheftel M.B., Malykh A.A., On classification of second-order PDEs possessing partner symmetries, J. Phys. A: Math. Theor. 42 (2009), 395202, 20 pages, arXiv:0904.2909.
  26. Tod K.P., Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Classical Quantum Gravity 12 (1995), 1535-1547, gr-qc/0105088.
  27. Vessiot E., Sur l'intégration des systèmes différentiels qui admettent des groupes continus de transformations, Acta Math. 28 (1904), 307-349.
  28. Yau S.T., Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977), 1798-1799.


Previous article  Next article   Contents of Volume 9 (2013)