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

SIGMA 12 (2016), 029, 28 pages      arXiv:1506.02473      https://doi.org/10.3842/SIGMA.2016.029

Flat $(2,3,5)$-Distributions and Chazy's Equations

Matthew Randall
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Received September 23, 2015, in final form March 14, 2016; Published online March 18, 2016

Abstract
In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split $G_2$ as their group of symmetries.

Key words: generic rank two distribution in dimension five; conformal geometry; Chazy's equations.

pdf (507 kb)   tex (26 kb)

References

  1. Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation and Schwarzian triangle functions, Asian J. Math. 2 (1998), 619-624.
  2. Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation from the self-duality equations, Stud. Appl. Math. 103 (1999), 75-88.
  3. Ablowitz M.J., Chakravarty S., Halburd R.G., Integrable systems and reductions of the self-dual Yang-Mills equations, J. Math. Phys. 44 (2003), 3147-3173.
  4. Ablowitz M.J., Fokas A.S., Complex variables: introduction and applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.
  5. An D., Nurowski P., Twistor space for rolling bodies, Comm. Math. Phys. 326 (2014), 393-414, arXiv:1210.3536.
  6. An D., Nurowski P., Symmetric $(2,3,5)$ distributions, an interesting ODE of 7$^{\rm th}$ order and Plebański metric, arXiv:1302.1910.
  7. Atiyah M., Hitchin N., The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.
  8. Bor G., Hernández Lamoneda L., Nurowski P., The dancing metric, $G_2$-symmetry and projective rolling, arXiv:1506.00104.
  9. Cartan E., Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109-192.
  10. Chakravarty S., Ablowitz M.J., Parameterizations of the Chazy equation, Stud. Appl. Math. 124 (2010), 105-135, arXiv:0902.3468.
  11. Chakravarty S., Ablowitz M.J., Clarkson P.A., Reductions of self-dual Yang-Mills fields and classical systems, Phys. Rev. Lett. 65 (1990), 1085-1087.
  12. Chazy J., Sur les équations différentielles dont l'intégrale générale est uniforme et admet des singularités essentielles mobiles, C. R. Acad. Sci. Paris 149 (1910), 563-565.
  13. Chazy J., Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes, Acta Math. 34 (1911), 317-385.
  14. Clarkson P.A., Olver P.J., Symmetry and the Chazy equation, J. Differential Equations 124 (1996), 225-246.
  15. Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), 1205-1222, math.DG/0102225.
  16. Dunajski M., Sokolov V., On the 7th order ODE with submaximal symmetry, J. Geom. Phys. 61 (2011), 1258-1262, arXiv:1002.1620.
  17. Goursat É., Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3-142.
  18. Leistner T., Nurowski P., Sagerschnig K., New relations between $G_2$-geometries in dimensions 5 and 7, arXiv:1601.03979.
  19. Nurowski P., Differential equations and conformal structures, J. Geom. Phys. 55 (2005), 19-49, math.DG/0406400.
  20. Nurowski P., Notes, 2012, available at http://www.mat.univie.ac.at/~cap/esiprog/Nurowski.pdf.
  21. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  22. Tod K.P., Metrics with SD Weyl tensor from Painlevé-VI, Twistor Newsletter 35 (1992), 5-7, available at http://people.maths.ox.ac.uk/lmason/Tn/35/TN35-04.pdf.
  23. Vidūnas R., Algebraic transformations of Gauss hypergeometric functions, Funkcial. Ekvac. 52 (2009), 139-180, math.CA/0408269.
  24. Willse T., Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy, Differential Geom. Appl. 33 (2014), suppl., 81-111, arXiv:1302.7163.
  25. Willse T., An explicit ambient metric of holonomy $G_2^*$, arXiv:1411.7172.

Previous article  Next article   Contents of Volume 12 (2016)