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

Matthew Randall

Symmetry, integrability and geometry: methods and applications, Tome 12 (2016)

Résumé

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.

Détail

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

Bibliographie

[1] Ablowitz M. J., Chakravarty S., Halburd R., “The generalized Chazy equation and Schwarzian triangle functions”, Asian J. Math., 2 (1998), 619–624 | DOI | MR | Zbl

[2] Ablowitz M. J., Chakravarty S., Halburd R., “The generalized Chazy equation from the self-duality equations”, Stud. Appl. Math., 103 (1999), 75–88 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[4] Ablowitz M. J., Fokas A. S., Complex variables: introduction and applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[5] An D., Nurowski P., “Twistor space for rolling bodies”, Comm. Math. Phys., 326 (2014), 393–414, arXiv: 1210.3536 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | MR | Zbl

[10] Chakravarty S., Ablowitz M. J., “Parameterizations of the Chazy equation”, Stud. Appl. Math., 124 (2010), 105–135, arXiv: 0902.3468 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[14] Clarkson P. A., Olver P. J., “Symmetry and the Chazy equation”, J. Differential Equations, 124 (1996), 225–246 | DOI | MR | Zbl

[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, arXiv: math.DG/0102225 | DOI | MR | Zbl

[16] Dunajski M., Sokolov V., “On the 7th order ODE with submaximal symmetry”, J. Geom. Phys., 61 (2011), 1258–1262, arXiv: 1002.1620 | DOI | MR | Zbl

[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 | MR

[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, arXiv: math.DG/0406400 | DOI | MR | Zbl

[20] Nurowski P., Notes, , 2012 http://www.mat.univie.ac.at/c̃ap/esiprog/Nurowski.pdf

[21] Olver P. J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[22] Tod K. P., “Metrics with SD Weyl tensor from Painlevé-VI”, Twistor Newsletter, 35 (1992), 5–7 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, arXiv: math.CA/0408269 | DOI | MR | Zbl

[24] Willse T., “Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy”, Differential Geom. Appl., 33, suppl. (2014), 81–111, arXiv: 1302.7163 | DOI | MR | Zbl

[25] Willse T., An explicit ambient metric of holonomy $G_2^*$, arXiv: 1411.7172