|DATE||January 24 (Wed), 2018|
|TITLE||Flat (2,3,5)-distributions and Chazy's equations|
|ABSTRACT||In the theory of generic 2-plane fields on 5-manifolds, or (2,3,5)-distributions, the local equivalence problem was solved by ?lie Cartan who also constructed the fundamental curvature invariant. For these distributions described by a single function of the form F(q), the vanishing condition for the curvature invariant is given by a 6th order nonlinear ODE. Furthermore, Daniel An and Pawel Nurowski have shown that this ODE is the Legendre transform of the nonlinear ODE that appeared in Gottfried Noth's thesis in 1904.
We show that the 6th order ODE can be reduced to a 3rd order nonlinear ODE that is a generalised Chazy equation with Chazy parameter 2/3. The ODE in Noth's thesis can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal 3/2. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2,3,5)-distributions. This talk is based on work available at arXiv:1506.02473 and arXiv:1607.04961.