Split signature 4-dimensional metrics, real totally null planes and (2,3,5) distributions

Pawel Nurowski

Abstract: This is a report on a joint work with Daniel An.

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null 2-planes of a given selfduality, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. If the metric g is not antiselfdual the distribution D is (2,3,5) in T(M), except at most 4 points in the fiber, which correspond to the algebraic degeneracy of the selfdual Weyl tensor. We argue, that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1-g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other without slipping and twisting, identifies the restricted velocity space for such a system, with the tautological distribution D on T(S1 x S2). Among the questions we address is the following one: which bodies can roll on each other without slipping and twisting in such a way that their restricted velocity space, identified with the distribution D on T(S1 x S2) has simple Lie group G2 as a group of its symmetries. Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find a 1-parameter family of surfaces S1 which when bounding a body rolling without slipping and twisting on a plane have D with the symmetry group G2. Although we've found the differential equations for the curvatures of S1 and S2 to define D with G2 symmetry, we are unable to solve them in full generality so far. We also consider the bundle T(M) and the corresponding distribution D for more general 4-manifolds then just a product of two surfaces. In the case of M being equipped with the Plebanski 2nd heavenly metric (not neccessarilly satisfying Plebanski equation), we are able to show that necessary condition for the symmetry of the distribution D on T(M) to be G2 is that the first derivative of the Plebanski's function hast to satisfy a certain 7th order ODE with remarkable properties. Under further assumption of high symmetries for the Plebanski function this equation is not only necessary but also sufficient for the symmetry G2 of D.