Local equivalence of some maximally symmetric $(2,3,5)$-distributions III: An-Nurowski surface rolling on a plane

Abstract

We consider the maximally symmetric $(2,3,5)$-distribution given by the An-Nurowski circle twistor bundle over the product of an An-Nurowski surface and the plane. This circle twistor distribution encodes the configuration space of an An-Nurowski surface rolling without slipping or twisting on the plane. We calculate the vector fields associated to this maximally symmetric $(2,3,5)$-distribution that define a split ${\mathfrak{g}}_2$ Lie algebra and by projection we obtain an action of $SL(3,\mathbb{R})$ on the submanifold ${\mathbb{R}}^2\times {\mathbb{S}}^1$ (the configuration space without a surface).

16 pages