Closed surface pairs with maximal local rolling symmetries

Abstract

In this dissertation, we will explore the relation between the geometry of rolling surfaces and the adjoint split-real form of the exceptional simple Lie group of rank 2. For every pair of Riemannian surfaces whose Gaussian curvatures never coincide, it is known that there is a (2, 3, 5)-distribution on a 5-dimensional configuration space whose elements describe “rolling (without slipping or twisting)” one surface along the other, and (2, 3, 5)-distributions happen to correspond to a specific type of parabolic Cartan geometry whose model group is given by this exceptional simple Lie group. Determining which pairs of surfaces admit “rolling distributions” with maximal local symmetry is a problem of particular interest, and our main result shows that the only such pairs of closed surfaces are those with constant Gaussian curvature in the ratio of either 1:9 or 9:1. Additionally, we present a novel, visual explanation for the well-known 1:3 ratio of radii required for the space of local symmetries for a pair of spheres rolling along each other to have maximal dimension.