G_2 and the "Rolling Distribution"

Abstract

Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphere. But if the ratio of radii of the spheres is 1:3 and if the distribution is lifted to the universal cover S^3 \times S^2 of M, then the symmetry group becomes much larger: the split real form of the Lie group G_2. This fact goes back to Cartan in a sense, and can be found in a paper by Bryant and Hsu. We prove this fact through two explicit constructions, relying on the theory of roots and weights for the Lie algebra of G_2, and on its 7-dimensional representation.