G₂ and the "Rolling Distribution"

2006/12/18 by Bor, Gil, Montgomery, Richard
#Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.math/0612469

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 SO₃ with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO₃'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 S³ × S² of M, then the symmetry group becomes much larger: the split real form of the Lie group G₂. 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 G₂, and on its 7-dimensional representation.

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