Exceptional rolling distributions for pairs of Riemannian surfaces

Determine which pairs of Riemannian surfaces admit rolling distributions with exceptional simple Lie group symmetries, thereby characterizing the broader class of surfaces that generalize the rolling-sphere 1:3-radius phenomenon.

Background

The paper studies rolling distributions associated with two round 2-spheres and explains why the radius ratio 1:3 or 3:1 produces the split real form of the exceptional simple Lie algebra of type Gâ‚‚. In the introduction, the authors situate this result within a broader research direction concerning rolling distributions for arbitrary pairs of Riemannian surfaces.

The cited work of An and Nurowski is described as addressing this broader, still-active question through conformal pseudo-Riemannian geometry, while recovering the spherical 1:3 ratio as a special case. The general classification of pairs of Riemannian surfaces yielding such exceptional symmetries is not resolved in the paper, whose results concern the pair of round 2-spheres.

References

Later, interested in the more general (and still active) question of when pairs of Riemannian surfaces can admit rolling distributions with such exceptional symmetries, An and Nurowski recovered the 1:3 ratio in using ideas from conformal pseudo-Riemannian geometry, also drawing inspiration from Cartan's work in .

— A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two  (2608.14442 - Erickson, 14 Aug 2026) in Section 1, Introduction