𝐺₂ and the rolling ball

Abstract

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a longstanding program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra g2 acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms, use the dot product and cross product of split octonions to describe the G2 incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.On the other hand, we can recover the imaginary split octonions from this variant of the 1:3 rolling ball via geometric quantization.Using a spinorial variant of the rolling ball system may seem odd, but it is essential if we want to see the hidden G 2 symmetry. In fact, we must consider three variants of the rolling ball system. The first is the ordinary rolling ball, which has configuration space S 2 × SO(3). This never has G 2 symmetry. We thus pass to the double cover, S 2 × SU(2), where such symmetry is possible. We can view this as the configuration space of a 'rolling spinor': a rolling ball that does not come back to its original orientation after one full rotation, but only after two. To connect this system with the split octonions, it pays to go a step further, and identify antipodal points of the fixed sphere S 2 . This gives RP 2 × SU(2), which is the configuration space of a spinor rolling on a projective plane.This last space explains why the 1:3 ratio of radii is so special. As mentioned, a spinor comes back to its original state only after two full turns. On the other hand, a point moving on the projective plane comes back to its original position after going halfway around the double cover S 2 . Consider a ball rolling without slipping or twisting on a larger fixed ball. What must the ratio of their radii be so that the rolling ball makes two full turns as it rolls halfway around the fixed one? Or put another way: what must the ratio be so that the rolling ball makes four full turns as it rolls once around the fixed one? The answer is 1:3.At first glance this may seem surprising. Isn't the correct answer 1:4? No: a ball of radius 1 turns R + 1 times as it rolls once around a fixed ball of radius R. One can check this when R = 1 using two coins of the same kind. As one rolls all the way about the other without slipping, it makes two full turns. Similarly, in our 365 1 4 day year, the Earth actually turns 366 1 4 times. This is why the sidereal day, the day as judged by the position of the stars, is slightly shorter than the ordinary solar day. The Earth is not rolling without slipping on some imaginary sphere. However, just as with the rolling ball, it makes an 'extra turn' by completing one full revolution ...

Singularities of tangent surfaces in Cartan’s split $G_2$-geometry

“…In this paper we show the following classification result of singularities: 3, 4, 5, 7), (2, 3, 4, 5, 7)), III : ( (2,3,5,7,8), (1,3,5,7,8)). …”

“…Since we work over R, we give a proof that G 2 acts transitively on Y, X and Z to make sure: First we remark that Y is a connected 5-dimensional manifold. In fact, the inner product on V is of index (3,4) and we see that Y is diffeomorphic to…”

“…Note that the same construction has been analyzed and utilized in the problem of "rolling balls" ( [1][4] [3]). From Theorems 1.1, 1.2 and 1.3, we observe that the tangent varieties for G 2 case project to that for C 2 and to that for A 2 .…”

“…We set G = G(C 2 ) = Sp(R 4 ), the symplectic group. Note that G is isomorphic the spinor group Spin(R I : ((1, 2, 3), (1, 2, 3)), II : ( (1,3,4), (2,3,4)), III : ( (2,3,5), (1,3,5)). …”

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

“…This family includes appropriate complexifications of the rolling distributions of two real surfaces of different nonzero constant curvature whose curvatures do not have ratio 9:1; for that ratio the rolling distribution is flat. See [1,7,8,10,32] for much more. -(Section 51) There is a single distribution with infinitesimal symmetry algebra isomorphic to so(3, C)⊕(so(2, C) C 2 ).…”