I don't have the energy to explain what this means - (see pages 87-88 in his book if you're curious - but in an 1910 paper Elie Cartan worked out an invariant of "(2,3,5) distributions" on a manifold, and when this invariant vanishes he showed the exceptional Lie group G2 acts on the manifold in question.
Montgomery shows that an example of this situation arises when we consider the phase space of a ball rolling on a table. This manifold is 5-dimensional: it takes 2 numbers to say where the center of the ball is, and 3 more to describe the rotational degrees of freedom of the ball. There is a 3-dimensional sub-bundle of the tangent bundle which describes the ways the ball can roll without slipping, and a 2-dimensional sub-bundle which describes the ways the ball can roll without slipping or "spinning", i.e. rotating about the vertical axis. Perhaps this enough for you to guess why this gives a "(2,3,5) distribution" in Cartan's terminology.
So, Montgomery poses the following open question:
Find an explicit geometric or physical description of the G2 action on the ball-table system.
This is interesting, but it's especially interesting to me because G2 is the automorphism group of the octonions! Answering Montgomery's question would for the first time give an example of the octonions appearing in real-world physics (as opposed to unproven theories of particle physics, like string theory).
But it's also puzzling, because the lowest-dimensional example I know of a manifold on which G2 acts nontrivially is 6-dimensional: the unit sphere in the imaginary octonions.
Of course, if you can answer Montgomery's question I'd be delighted. But I'd settle for any example of a 5-dimensional manifold on which G2 acts in a nontrivial way.
For the compact real form of G2, I can believe that SU(3) is the proper subgroup of maximal dimension. For the split form, or for G2(C), the group has maximal parabolics P of dimension 9, so codimension 14-9=5. Fulton and Harris discuss these maximal parabolics around page 391 of their book _Representation Theory: A First Course_. For one of them, G/P is a quadric surface in P^6, which is presumably the octonions o such that o^2=0 (that is, Tr(o)=Norm(o)=0), modulo scaling.Just to clarify: as you mentioned in your email to me, these "octonions" are really the split octonions if we're working with the split form of G2, or the complexified octonions (aka bioctonions) if we're working with the complex form G2(C).
This seems like a reasonable guess for John Baez's 5-manifold. Fulton-Harris indicates that the other G/P is harder to get one's hands on.Naively I'd guess that for the split octonions, the the equation Norm(o) = 0 amounts to something like
a^2 + b^2 + c^2 + d^2 - e^2 - f^2 - g^2 - h^2 = 0
where (a,b,c,d,e,f,g,h) are an 8-tuple of real numbers. The condition Tr(o) = 0 says the "real part" of our split octonion vanishes, and naively I'd guess this gives us an equation like
b^2 + c^2 + d^2 - e^2 - f^2 - g^2 - h^2 = 0
in 7 variables. If we projectivize this we get a manifold diffeomorphic to (S^2 x S^3) / Z_2 . It seems a bit odd that this is the same projective quadric we get from G/P where G = SO(4,4) and P is a maximal parabolic, so maybe I'm screwing up, or maybe this is one of those coincidences like how S^7 is a homogenous space of both SO(8) and the compact real form of G2.
Anyway, this space (S^2 x S^3) / Z_2 is a bit different, but not *drastically* different, from the phase space of a ball rolling on the plane - namely R^2 x (S^3 / Z_2). So, maybe we're pretty close!
I'll work through it more carefully sometime and straighten it out. Thanks a million!