hmm, maybe i'm finally beginning to understand the relationship between "cartan geometries" and a generalized sort of "rolling" that derek tried to explain to me...

when derek explained to me the definition of "cartan geometry modeled on the homogeneous space g/h", i found it a useful conceptual simplification to reformulate it as what i call an "h-flat g-torsoroid". a g-torsoroid is similar to a g-torsor in that it's a manifold of the same dimension as the lie group g and equipped with a "maurer-cartan t_1(g)-valued 1-form", except that the 1-form isn't required to be flat. so infinitesimally the torsoroid has a g-valued "distance" or "displacement" function, but unlike a torsor the displacements don't satisfy the triangle equality (aka the homomorphism law for the principal action of g). being "h-flat" though makes it possible to "mod out by h", giving a quotient space (that i'll call "x") that resembles g/h but is wobblier.

like i say, i found that to be a useful simplification that helped me understand what a cartan geometry really is. on the other hand though it may have delayed my understanding of what derek meant by "generalized rolling". in the standard approach to cartan geometries the quotient space x is the central focus, and the torsoroid arises as its "h-frame" bundle. my approach was to jettison x because i could reconstruct it from its h-frame bundle, but because of that i also jettisoned a convenient piece of structure on x, namely the t_1(g)-valued 1-form that it manages to inherit from its h-frame bundle.

so i think that derek's point is that you can interpret this 1-form as a connection on the trivial g-torsor bundle over x, and take the "horizontal distribution" of that connection and interpret it as a "generalized rolling distribution" on the total space of that trivial bundle. and this really does generalize the classical sort of "rolling distribution" that i've been studying in connection with g2 and the octonions and the rolling ball.

i think that part of what confused me here all along is that i didn't manage to distinguish clearly enough between two ways of getting an ehresmann connection from a cartan connection: you can get one on the h-frame bundle of x (and this might be the way that allegedly led to ehresmann's invention of ehresmann connections?), but pehaps more importantly, you can get one on the trivial g-torsor bundle over x, and this is the way that gives a "generalized rolling distribution" as its horizontal distribution.

derek sometimes mentions the (actually pretty visually intuitive) idea that "rolling" of one surface on another doesn't require that _either_ surface be homogeneous. i'm not sure to what extent derek or others may have worked out this idea, but anyway it suggests viewing the configuration space of the "generalized rolling" system in a more symmetrical way wrt the roles played by the homogeneous surface and the non-homogeneous one. there's the "point of contact" on each of the two surfaces, and there's the h-torsor of structure-preserving isomorphisms between the tangent spaces of the two contact points... ??...

it's funny that i didn't understand this stuff until i decided that i wanted to try to generalize the "rolling" interpretation of the g2 incidence geometry to other simple lie groups...