Astronomy
Sci Advisor
PF Gold
P: 23,215

The following came up in Lethe's forms thread, but I'll separate it out and add to it to use as a footnote in Gravitivity threadgroup action, differential forms, bundles all come up in a classical treatment of gravity.
A differential geometry book (Bishop and Crittenden) that I happened to pick up defines a "principal bundle" as a triple (P, G, M) where P and M are smooth manifolds and G is a Lie group
(1) G acts freely on P, GxP > P (they choose a right action, it could be left)
(2) M is the quotient space of P mod equivalence by G
the projection map is ξ:P > M
G acts transitively on the fiber ξ^{1}(m) over any point m in M
(3) P is locally trivial. that means that around any point m in M there is a neighborhood U ( picture a disk) such that the part of P that is over U ( picture a cylinder over the disk), namely
ξ^{1}(U), is diffeomorphic to the cartesian product
U x G ( picture a second cylinder U x G, with U a disk and G a vertical line).
The diffeomorphism ξ^{1}(U) > UxG takes a point p to (ξ(p), F_{U}(p)) and this map F_{U}: ξ^{1}(U) > G satisfies an equation F_{U}(gp) = gF_{U}(p).
The equation says you can do the group action first and then do F, or you can do F first and then do the group action, same result. In other words F "commutes with the group action."

Examplethe "bundle of bases":
Let M be a smooth manifold and B(M) be the set of N+1tuples
(m, e1,...,eN) where m ε M and {e1,...,eN] is a basis of the tangent space at m.
Let ξ: B(M) > M be the projection ξ(m, e1,...,eN) = m
GL(N, R) acts on the left on B(M), lets define the action,
g ε GL(N, R) can be viewed as a matrix (g_{ij})
so just let its rows specify linear combinations of {e1,...,eN}
and you have a new basis {...Σg_{ij}ej...)
I see no ambiguity here.
(m, e1,...,eN) > (m, Σg_{1j}ej,...,Σg_{Nj}ej)
So far there has been no arbitrary choice. R^{N} has a natural basis (1,0,...), (0,1,0,...) and so on and the matrix is with respect to that but we can apply it to the {e} basis of the tangent space
Now to make this a principal bundle we need to be able to coordinatize it and to say, for any coordinate patch neighborhood U of a point m, what the locally trivializing maps
F_{U} are.
F_{U}: ξ^{1}(U) > G
So suppose we have coordinates x1,..,xN defined in a patch U around a point m
and suppose the point m' is in the neighborhood U
and (m', f1,...,fN) is in the fiber ξ^{1}(m') over m'
Now {f1,...,fN} is a basis of the tangent space
we have to get a NxN matrix in the group G, this being what
F_{U} does. Well define
F_{U}(m', f1,...,fN) = (g_{ij}) = (dx_{j}f_{i})
this provides a coordinate system for ξ^{1}(U)
the system says, map it by (ξ, F_{U}) into UxG
and then use the x coords in U and the standard R^{N} basis coords to give the matrix. This way you get N + NxN numbers. But it doesnt even look bad. No sweat. Bundles must be good language.

Now I am thinking that if we just took Minkowski space in place of
R^{N} and the Lorentz group in standard matrix form for our G in place of GL(N,R), we could have a bundleofbases as a principal Gbundle over a 4D manifold M. The bundle would consist of 5tuples (m, e0,...,e3) where the e0,...,e3 are a basis of the TANGENT space...there are some details to fill in and maybe this will not work as is. I would like to see if this model can apply to streamline what rovelli is saying. Be back later.

