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marcus
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Aug11-03, 04:24 PM
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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 thread---group 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), FU(p)) and this map FU: ξ-1(U) --> G satisfies an equation FU(gp) = gFU(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."
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Example---the "bundle of bases":

Let M be a smooth manifold and B(M) be the set of N+1-tuples
(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 (gij)
so just let its rows specify linear combinations of {e1,...,eN}
and you have a new basis {...Σgijej...)
I see no ambiguity here.
(m, e1,...,eN) --> (m, Σg1jej,...,ΣgNjej)

So far there has been no arbitrary choice. RN 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
FU are.
FU: 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
FU does. Well define
FU(m', f1,...,fN) = (gij) = (dxjfi)

this provides a coordinate system for ξ-1(U)
the system says, map it by (ξ, FU) into UxG
and then use the x coords in U and the standard RN 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.
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Now I am thinking that if we just took Minkowski space in place of
RN and the Lorentz group in standard matrix form for our G in place of GL(N,R), we could have a bundle-of-bases as a principal G-bundle over a 4D manifold M. The bundle would consist of 5-tuples (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.
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