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marcus

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**Principal G-bundle and "bundle of bases" (footnote)**

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), 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."

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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 (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.

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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 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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