
#1
Mar112, 04:45 PM

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Ok, so I don't have much of an intuition for frame bundles, so I have some basic questions.
A frame bundle over a manifold M is a principle bundle who's fibers are the sets of ordered bases for the vector fields on M right. 1) This means that any point in the fiber (say, over a point m in M) is literally a set of ordered bases right? 2) Since the frame bundle is a principle fiber bundle, each fiber has to be isomorphic to its structure group, which I gather is GL(n,R) right. So, a frame bundle over a 4d manifold is 16 dimensional? Why so many dimensions? 3) What do these dimensions mean? Going "in a different direction" in this fiber corresponds to doing what to my ordered bases? This stuff seems really confusing to me... 



#2
Mar112, 06:15 PM

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I will try to clear things up a little for you, if that is possible.
[tex]w=\sum_i a_iv_i[/tex] In particular, if (w_{j}) is another basis of V, then this corresponds to n² numbers (n numbers for each w_{j}), which you can arrange in a matrix by declaring that the first row is to be made up of the n coordinates of w_{1} and so on. Then the fact that these (wj) are linearly independant is equivalent to saying that the matrix thus constructed has nonvanishing determinant. That is, it is a matrix in GL(n,R). So you see that there is a (non canonical) bijective correspondance between the frames F(V) of V and GL(n,R). Use that to transfer the smooth structure of GL to F(V). You can verify that this will be independant of the choice of bijection and so puts a welldefined canonical smooth structure on F(V) such that given any choice of basis in V resulting as above in a bijection F(V)<>GL, this bijection is a diffeomorphism. So that is how each fiber of the frame bundle is (non canonically) diffeomorphic to GL(n,R). 



#3
Mar212, 07:37 AM

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In particular, now we see "why" there are 16 dimensions to F(T_{m}M) when dim(M)=4. Namely, once a basis (v_{i}) for T_{m}M has been fixed, there is one degree of freedom/dimension corresponding to each of the 4²=16 ways to choose each coordinate a_{ji} of each vector w_{j} of another basis (w_{j}=Ʃa_{ji}v_{i}).




#4
Mar212, 03:17 PM

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Frame bundle
Sorry, I got a little busy. Gimme a little bit of time to digest your post and then I'll respond with more questions. =D




#5
Mar212, 03:42 PM

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"More precise would be that the frame bundle over a manifold M is a principle GL(n,R)bundle who's fiber over p is the set of ordered bases for the tangent space of M at p. (where n=dim(M))" 



#6
Mar312, 01:24 AM

P: 2,045

Ok, I had some time to look at your post and digest, I get it now thanks. I had a wrong conception that it would only take 4 numbers to specify a basis in 4D because you only need 4 vectors, but I forgot that each vector actually consists of 4 numbers.
Thanks for the help. =] 



#8
Mar812, 04:18 PM

P: 2,045

As a follow up, frame bundles, and principle bundles in general, are not vector bundles right? Since GL(n,R) doesn't seem to form a vector space. I'm a little confused because this book I'm reading is discussing the "distribution on P of vertical vectors...", where P is a principle bundle, but it seems to me that a principle bundle is not a vector bundle so where are the vectors?




#9
Mar812, 05:07 PM

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in the tangent to P!
Let [itex]\pi[/itex]:P>B be a principal Gbundle. Then consider the derivative [itex]\pi_*[/itex]:TP>TB and for each point p in P, form the subspace V_p:=Ker([itex]\pi_{*,p}[/itex])[itex]\subset T_pP[/itex]. Since p is a submersion, this has dimension dim(G) and corresponds to all the directions (tangent vectors) along the fiber through p. According to the standard picture of fiber bundles as a bunch of G's lined up vertically against a horizontally drawn base space, V_p is called the vertical space at p since it is tangent to the fibers. The collection of all the V_{p}'s form a subbundle (aka a tangent distribution!) of TP called the vertical subbundle V. By the way... a connexion in a principal Gbundle is, by definition, a choice of a subbundle H complementary to V which we call a horizontal subbundle. Such an H is isomorphic to TM via [itex]\pi_*[/itex] and so we think of it as giving the direction "along M" in P. This allows one to define a notion of parallel transport: given a point p in P and a path c(t) in M, to parallelly transport p along c, first lift the velocity vector field v(t) of c up to H via [itex]\pi_*[/itex]. Then use this to lift c(t) up to the unique path c'(t) in P such that a) c'(0)=p b) c' is an integral curve of the lifted velocity field Then by definition, c'(t) is the parallell translate of p along c. Hence, the name "connexion" is justified. And of course, when the bundle is a vector bundle, it can be shown that this definition of connecxon is equivalent to the more common one in terms of specifying an operator on sections [itex]\nabla[/itex]. 



#10
Mar812, 06:06 PM

P: 2,045

Wow, you explain it a lot better than this book I got haha.
Can you explain a little more your last sentence regarding "when the bundle is a vector bundle..."? Can vector bundles be principle bundles? This seems odd to me because a vector space needs the 0 vector, but the structure group which is usually GL(n,R) (or some subset thereof) for vector bundles cannot contain the 0 matrix. Or does the above construction of a connexion work even for vector bundles? 



#11
Mar812, 07:47 PM

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#12
Mar812, 11:54 PM

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Interesting, because I assumed that the connection on the tangent bundles we are used to would arise out of some kind of induced connexion on the principle bundle it is associated with. hmm.
I'm only "interested" in principle bundles in so far as that's what the book I'm reading seems to be concerned with. I had assumed that that was so because it "leads" to the regular connection due to some kind of inducement. EDIT: The book I'm using (Bishop and Crittenden) seems to always suggest that V and H are in P not TP, it'd be pretty weird if they were just blatantly wrong, is there some subtleties there? Or are they just wrong and it should be that V and H live in TP? 



#13
Mar912, 08:12 AM

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Well, on page 74, they say V is a distribution on P. By definition, this means it is a vector subbundle of TP. Also, note that they are using the notation P_{p} to mean T_{p}P. Also, they use the notation [itex]d\pi[/itex] while I use [itex]\pi_*[/itex]. Other than that, their definition of V is exactly the same as mine. They put
[tex]V_p:=\{t\in T_pP  \pi_*(t)=0\}=\mathrm{Ker}(\pi_{*,p})[/tex] 



#14
Mar912, 02:29 PM

P: 2,045

Oh, so it's just notation...ok. @_@
You've been of great help thanks. =] 



#15
Mar912, 05:23 PM

P: 2,045

Some follow up questions. This formulation of parallel transport is pretty foreign to me.
Let's restrict ourselves to vector bundles since that is ultimately my interest as a physicist. You said that with this construction, I can get a unique curve c'(t) which is the horizontal lift of c(t) given c'(0)=p and c'(t) being the integral curve of the horizontally lifted tangent vectors to c(t). Does this uniqueness define a unique way for me to define parallel transport? The way that I've always learned about parallel transport and connections is that there is some degree of arbitrariness to them. If I am not given a metric structure on my manifold, then I can have connections with torsion or without torsion, for example, and there was no way for me to a priori define which matrices in GL(n,R) to act on my tangent spaces to transform my vectors as they move along a path. This seems to clash with the uniqueness of the horizontal lift. Does this uniqueness imply that given a vector bundle E over a manifold M, I have only 1 unique way to define parallel transport of a vector in E along a curve in M? 



#16
Mar912, 05:50 PM

P: 2,045

Another follow up question. This book is now talking about how the connexion in the principle bundle will "induce" in some special way a connexion in its associated fiber bundle. Is this construction the same as one would get if one simply redid the construction, as you said was possible, on this associated fiber bundle without first doing the construction on the principle fiber bundle?
For example, if I have a connexion on my frame bundle to M, it should induce a connexion on my tangent bundle TM. If I had, instead, just started with TM as a vector bundle without any regard to the fact that it was associated with the frame bundle and did the construction of vertical distributions, etc., would I have arrived at the same result? 



#17
Mar912, 06:09 PM

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#18
Mar912, 06:27 PM

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Hmmm, can you elaborate a bit on that?
I can see that if H is arbitrary, then my definition of what is a horizontal lift is aribtrary, and that would mean arbitrariness in parallel transport. But I can't see how H would be arbitrary, doesn't it have to encompass every "direction" that V doesn't in order to be a compliment to it? I mean dim(V)=dim(G) and dim(H)=dim(M) and dim(P)=dim(G)+dim(M) right, so I'm not seeing how it can be arbitrary. Perhaps a simple example would help? Or perhaps one could try to explain, for example, if my manifold is now endowed with a metric, how that limits my choice of H to be compatible somehow. 


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