What Are Frame Bundles on Manifolds and Why Are They Important?

[quasar987]

No reference to algebraic dimension. :)

 

[Bacle2]

Sorry, Quasar,
I misunderestimated your question :). Nice post, btw.

 

[Bacle2]

Sorry again for my slowness, Quasar. I understand that Gl(n,R)=Det^-1(ℝ\{0}), which is open, yada, yada; I realized where my confusion lay.

Anyway, another dumb question; I am also trying to teach myself some bundles: (my apologies, my quoting button is disabled for some reason): say you are considering the case of a vector bundle ( thinking mostly of tangent bundle), in a situation where you have a metric defined. It would then make sense to define the complement bundle to be the ortho-complement (with respect to this metric), right? Is there anything special to this choice?

 

[quasar987]

I'm not sure I follow you. You are thinking of the case where (M,g) is a riemannian manifold. Then you have the projection map pr:TM-->M and its differential pr_*:T(TM)-->TM and you want a complementary subbundle to V:=ker(pr_*). You cannot just pullback g to T(TM) via pr, and use that to define H:=V^{\perp} because pr^*g is very degenerate: pr_* vanishes on V and so V^{\perp}=T(TM).

Is this what you were thinking?

 

[Bacle2]

Yes, got it, thanks. I just need to read more carefully. Sorry.

 

[Bacle2]

For the sake of background: I ended up studying bundles by mistake:

My girlfriend wanted to take a class in cosmetology, but she misread the

instructions in the webpage, and ended up registering for cosmology instead

. Since there were no refunds, and she knew no math,I had to help her.

Then I became interested in bundles.

 

[quasar987]

haha, funny story. :)

But how do bundles pop-up in cosmology?!?

 

[Matterwave]

Even in graduate GR, I never had to use bundles for cosmology, seems weird. XD

 

[Bacle2]

Not really no bundles that I know of in cosmology ; just a contrivance to do a bad joke.

 

[quasar987]

Ah I see. We sure fell for it. :P