Exploring Frame Bundles on Manifolds

In summary: M is a principle GL(n,R)-bundle who's fibers are the sets of ordered bases for the tangent space of M at p. (where n=dim(M))"1) This means that any point in the fiber (say, over a point m in M) is literally a set of ordered bases right?Yes, it is literally an n-tuple of vectors spanning TmM.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 4-d manifold is 16 dimensional? Why so many dimensions
  • #36
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.
 
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  • #37
haha, funny story. :)

But how do bundles pop-up in cosmology?!?
 
  • #38
Even in graduate GR, I never had to use bundles for cosmology, seems weird. XD
 
  • #39
Not really no bundles that I know of in cosmology ; just a contrivance to do a bad joke.
 
  • #40
Ah I see. We sure fell for it. :P
 
<h2>1. What is a frame bundle?</h2><p>A frame bundle is a mathematical object that describes the collection of all possible coordinate frames at each point on a manifold. It is used to study the geometry and topology of manifolds.</p><h2>2. How is a frame bundle related to tangent bundles?</h2><p>A frame bundle is closely related to a tangent bundle, as it is a principal bundle whose fibers are isomorphic to the tangent space at each point on a manifold. The frame bundle provides a way to describe the tangent bundle in a more geometric and coordinate-independent manner.</p><h2>3. What is the structure group of a frame bundle?</h2><p>The structure group of a frame bundle is the group of all linear transformations that preserve the frame at each point on the manifold. In other words, it is the group of all possible changes of basis for the tangent space at each point.</p><h2>4. How can frame bundles be used to study vector fields on manifolds?</h2><p>Frame bundles can be used to study vector fields on manifolds by providing a way to describe and manipulate the components of a vector field in a coordinate-independent manner. This allows for a more geometric understanding of vector fields and their properties.</p><h2>5. Are there any applications of frame bundles in physics?</h2><p>Yes, frame bundles have many applications in physics, particularly in the field of general relativity. They are used to study the geometry of spacetime and describe the behavior of particles and fields in curved spacetime. They are also used in gauge theories to describe the symmetries and transformations of physical systems.</p>

1. What is a frame bundle?

A frame bundle is a mathematical object that describes the collection of all possible coordinate frames at each point on a manifold. It is used to study the geometry and topology of manifolds.

2. How is a frame bundle related to tangent bundles?

A frame bundle is closely related to a tangent bundle, as it is a principal bundle whose fibers are isomorphic to the tangent space at each point on a manifold. The frame bundle provides a way to describe the tangent bundle in a more geometric and coordinate-independent manner.

3. What is the structure group of a frame bundle?

The structure group of a frame bundle is the group of all linear transformations that preserve the frame at each point on the manifold. In other words, it is the group of all possible changes of basis for the tangent space at each point.

4. How can frame bundles be used to study vector fields on manifolds?

Frame bundles can be used to study vector fields on manifolds by providing a way to describe and manipulate the components of a vector field in a coordinate-independent manner. This allows for a more geometric understanding of vector fields and their properties.

5. Are there any applications of frame bundles in physics?

Yes, frame bundles have many applications in physics, particularly in the field of general relativity. They are used to study the geometry of spacetime and describe the behavior of particles and fields in curved spacetime. They are also used in gauge theories to describe the symmetries and transformations of physical systems.

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