Show me a book about hyper-sphere (n-sphere)

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In summary, the group structure of n-sphere is SO(n+1)/SO(n), which can be realized as a group acting on a set modulo a subgroup fixing certain points. The only spheres that are Lie groups are S^1 and S^3, and this result places a restriction on the possible division algebras. S^1, S^3, and S^7 are the only spheres that can be turned into groups, with S^7 being the octonians, which are non-associative. This phenomenon has something to do with division algebras, and the proof for this result should proceed along those lines. Additionally, these spheres cannot be topological groups with the usual topology.
  • #1
thanhsonsply
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I find that group structure of n-sphere is SO(n+1)/SO(n) (at http://en.wikipedia.org/wiki/N_sphere). So I want to find a book show that.

Please, show me a title of book and author. Thanks!
 
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  • #2
many spaces can be realized as a group acting on a set, modulo a subgroup fixing certain points. I.e. if a group acts on a set transitively, and a point has a certain subgroup fixing it, then the set is the group mod the subgroup in some sense. probably this is what is going on. rather than read some book, think about this phenomenon.

e.g. is the sphere in three space somehow equal to the rotation group on three space, modulo the subgroup of rotations fixing one point of the sphere? i.e. is it SO(3)/SO(2)?

I'm not saying it is, just that you will learn more by deciding it for yourself.
 
  • #3
John Lee's Introduction the Smooth manifold. Look for one of the examples in chapter 9.
 
  • #4
Most spheres are not Lie groups. You could always turn the underlying set into a group, but then there's no point in it being a sphere.

I think this has something to do with division algebras, maybe. I think if it were a Lie group, maybe you could take the group algebra and get a division algebra. Division algebras only exist in certain dimensions: 1, 2, 4, and 8. So you get spheres of dimension 1 less: 1, 3, and 7. And 7 doesn't work because the octonians, the dimension 8 division algebra, are non-associative, so you can't get a group by taking the unit octonians.

So the only spheres that are groups are S^1 and S^3.

Don't quote me on this, though, since I haven't quite thought it through. Just came up with it off the top of my head. I think it's right, though.

The end result is correct, anyway:

http://en.wikipedia.org/wiki/3-sphere
 
  • #5
homeomorphic said:
Most spheres are not Lie groups. You could always turn the underlying set into a group, but then there's no point in it being a sphere.

I think this has something to do with division algebras, maybe. I think if it were a Lie group, maybe you could take the group algebra and get a division algebra. Division algebras only exist in certain dimensions: 1, 2, 4, and 8. So you get spheres of dimension 1 less: 1, 3, and 7. And 7 doesn't work because the octonians, the dimension 8 division algebra, are non-associative, so you can't get a group by taking the unit octonians.

So the only spheres that are groups are S^1 and S^3.

Don't quote me on this, though, since I haven't quite thought it through. Just came up with it off the top of my head. I think it's right, though.

The end result is correct, anyway:

http://en.wikipedia.org/wiki/3-sphere
You kind of have it the other way around. If we can endow ##\mathbb R^n## with some "sufficiently nice" multiplication operation, then ##S^{n-1}## will be a group. But the converse isn't clear - why can't we have a group operation on ##S^{n-1}## that doesn't necessarily extend to a nice multiplication on ##\mathbb R^n##?

Anyway, the OP should really try to understand mathwonk's post, which is on the money.
 
  • #6
You kind of have it the other way around. If you can endow Rn with some nice "multiplication" operation, then Sn−1 will be a nice group. But the converse isn't clear - why can't we have a nice group operation on Sn−1 that doesn't necessarily extend to a nice multiplication on Rn?

It isn't clear, but according to wikipedia and probably some other sources I remember, it is true that only S^1 and S^3 are Lie groups. So, my speculation was that given a group operation on S^n-1, you can construct a division algebra of dimension n somehow. But it's not clear how. The wikipedia article also seems to indicate that the proof should proceed along these lines.
 
  • #7
homeomorphic said:
It isn't clear, but according to wikipedia and probably some other sources I remember, it is true that only S^1 and S^3 are Lie groups. So, my speculation was that given a group operation on S^n-1, you can construct a division algebra of dimension n somehow. But it's not clear how. The wikipedia article also seems to indicate that the proof should proceed along these lines.
Yes, that result is true: S^1 and S^3 (and S^0) are the only spheres that are Lie groups.

Most proofs I know don't mention real division algebras; it typically goes the other way around: once you have this result, you get a restriction on the possible division algebras.
 
  • #8
Most proofs I know don't mention real division algebras; it typically goes the other way around: once you have this result, you get a restriction on the possible division algebras.

That only works for associative division algebras, though, which is a bit unfair to the octonians, but I guess the result is really that those are the only spheres that are H-spaces, which have some kind of product that's not necessarily associative.

It should also be mentioned the they can't even be topological groups (with the usual topology).

This might also be of interest:

http://www.unizar.es/acz/05Publicaciones/Revistas/Revista62/p075.pdf
 

FAQ: Show me a book about hyper-sphere (n-sphere)

What is a hyper-sphere (n-sphere)?

A hyper-sphere, also known as an n-sphere, is a generalization of a sphere to higher dimensions. It is a geometric shape in n-dimensional space that is defined as the set of all points that are a fixed distance from a central point.

How is a hyper-sphere (n-sphere) different from a regular sphere?

A hyper-sphere differs from a regular sphere in that it exists in n-dimensional space instead of just three-dimensional space. This means that it has n+1 dimensions, while a regular sphere only has three dimensions (length, width, and height).

What are the properties of a hyper-sphere (n-sphere)?

The properties of a hyper-sphere include its radius, diameter, volume, and surface area. These properties can be calculated using formulas specific to n-dimensional space.

How is a hyper-sphere (n-sphere) used in mathematics and science?

Hyper-spheres have many applications in mathematics and science, particularly in geometry, topology, and physics. They are used to model higher-dimensional spaces, such as in string theory, and in studying the behavior of particles in n-dimensional space.

Are there any real-world examples of hyper-spheres (n-spheres)?

While it is difficult to imagine or visualize a hyper-sphere in higher dimensions, there are some real-world examples that can help us understand the concept. For instance, a 3-sphere can be thought of as a four-dimensional ball, and a 4-sphere can be thought of as a five-dimensional hypersphere. However, it should be noted that these are just analogies and not actual hyper-spheres in our three-dimensional world.

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