Fields that are also Manifolds

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  • #1
WWGD
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Main Question or Discussion Point

Does anyone know of fields (with additional structure/properties) other than either ##\mathbb R, \mathbb C ## that are "naturally" manifolds?

Thanks.
 

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  • #2
WWGD
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Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.
 
  • #3
lavinia
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Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.
Every Lie group is a manifold - in fact a differentiable manifold. I dont think there are any other fields that are a Euclidean space.
 
  • #4
WWGD
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Yes, thanks, that is where I was going with the comment. I don't know if there are similar results for fields; I was actually thinking of finite-dimensional vector spaces as manifolds, given that an f.d vector space over a field ##\mathbb F ## is isomorphic to ##\mathbb F^n ##
 
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Hint: every "Lie field" must also be a commutative Lie group. And the connected commutative Lie groups are exactly of the form ##S^1\times ... \times S^1\times \mathbb{R}^n##.

More generally, you could be interested in topological fields. It turns out that the only connected, locally compact fields are ##\mathbb{R}## and ##\mathbb{C}##, but this is a tad more difficult to prove than the Lie case.
 
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WWGD
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Do you mean that the restriction of the Lie field to the additive (Abelian) group is a Lie group? And aren't ##S^3, S^7 ## also connected, commutative Lie groups? EDIT: Never mind, these are not Abelian.
 
  • #7
lavinia
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Do you mean that the restriction of the Lie field to the additive (Abelian) group is a Lie group? And aren't ##S^3, S^7 ## also connected, commutative Lie groups?
they are not commutative
 
  • #8
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Yes, I just remembered and edited, sorry.
 

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