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Fields that are also Manifolds

  1. Jan 16, 2015 #1

    WWGD

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

    Thanks.
     
  2. jcsd
  3. Jan 18, 2015 #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.
     
  4. Jan 18, 2015 #3

    lavinia

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    Every Lie group is a manifold - in fact a differentiable manifold. I dont think there are any other fields that are a Euclidean space.
     
  5. Jan 18, 2015 #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 ##
     
  6. Jan 19, 2015 #5
    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.
     
    Last edited: Jan 19, 2015
  7. Jan 23, 2015 #6

    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.
     
  8. Jan 23, 2015 #7

    lavinia

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    they are not commutative
     
  9. Jan 23, 2015 #8

    WWGD

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    Yes, I just remembered and edited, sorry.
     
  10. Jan 23, 2015 #9

    lavinia

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