Fields that are also Manifolds

[WWGD]

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.

 

[WWGD]

Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.

 

[lavinia]

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.

 

[WWGD]

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 $\mathbb F^n$

 

[micromass]

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.

 

[WWGD]

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.

 

[lavinia]

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

 

[WWGD]

Yes, I just remembered and edited, sorry.

 

[lavinia]

Yes, I just remembered and edited, sorry.

also the octonians are not a group.

Here is a link that has a proof that the only spheres that are Lie groups are the circle and the 3 sphere.

http://math.stackexchange.com/questions/12453/is-there-an-easy-way-to-show-which-spheres-can-be-lie-groups