# Fields that are also Manifolds

1. Jan 16, 2015

### WWGD

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

Thanks.

2. Jan 18, 2015

### WWGD

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

3. Jan 18, 2015

### lavinia

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. Jan 18, 2015

### 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 $\mathbb F^n$

5. Jan 19, 2015

### micromass

Staff Emeritus
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
6. Jan 23, 2015

### 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.

7. Jan 23, 2015

### lavinia

they are not commutative

8. Jan 23, 2015

### WWGD

Yes, I just remembered and edited, sorry.

9. Jan 23, 2015