Which Fields Are Naturally Manifolds Beyond ##\mathbb{R}## and ##\mathbb{C}##?

In summary, the conversation discusses the possibility of other fields being "naturally" manifolds besides ##\mathbb R## and ##\mathbb C##, and the possibility of them being Lie groups. It is mentioned that every Lie group is a differentiable manifold, but there may not be any other fields that are a Euclidean space. However, there is a suggestion to look at finite-dimensional vector spaces as manifolds. The conversation then delves into the idea of "Lie fields" and topological fields, and the proof that the only connected, locally compact fields are ##\mathbb R## and ##\mathbb C##. It is also noted that the only spheres that are Lie groups are the circle and
  • #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.
 
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  • #2
Just a comment that we could look at the two as groups and then the issue is whether these are Lie groups.
 
  • #3
WWGD said:
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 don't think there are any other fields that are a Euclidean space.
 
  • #4
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
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:
  • #6
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
WWGD said:
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
Yes, I just remembered and edited, sorry.
 

FAQ: Which Fields Are Naturally Manifolds Beyond ##\mathbb{R}## and ##\mathbb{C}##?

What is a field that is also a manifold?

A field that is also a manifold refers to a physical quantity or phenomenon that can be described by a mathematical function that varies continuously in space and time. This function can be represented as a smooth, continuous surface, which is known as a manifold.

How do fields and manifolds relate to each other?

Fields and manifolds are closely related in mathematics and physics. A field can be described as a function that assigns a value to every point in space, while a manifold is a space that can be described by a set of coordinates. This means that a field can be represented as a manifold, where each point on the manifold corresponds to a specific value of the field.

What are some examples of fields that are also manifolds?

Some common examples of fields that are also manifolds include gravitational fields, electromagnetic fields, and temperature fields. These fields can all be described by mathematical functions that vary continuously in space and time, and can be represented as smooth, continuous surfaces.

What is the significance of studying fields that are also manifolds?

Studying fields that are also manifolds allows us to better understand and describe physical phenomena. These fields play a crucial role in many areas of physics, such as electromagnetism, thermodynamics, and general relativity. By studying these fields, we can gain insights into the behavior of complex systems and make predictions about their behavior.

Are there any real-life applications of fields that are also manifolds?

Yes, there are many real-life applications of fields that are also manifolds. For example, in engineering, these fields are used to model and analyze complex systems, such as fluid flow in pipes or heat transfer in engines. In physics, they are used to study the behavior of particles and waves, and in geology, they are used to model the Earth's magnetic field and seismic activity.

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