How to say a given space is a manifold?

In summary, to say a given space is a manifold, you must first satisfy the definition and then break the space up into its covering set. After that, you must find the charts that correspond to each open set.
  • #1
guhan
43
1
How to say a given space is a manifold?

The only thing that props in my mind is to check if every open set has a euclidean coordinate chart on it. But, what if the space I am dealing with is not fully understood apriori?

As in, how were the spaces of thermodynamic equilibrium states, phase and configuration spaces in classical mechanics, space-time (of general relativity) etc given a manifold structure? How to justify that every open set of these spaces correspond to an euclidean space?
 
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  • #2
guhan said:
How to say a given space is a manifold?

The only thing that props in my mind is to check if every open set has a euclidean coordinate chart on it. But, what if the space I am dealing with is not fully understood yet?

As in, how were the spaces of thermodynamic equilibrium states, phase and configuration spaces in classical mechanics, space-time (of general relativity) etc given a manifold structure? How to justify that every open set of these spaces correspond to an euclidean space?

You just need to satisfy the definition of a manifold for that space.

First, you probably need to guess the dimension. How many degrees of freedom does the space have after you've applied all the constraints? Once you understand the dimension, it should usually be clear how to break the space up (really, you can probably break it up in number of ways, as long as you cover the space. Once you've done that, you have a manifold. I image the hard part would be to show the manifold is smooth or satisfies other properties.
 
  • #3
Even after I break it up into its covering set (meaning, I have laid out the open sets), how to justify that each of these open sets correspond to an euclidean chart?

Do we have to worry about differentiable-ness, because once I have assigned one chart to each open set I can then assign charts that will be [tex]C^\infty[/tex]-compatible with the original and call it as the atlas.
 
  • #4
guhan said:
Even after I break it up into its covering set (meaning, I have laid out the open sets), how to justify that each of these open sets correspond to an euclidean chart?

When picking out the charts, there is no real "justification" to do. You simply need to provide the chart itself. Find the function, x:M->R^n, which gives coordinates to each point. Then find the inverse function, x^-1:R^n->M, which maps points to coordinates.

Do we have to worry about differentiable-ness, because once I have assigned one chart to each open set I can then assign charts that will be [tex]C^\infty[/tex]-compatible with the original and call it as the atlas.

Yes. If you want a smooth manifold, you must all the transition maps are smooth.

What kind of spaces are you working with in particular?
 
  • #5
I don't have a specific space in mind. Let us take the example of thermodynamic systems (the axiomatic formulation of Caratheodory)...
The assumption that each state (or point in manifold) of the system be described by some variables (euclidean coordinates) is fine. But, the requirement that there be a neighbourhood of each such point which can be described by the same euclidean chart isn't obvious to me.

Or even talking about the generic space-time, it is fine to talk of 4 coords to represent each point in space-time, but what is the basis for assuming the same for an open neighbourhood of each such point?
 

Related to How to say a given space is a manifold?

1. What is a manifold?

A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space. In simpler terms, it is a space that can be smoothly and consistently mapped onto Euclidean space, which allows for the use of standard calculus techniques to study the space.

2. How do you determine if a space is a manifold?

To determine if a space is a manifold, you must check if it satisfies two main criteria: it must be locally similar to Euclidean space, and it must be Hausdorff, meaning that any two points in the space can be separated by distinct neighborhoods. Additionally, the space must also be second-countable, meaning that it has a countable basis of open sets.

3. What are the types of manifolds?

There are several types of manifolds, including smooth manifolds, topological manifolds, and differentiable manifolds. Smooth manifolds are the most commonly studied and are characterized by having a smooth structure, meaning that there is a consistent way to define tangent spaces at each point. Topological manifolds are more general and do not necessarily have a smooth structure. Differentiable manifolds are a type of smooth manifold that also has a differentiable structure, allowing for the use of differential calculus.

4. How do you describe a manifold?

A manifold can be described in terms of its dimension, which is the number of coordinates needed to describe a point in the space. For example, a two-dimensional manifold could be a flat surface like a plane, a sphere, or a torus. Manifolds can also be described by their topological properties, such as connectedness, compactness, and orientability.

5. What are some examples of manifolds?

Manifolds can be found in many places, including in nature and in mathematics. Some examples of manifolds in nature include the surface of the Earth, the shape of a mountain, or the surface of a body of water. In mathematics, examples of manifolds include spheres, tori, and projective spaces. Manifolds are also commonly used to study higher-dimensional spaces and abstract concepts in physics and other scientific fields.

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