Manifold Explained in Simple Terms

  • Thread starter KFC
  • Start date
  • Tags
    Manifold
In summary, a manifold is a geometric object that is locally Euclidean. This means that you can define a local coordinate system anywhere, but not necessarily a global coordinate system. A "manifold" is a geometric object that is locally Euclidean, and we require that if p is any point in the manifold there exist at least one \alpha such that p is in M_\alpha ( the open sets cover the manifold). Additionally, in the intersection of two such sets, M_\alpha and M_\beta, \phi_\alpha o\phi_\beta be a homeomorphism from Rn to itself. Finally, a manifold must be differentiable if you want to be able to describe your
  • #1
KFC
488
4
Hi there,
I find that the term 'manifold' appears in many book of statistical physics or classical mechanics while talking about phase space. I try to search the explanation on online but it is quite abstract and hard to understand what's manifold really refers to. Can anyway explain this a bit with a simplest picture? Thanks
 
Physics news on Phys.org
  • #2
An n-dimensional manifold, in simplest terms, is a representation of non-Euclidean (in general) n-dimensional space that is locally Euclidean* at any point. In other words, you can define a local coordinate system anywhere, but not necessarily a global coordinate system.

Keep in mind, I am oversimplifying a bit. For formal definition, look up pretty much any topology text.
 
  • #3
One can think of a Manifold as any set of objects which can be described by a set of coordinate charts (called an atlas). A coordinate chart is a mapping between your set of objects to some Euclidean space R^n (n is then the dimension of your manifold). There's really nothing much more to it than that. If you can describe your set of objects in such a way, then it's a manifold.
 
  • #4
K^2 said:
An n-dimensional manifold, in simplest terms, is a representation of non-Euclidean (in general) n-dimensional space that is locally Euclidean* at any point. In other words, you can define a local coordinate system anywhere, but not necessarily a global coordinate system.

Keep in mind, I am oversimplifying a bit. For formal definition, look up pretty much any topology text.

I am a bit confused. Isn't it the "definition" of differentiable manifold instead of manifold?
 
  • #5
agostino981 said:
I am a bit confused. Isn't it the "definition" of differentiable manifold instead of manifold?
Could be. Did I make an assumption that mapping is differentiable when I said that coordinate system could be defined? I think you might be right about that. I'm sure the OP would be dealing with such, but I probably shouldn't have defined it so narrowly anyways. Matterwave's explanation is a lot closer to a formal, general definition of a manifold.
 
  • #6
A "manifold" is a geometric object that is locally Euclidean. More precisely, an n dimensional manifold is a topological space, together with a set of "pairs", [itex]\{(M_\alpha, \phi_\alpha)\}[/itex], in which one member of each pair, [itex]M_\alpha[/itex], is an open set and the other member, [itex]\phi_\alpha[/itex], is a continuous function from [itex]M_\alpha[/itex] to Rn.

We require, further, that if p is any point in the manifold there exist at least one [itex]\alpha[/itex] such that p is in [itex]M_\alpha[/itex] ( the open sets cover the manifold). We require, further, that in the intersection of two such sets, [itex]M_\alpha[/itex] and [itex]M_\beta[/itex], [itex]\phi_\alpha o\phi_\beta[/itex] be a homeomorphism from Rn to itself.

In order to have a differentiable manifold, we require that, in the intersection of [itex]M_\alpha[/itex] and [itex]M_\beta[/itex], both [itex]\phi_\alpha o \phi_\beta[/itex] and [itex]\phi_\beta o\phi_\alpha[/itex] be differentiable.
 
  • #7
Halls, OP states that he looked at former definitions. I think he was looking for a simpler illustration.
 
  • #8
HallsofIvy said:
We require, further, that in the intersection of two such sets, [itex]M_\alpha[/itex] and [itex]M_\beta[/itex], [itex]\phi_\alpha o\phi_\beta[/itex] be a homeomorphism from Rn to itself.
If 'o' here just means the usual composition of functions, there's some inversion missing. But then the functions must be 1-1 onto.
As I read it, the individual functions are required to be homeomorphisms. The composition of one with the inverse of another then is automatically a homeomorphism.

Going back to the original question, maybe some examples help. The mapping of the surface of the Earth to flat maps by carving it into overlapping regions is an obvious one.
 

1. What is a manifold?

A manifold is a mathematical concept that refers to a space that is locally similar to Euclidean space. It can be visualized as a surface that can be smoothly bent and stretched in different directions without tearing or folding.

2. How is a manifold different from a regular space?

A regular space, such as a flat plane, has a constant curvature and can only be bent or stretched in certain ways. A manifold, on the other hand, can have varying curvature and can be bent and stretched in more complex ways.

3. What are some examples of manifolds?

Some common examples of manifolds include spheres, cylinders, and tori. In higher dimensions, manifolds can be more complex and may not have a physical representation.

4. How are manifolds used in science?

Manifolds are used in many areas of science, including physics, engineering, and computer science. They can help describe the structure of physical systems, model complex data sets, and optimize algorithms.

5. Are there different types of manifolds?

Yes, there are different types of manifolds, such as smooth manifolds, topological manifolds, and differentiable manifolds. These types differ in their level of smoothness and the type of functions that can be defined on them.

Similar threads

Replies
5
Views
314
Replies
2
Views
907
  • Special and General Relativity
Replies
28
Views
2K
  • Topology and Analysis
2
Replies
38
Views
4K
Replies
16
Views
792
  • Classical Physics
Replies
21
Views
989
Replies
8
Views
1K
  • Other Physics Topics
Replies
2
Views
13K
Replies
5
Views
844
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
996
Back
Top