Compact Smooth Manifolds in n-dimensional Euclidean Space

In summary, the conversation discusses the topic of smooth manifolds in n-dimensional euclidean space and whether they can be compact or not. The answer is no, as R itself is a smooth manifold that is not compact. However, if "manifolds with boundary" are included, then there are compact smooth manifolds. The conversation also touches on the concept of boundaries in manifolds and the need for further study in this subject.
  • #1
gotjrgkr
90
0
Hi!
I want to know if any smooth manifold in n-dimensional euclidean space can be compact or not.
If it is possible, then could you give me an example about that?

I also want to comfirm whether a cylinder having finite volume in 3-dimensional euclidean space can be a smooth manifold.

I hope to receive your reply soon. Thank you . Have a nice day!:smile:
 
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  • #2
I'm not sure what you mean by "any smooth manifold in n-dimensional euclidean space can be compact". If you mean "is every smooth manifold compact?", the answer is trivially "no". R itself is a smooth manifold but is not compact because it is not bounded.

If you mean "do there exist compact smooth manifolds?", then the answer depends upon whether you are including "manifolds with boundary" in "smooth manifolds". If you are then any closed and bounded segement or R is compact so the answer is yes. If not then the answer is "no" again. The only closed smooth manifolds in [itex]R^n[/itex] without boundaries are the Rn themselves which are not bounded.
 
  • #3
I actually haven't studied the subject related with manifolds,yet. But I will study it, sooner or later.
I don't know whay you mean "manifolds with boundary".
Do you mean that it implies an union of a set called "manifolds" and its boundary?
Could you explain about it more??
 

Related to Compact Smooth Manifolds in n-dimensional Euclidean Space

1. What is a compact smooth manifold?

A compact smooth manifold is a geometric object that can be described as a topological space that is locally similar to n-dimensional Euclidean space, but may have a more complicated overall shape. It is a smooth manifold if it has a differentiable structure, meaning that it can be described using smooth coordinate charts.

2. What is the significance of a compact smooth manifold in n-dimensional Euclidean space?

A compact smooth manifold is important in n-dimensional Euclidean space because it allows us to study and understand complex geometric objects in a simpler and more structured setting. It also has many applications in fields such as physics, engineering, and mathematics.

3. How is a compact smooth manifold different from a non-compact smooth manifold?

A compact smooth manifold is a closed and bounded subset of n-dimensional Euclidean space, meaning that it has finite size and no holes or boundary. A non-compact smooth manifold, on the other hand, may have infinite size or have a boundary. This can affect the behavior and properties of the manifold.

4. What are some examples of compact smooth manifolds in n-dimensional Euclidean space?

Some common examples of compact smooth manifolds in n-dimensional Euclidean space include spheres, tori, and projective spaces. Other examples can include surfaces of higher genus such as the double torus or the Klein bottle.

5. What are some applications of compact smooth manifolds in n-dimensional Euclidean space?

Compact smooth manifolds have numerous applications in various fields, including physics, engineering, and mathematics. They are used to study and model physical systems, such as in general relativity and quantum mechanics. They also have applications in computer graphics, data analysis, and optimization problems.

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