Compact Smooth Manifolds in n-dimensional Euclidean Space

[gotjrgkr]

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!

 

[HallsofIvy]

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 $R^n$ without boundaries are the Rⁿ themselves which are not bounded.

 

[gotjrgkr]

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??