Proving the Compactness of O(3) as a 3-Manifold in R^9

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In summary, the conversation discusses how to show that O(3) is a compact 3-manifold in R^9. The first and last two steps are straightforward, but there is some confusion about the details of the proof and how to show it is closed and bounded. It is explained that O(3) is a subset of R^9 and is defined by equations in the coordinates of M(3). These equations make it clear that O(3) is both closed and bounded, and since it is also a group, it is a homogeneous variety, making it a manifold.
  • #1
dtkyi
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Hi folks,

How would I go about showing that O(3) (the set of all orthogonal 3x3 matrices) is a compact 3-manifold (without boundary) in R^9?
 
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  • #2
Which part is troubling you? That is in R^9, is a manifold, or is compact? The first and the last two are straightfoward - it is a closed, bounded subset of R^9 (3x3 matrices form a 9-dim real space), and the inclusion makes it a manifold naturally.
 
  • #3
I understand that it is in R^9. "Intuitively", it makes sense that it's a 3-manifold, but I'm not entirely clear on the details of the proof. Also, how do you show it is closed and bounded? I guess the fact that we are dealing w/ matrices rather than real numbers is troubling me here...

Thanks for your help :smile:
 
  • #4
M(3)=R^9, is, in coordinates {m_ij : 1<=i,j<=3}. O(3) is given by equations in the m_ij. E.G. M is in O(3) if and only if MM^t=I, giving equations the m_ij must satisfy. What are the equations that define O(3)? These make it clear that the set is closed, and bounded. O(3) is just cut out by some (nonsingular) equations, so it's a manifold (even a variety).
 
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  • #5
if its a variety its a manifold, since its a group, hence homogeneous.
 

1. What is an O(3) 3-manifold in R^9?

An O(3) 3-manifold in R^9 is a three-dimensional space embedded in a nine-dimensional Euclidean space. It is a type of topological space that can be described as a collection of points and their relationships to each other.

2. How is an O(3) 3-manifold different from a regular three-dimensional space?

An O(3) 3-manifold is different from a regular three-dimensional space in that it is embedded in a higher-dimensional space. This means that it has additional dimensions that are not visible or accessible in its three-dimensional form.

3. What properties does an O(3) 3-manifold possess?

An O(3) 3-manifold has several important properties, including being orientable, compact, and connected. It also has a well-defined metric and can be smoothly deformed without tearing or ripping.

4. How is an O(3) 3-manifold studied and understood?

An O(3) 3-manifold is studied and understood using techniques from topology, geometry, and differential equations. These fields allow for the exploration of the manifold's structure, properties, and relationships with other mathematical objects.

5. What are the real-world applications of studying O(3) 3-manifolds?

The study of O(3) 3-manifolds has many real-world applications, including in physics, engineering, and computer science. For example, they are used in the simulation of physical systems, in the design of complex structures, and in the development of algorithms for data analysis and machine learning.

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