S0(3)/SO(2) is isomorphic to the projective plane

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In summary, for the given assignment, the goal is to show that S0(3)/SO(2) is isomorphic to the projective plane. However, there seems to be some confusion as a textbook claims that SO(3)/SO(2) is isomorphic to the 2-sphere. It is clarified that both statements are correct, as S^n \cong SO(n+1)/SO(n) and \mathbb{R}P^n \cong SO(n+1)/O(n). Therefore, SO(3)/SO(2) is isomorphic to the 2-sphere and SO(3)/O(2) is isomorphic to the projective plane. This makes sense intuitively due
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eddo
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For an assignment, my prof asked that we show that S0(3)/SO(2) is isomorphic to the projective plane (ie the 2-sphere with antipodal points identified). Here's my problem. I checked in a textbook for some help, and it claimed that SO(3)/SO(2) is isomorphic to the 2-sphere. So which one is right? It would be nice to know what I should be trying to prove before I put too much work in. Thank you.
 
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eddo said:
For an assignment, my prof asked that we show that S0(3)/SO(2) is isomorphic to the projective plane (ie the 2-sphere with antipodal points identified). Here's my problem. I checked in a textbook for some help, and it claimed that SO(3)/SO(2) is isomorphic to the 2-sphere. So which one is right? It would be nice to know what I should be trying to prove before I put too much work in. Thank you.

[tex]S^n \cong SO \left( n+1 \right)/SO \left( n \right)[/tex]

and

[tex]\mathbb{R}P^n \cong SO \left( n+1 \right)/O \left( n \right),[/tex]

so [itex]SO \left( 3 \right)/SO \left( 2 \right)[/itex] is isomorphic to the 2-sphere, and [itex]SO \left( 3 \right)/O \left( 2 \right)[/itex] is isomorphic to the projective plane.
 
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Thank you. Intuitively that's what I expected to be the case, because of the relationship between O(n) and SO(n). It makes sense that antipodal points get identified if you mod out by O(n) because the reflections that O(n) has over SO(n) would identify these points. But you can't always count on intuition so thank you for verifying this.
 

FAQ: S0(3)/SO(2) is isomorphic to the projective plane

What is the meaning of "isomorphic" in this context?

In mathematics, isomorphism is a term used to describe a one-to-one correspondence between two mathematical structures that preserves their underlying structure. In this case, it means that S0(3)/SO(2) and the projective plane have the same structure, despite being represented in different ways.

Can you explain the concept of S0(3)/SO(2) and how it relates to the projective plane?

S0(3) is the special orthogonal group in three dimensions, which is a group of rotations in three-dimensional space. SO(2) is the special orthogonal group in two dimensions, which is a group of rotations in two-dimensional space. S0(3)/SO(2) is the quotient group formed by dividing S0(3) by SO(2), and it represents the set of all possible orientations in three-dimensional space. This is isomorphic to the projective plane, which is a geometric space that represents the set of all possible lines passing through the origin in three-dimensional space.

What is the significance of this isomorphism between S0(3)/SO(2) and the projective plane?

This isomorphism is significant because it allows us to view the projective plane as a group, which makes it easier to study and understand. It also helps us to better understand the structure of rotations in three-dimensional space.

How is this isomorphism used in practical applications?

This isomorphism is used in applications such as computer graphics, robotics, and physics, where rotations in three-dimensional space are important. It allows for more efficient calculations and provides a deeper understanding of the underlying principles involved.

Are there any other mathematical concepts related to this isomorphism?

Yes, there are several other mathematical concepts that are related to this isomorphism, such as group theory, projective geometry, and topology. These concepts help to further explain and explore the relationship between S0(3)/SO(2) and the projective plane.

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