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

[eddo]

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.

 

[George Jones]

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.

$$S^n \cong SO \left( n+1 \right)/SO \left( n \right)$$

and

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

so $SO \left( 3 \right)/SO \left( 2 \right)$ is isomorphic to the 2-sphere, and $SO \left( 3 \right)/O \left( 2 \right)$ is isomorphic to the projective plane.

 

[eddo]

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.