Given $ \ O(2) = \{A \in M_{2 \times 2 } : A^TA = I_2 \ \} \ $
prove that $ \ O(2) \ $ is diffeomorphic to $ \ S^1 \sqcup S^1 $
with $ \sqcup \ $ the disjoint union and $ \ S^1 \ $ the 1-sphere .
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Is there any help ?
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no, $O(2)$ is one-dimensional – edm Aug 15 '17 at 03:50
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$O(2)$ is a 1-dim manifold. – Randall Aug 15 '17 at 03:50
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"Since they are finite dimensional , there exists an Isomorphism between them" might be true for something like vector spaces, but not manifolds. – Randall Aug 15 '17 at 03:53
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I know that $O(2)$ is a $2$-dimensional manifold.
No it isn't.
Hence $S^1\sqcup S^1$ is a $2$-dimensional manifold.
No, no, no.
Since they are finite dimensional , there exists an Isomorphism between them .
NO, STAHP.
Saying two spaces have the same number of dimensions does not imply they are isomorphic as groups - in fact, manifolds may not be groups at all! How can you say there exists an isomorphism when you haven't specified any group structure on $S^1\sqcup S^1$?
The Lie group $\mathrm{O}(2)$ is a one-dimensional manifold with two connected components. The component containing the identity matrix is the group $\mathrm{SO}(2)$ of all rotations of the plane. The other component contains all reflections across lines.
The space $S^1\sqcup S^1$ is a pair of disjoint circles, which is also evidently one-dimensional.
So your task is twofold: (a) find a nice bijection between rotations and points on the circle, (b) find a nice bijection between reflections across lines and points on the circle. For the latter, it may help to know why the real projective line is diffeomorphic to a circle, $\mathbb{RP}^1\simeq S^1$. Then check the bijections are diffeomorphisms.
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