Show that $SO(3)$ in an embedded submanifold of $\mathbb{R}^{3 \times 3}$

Question

How does one go about proving that $SO(3)$ is an embedded submanifold of $\mathbb{R}^{3 \times 3}$? I know that there is a definition of embedded submanifold in terms of the induced topology, and I know the Cayley construction of charts for $SO(3)$. I'm looking for a simpler method.

Answer 1

Hint A square matrix $A$ is orthogonal iff $A^T A = I$, so $O(n)$ is the preimage $F^{-1}(I)$ of the map $$F : \Bbb R^{n \times n} \to S(n, \Bbb R) , \qquad F : A \mapsto A^T A.$$ (Here, $S(n, \Bbb R)$ is the set of all symmetric $n \times n$ matrices.) Thus, to establish that $O(n, \Bbb R)$ is an embedded submanifold of $\Bbb R^{n \times n}$ it suffices to show that $I$ is a regular value of $F$.