Real and complex projective spaces as homogeneous spaces

The actions on lines are smooth: in an affine projective chart where one coordinate is nonzero, the transformed line coordinates are ratios of linear functions with a nonvanishing denominator. Given two real lines, choose unit generators and extend them to oriented orthonormal bases; the resulting element of $SO(n+1)$ carries one line to the other. Given two complex lines, extend unit generators to unitary bases; the resulting element of $U(n+1)$ does the same. Hence both actions are transitive.

The stabilizer in $SO(n+1)$ of the line $\mathbb{R}{e}_0$ preserves its orthogonal complement and is therefore $S(O(1)\times O(n))=\{\mathrm{diag}(\epsilon ,A):\epsilon =\pm 1,A\in O(n),\epsilon \det A=1\}\mathrm{.}$ Conversely every such block matrix fixes the line. The stabilizer in $U(n+1)$ of $\mathbb{C}{e}_0$ is exactly the block subgroup $U(1)\times U(n)$: unitarity forces preservation of the orthogonal complement, and every such block matrix fixes the line.

Apply [A1] to the base lines. It yields the two displayed equivariant diffeomorphisms. The determinant-one condition in the real stabilizer is essential; replacing it by $O(1)\times O(n)$ would not be a subgroup of $SO(n+1)$. For $n=0$, both projective spaces are a point and the analogous quotient is trivial. Countable choice is used through [A1].