Example 1

Let $n$ be an integer $\geqslant 0$, let $\mathbf{P}_n(\mathbf{R})$ be the projective space of dimension $n$ (TG, VI, p. 13) and let $\mathbf{S}_n$ be the unit sphere of $\mathbf{R}^{n+1}$ (TG, VI, p. 9). The canonical mapping of $\mathbf{S}_n$ onto $\mathbf{P}_n(\mathbf{R})$ makes $\mathbf{S}_n$ into a principal covering of group $\{1,-1\}$, this group operating by homotheties; the orbits are the pairs of diametrically opposite points.