For the quaternionic unit ball $\mathbb{ B}$, let us denote by $\mathcal{ M}(\mathbb{ B})$ the set of slice regular M\"obius transformations mapping $\mathbb{ B}$ onto itself. We introduce a smooth manifold structure on $\mathcal{ M}(\mathbb{ B})$, for which the evaluation(-action) map of $\mathcal{ M}(\mathbb{ B})$ on $\mathbb{ B}$ is smooth. The manifold structure considered on $\mathcal{ M}(\mathbb{ B})$ is obtained by realizing this set as a quotient of the Lie group $\mathrm{ Sp}(1,1)$, Furthermore, it turns out that $\mathbb{ B}$ is a quotient as well of both $\mathcal{ M}(\mathbb{ B})$ and $\mathrm{ Sp}(1,1)$. These quotients are in the sense of principal fiber bundles. The manifold $\mathcal{ M}(\mathbb{ B})$ is diffeomorphic to $\mathbb{ R}^4 \times S^3$.
Categories: math.CV math.DG