Geometry of the slice regular Möbius transformations of the quaternionic unit ball
Raul Quiroga-Barranco
TL;DR
This work constructs a smooth manifold structure for the space of slice regular Möbius transformations $\mathcal{M}(\mathbb{B})$ on the quaternionic unit ball by realizing it as a quotient of the Lie group $\mathrm{Sp}(1,1)$ via the nonnormal subgroup $\mathrm{Sp}(1)I_2$, and shows that $\mathcal{M}(\mathbb{B})$ is diffeomorphic to $\mathbb{R}^4 \times S^3$ with a natural principal $\mathrm{Sp}(1)$-bundle over it. It also connects $\mathcal{M}(\mathbb{B})$ to the unit ball $\mathbb{B}$ through smooth, transitive evaluation actions and inverse-orbit quotients, yielding doubly-quotient realizations of $\mathbb{B}$ as both a quotient of $\mathcal{M}(\mathbb{B})$ and a double coset of $\mathrm{Sp}(1,1)$. The results bridge slice regular hyperholomorphic function theory with quaternionic hyperbolic geometry, providing a differential-geometric framework for slice regular Möbius transformations and their action on $\mathbb{B}$.
Abstract
For the quaternionic unit ball $\mathbb{B}$, let us denote by $\mathcal{M}(\mathbb{B})$ the set of slice regular Möbius 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 $\mathbb{R}^4 \times S^3$.