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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 \(\mathbb {R}^4 \times S^3\) .

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Geometry of the Slice Regular Möbius Transformations of the Quaternionic Unit Ball

  • 会议论文

  • 发表日期：

    2026年

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  • Raul Quiroga-Barranco

摘要

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 \(\mathbb {R}^4 \times S^3\) .