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arXiv · 2502.18669

Lie theory of the slice Riemannian geometry on the quaternionic unit ball

Raul Quiroga-Barranco
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Abstract

The quaternionic unit ball carries a Riemannian metric built using regular M\"obius transformations: the slice Riemannian metric. We prove that the geometry induced by this metric is strongly related to the group $\mathrm{Sp}(1,1)$. We also develop the foundations for a Lie theoretic study of the slice Riemannian metric. In particular, we compute its isometry group and prove that it is built from symmetries of the Lie group $\mathrm{Sp}(1,1)$. We also compare the slice Riemannian geometry with the quaternionic Poincar\'e geometry, where the latter is considered within the setup of Riemannian symmetric spaces.

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Raul Quiroga-Barranco. 2025-02-25. Lie theory of the slice Riemannian geometry on the quaternionic unit ball. https://arxiv.org/abs/2502.18669

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