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

Raul Quiroga-Barranco

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

Differential Geometry 2025-02-27 v1 Complex Variables

Authors: Raul Quiroga-Barranco

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.

Keywords

quaternionic unit ball slice Riemannian metric Lie theory isometry group Sp(1,1) quaternionic Poincaré geometry

Cite

@article{arxiv.2502.18669,
  title  = {Lie theory of the slice Riemannian geometry on the quaternionic unit ball},
  author = {Raul Quiroga-Barranco},
  journal= {arXiv preprint arXiv:2502.18669},
  year   = {2025}
}