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

TL;DR

This paper explores the Lie group structure underlying the slice Riemannian geometry on the quaternionic unit ball, establishing its relation to symmetries of the group Sp(1,1) and comparing it with quaternionic Poincaré geometry.

Contribution

It develops a Lie theoretic framework for the slice Riemannian metric on the quaternionic ball, including computing its isometry group and relating it to Sp(1,1).

Findings

The isometry group is generated by symmetries of Sp(1,1).

The slice Riemannian geometry is closely related to quaternionic Poincaré geometry.

A foundation for Lie algebraic analysis of the metric is established.

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.