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

Raul Quiroga-Barranco

TL;DR

This work develops a Lie-theoretic framework for the slice Riemannian metric on the quaternionic unit ball $\mathbb{B}$ and connects it to the classical quaternionic Poincaré geometry governed by $Sp(1,1)$. By defining slice regularity and regular Möbius transformations, the authors realize $\mathbb{B}$ via double coset quotients of $Sp(1,1)$ and compute the isometry group of $(\mathbb{B},g)$, relating its symmetries to centralizers in $Sp(1,1)$. A detailed comparison with the quaternionic Poincaré metric $(\mathbb{B},\widehat{g})$ highlights both structural similarities and fundamental differences arising from quaternion noncommutativity, including orbit structures and symmetry groups. The results provide foundational tools for further Lie-theoretic studies of slice geometry and illuminate how regular and classical Möbius structures intertwine on $\mathbb{B}$.

Abstract

The quaternionic unit ball carries a Riemannian metric built using regular Möbius 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 Poincaré geometry, where the latter is considered within the setup of Riemannian symmetric spaces.

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

TL;DR

This work develops a Lie-theoretic framework for the slice Riemannian metric on the quaternionic unit ball and connects it to the classical quaternionic Poincaré geometry governed by . By defining slice regularity and regular Möbius transformations, the authors realize via double coset quotients of and compute the isometry group of , relating its symmetries to centralizers in . A detailed comparison with the quaternionic Poincaré metric highlights both structural similarities and fundamental differences arising from quaternion noncommutativity, including orbit structures and symmetry groups. The results provide foundational tools for further Lie-theoretic studies of slice geometry and illuminate how regular and classical Möbius structures intertwine on .

Abstract

The quaternionic unit ball carries a Riemannian metric built using regular Möbius transformations: the slice Riemannian metric. We prove that the geometry induced by this metric is strongly related to the group . 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 . We also compare the slice Riemannian geometry with the quaternionic Poincaré geometry, where the latter is considered within the setup of Riemannian symmetric spaces.

Paper Structure

This paper contains 9 sections, 20 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Riemannian symmetric space structure of $\mathbb{B}$
  3. Riemannian symmetric spaces
  4. The quaternionic unit ball $\mathbb{B}$ as Riemannian symmetric space
  5. Slice regularity and Riemannian geometry on $\mathbb{B}$
  6. Slice regular functions on $\mathbb{B}$
  7. The slice Riemannian metric on $\mathbb{B}$
  8. Lie theoretic symmetries of the slice Riemannian metric on $\mathbb{B}$
  9. Riemannian symmetric vs slice Riemannian on $\mathbb{B}$

Key Result

Proposition 2.1.2

Let $M$ be a connected Riemannian manifold. Then, $M$ is a Riemannian symmetric space if and only if it is homogeneous, as a Riemannian manifold, and for some $p_0 \in M$ there is an isometry of $M$ that has $p_0$ as an isolated fixed point. Furthermore, in this case $M$ is a complete Riemannian man

Theorems & Definitions (43)

  • Definition 2.1.1
  • Proposition 2.1.2
  • proof
  • Proposition 2.1.3
  • Remark 2.1.4
  • Proposition 2.1.5
  • Remark 2.1.6
  • Proposition 2.1.7
  • Remark 2.1.8
  • Proposition 2.1.9
  • ...and 33 more