On the sub-Riemannian geometry of the quaternionic Heisenberg group

TL;DR

This paper develops the sub-Riemannian geometry of the 7-dimensional quaternionic Heisenberg group by leveraging its quaternionic contact (QC) structure. It introduces a Riemannian approximation $g_L$ that rescales vertical directions and analyzes the limits to obtain explicit Carnot-Carathéodory (CC) geodesics, the CC distance, and CC-spheres, together with a general formula for horizontal mean curvature. The work also computes the full Riemannian connection and curvatures for the approximants, derives CC-geodesics as $L oty$, and demonstrates a quasi-isometry between the CC metric and the Korányi metric. Overall, the results provide foundational tools for isoperimetric problems and horizontal geometry on the quaternionic Heisenberg group, enabling concrete descriptions of geodesics, spheres, and minimal surfaces in this QC setting.

Abstract

Utilizing the framework of quaternionic contact geometry, we define a sequence of Riemannian metrics $\{g_L\}$ on the quaternionic Heisenberg group $\mathfrak{H}_{\mathbb{H}}$ by rescaling the vertical directions. By analyzing the limit of this sequence, we characterize the Carnot-Carathéodory geodesics and provide the explicit description of the Carnot-Carathéodory distance and spheres in $\mathfrak{H}_{\mathbb{H}}$ . Furthermore, we derive a general formula for the horizontal mean curvature of hypersurfaces.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Proposition 4.1
• Corollary 4.2