On the sub-Riemannian geometry of the quaternionic Heisenberg group

TL;DR

This paper explores the sub-Riemannian geometry of the quaternionic Heisenberg group by defining rescaled metrics, analyzing geodesics, and describing distances, spheres, and curvature within this geometric framework.

Contribution

It introduces a new family of Riemannian metrics on the quaternionic Heisenberg group and characterizes its sub-Riemannian geodesics, distances, and curvature properties.

Findings

Explicit description of Carnot-Carathéodory distance and spheres.

Characterization of geodesics in the quaternionic Heisenberg group.

Formula for horizontal mean curvature of hypersurfaces.

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-Carath\'eodory geodesics and provide the explicit description of the Carnot-Carath\'eodory distance and spheres in $\mathfrak{H}_{\mathbb{H}}$ . Furthermore, we derive a general formula for the horizontal mean curvature of hypersurfaces.