On the sub-Riemannian geometry of the quaternionic Heisenberg group

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.

18 pages