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arXiv · 2601.11312

On the sub-Riemannian geometry of the quaternionic Heisenberg group

Joonhyung Kim·Ioannis D. Platis·Li-Jie Sun
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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.

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Joonhyung Kim, Ioannis D. Platis, Li-Jie Sun. 2026-01-16. On the sub-Riemannian geometry of the quaternionic Heisenberg group. https://arxiv.org/abs/2601.11312

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