Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

TL;DR

This paper explores the geometry of quaternionic Heisenberg groups, revealing their naturally reductive homogeneous space structure, their canonical connection, and unique spinorial properties, especially in 7 dimensions.

Contribution

It introduces a canonical connection on quaternionic Heisenberg groups that preserves key structures and links to cocalibrated G2 structures in 7 dimensions, highlighting their unique spinorial features.

Findings

Quaternionic Heisenberg groups admit an almost 3-contact metric structure.

The canonical connection preserves the quaternionic contact structure.

The 7-dimensional case relates to cocalibrated G2 structures and has unique spinorial properties.

Abstract

In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the $3$-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on the $7$-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated $G_2$ structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.