Quaternionic Heisenberg groups as naturally reductive homogeneous spaces
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 -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 -Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on the -dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated 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.
Keywords
Cite
@article{arxiv.1503.08350,
title = {Quaternionic Heisenberg groups as naturally reductive homogeneous spaces},
author = {Ilka Agricola and Ana Cristina Ferreira and Reinier Storm},
journal= {arXiv preprint arXiv:1503.08350},
year = {2015}
}