Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

Abstract

We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kahler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kahler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kahler with either a closed fundamental four form or fundamental two forms defining a differential ideal.