Explicit Quaternionic Contact Structures and Metrics with Special Holonomy

TL;DR

This paper constructs explicit quaternionic contact structures on Lie groups, explores their curvature properties, and derives new complete metrics with special holonomy, expanding the understanding of quaternionic geometries.

Contribution

It provides explicit examples of quaternionic contact structures with various torsion and curvature properties, and introduces new complete metrics with special holonomy on certain manifolds.

Findings

Existence of quaternionic contact structures not conformal to the quaternionic sphere.

Explicit complete quaternionic Kähler and Spin(7) metrics on specific Lie groups.

Construction of non-compact almost quaternion Hermitian manifolds with special properties.

Abstract

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. We present a left invariant quaternionic contact structure on a seven dimensional non-nilpotent Lie group, and show that this structure is locally quaternionic contact conformal to the flat quaternionic contact structure on the quaternionic Heisenberg group. On the product of a seven dimensional Lie group, equipped with a quaternionic contact structure, with the real line we determine explicit complete quaternionic Kaehhler metrics and $Spin(7)$-holonomy metrics which seem to be new. We give explicit complete non-compact eight dimensional almost quaternion hermitian manifolds with closed fundamental four form which are not quaternionic K\"ahler.