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

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

2010-09-14

arxiv.org/abs/1009.2745v1

Mathematics

Differential Geometry

This paper is an expanded version of sections 1,2,3,4,5 and 6 of arXiv:0903.1398

Scientific paper

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 Kaehler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kaehler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds with either a closed fundamental four form or fundamental two forms defining a differential ideal that are not quaternionic Kaehler.

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