No abstract

We consider 5-manifolds with a contact form arising from a hypo structure [9], which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU (3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU (3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a hypo-contact structure. This allows us to exhibit examples of Sasakian η-Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU (3) and G2.

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 Kähler metric as well as a metric with holonomy contained in Spin (7). As a consequence we determine explicit quaternionic Kähler 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 Kähler.

No abstract

We present a simple explicit construction of hyper-Kähler and hyper-symplectic (also known as neutral hyper-Kähler or hyper-parakähler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kähler and hyper-symplectic structures in dimension four.In this section we recover some of the known hyper-Kähler metrics in dimension four. To this end, we lift the special structure on the non-Euclidean Bianchi type groups of class A to a hyper-Kähler metric on its product with (an interval in) the real line.