Jose A. Santisteban | Searcharxiv

arXiv2010

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For non-unimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.

math.DG↗

arXiv2009

Explicit Quaternionic Contact Structures and Metrics with Special Holonomy

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 Kähler.

math.DG↗