Contact 5-manifolds with SU(2)-structure

TL;DR

This paper studies 5-manifolds with a special contact structure derived from hypo structures, linking them to 6-manifolds with half-flat SU(3)-structures, and explores their geometric properties and examples with special holonomy.

Contribution

It establishes conditions for hypo-contact structures on 5-manifolds within 6-manifolds with half-flat SU(3)-structures and classifies solvable Lie algebras admitting such structures, leading to new metrics with special holonomy.

Findings

Existence conditions for hypo-contact structures on 5-manifolds.

Classification of solvable Lie algebras with hypo-contact structures.

Construction of new metrics with SU(3) and G_2 holonomy.

Abstract

We consider 5-manifolds with a contact form arising from a hypo structure, which we call \emph{hypo-contact}. We provide conditions which imply that there exists 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 \emph{hypo-contact} structure. This allows us exhibit examples of Sasakian $\eta$-Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and to new metrics with holonomy $G_2$.