Natural SU(2)-structures on tangent sphere bundles

Abstract

We define and study natural SU(2)-structures, in the sense of Conti–Salamon, on the total space S of the tangent sphere bundle of any given oriented Riemannian 3-manifold M. We recur to a fundamental exterior differential system of Riemannian geometry. Essentially, two types of structures arise: the contact-hypo and the non-contact and, for each, we study the conditions for being hypo, nearly-hypo or double-hypo. We discover new double-hypo structures on S^3×S^2, of which the well-known Sasaki–Einstein are a particular case. Hyperbolic geometry examples also appear. In the search of the associated metrics, we find a theorem, useful for explicitly determining the metric, which applies to all SU(2)-structures in general. Within our application to tangent sphere bundles, we discover a whole new class of metrics specific to 3d-geometry. The evolution equations of Conti–Salamon are considered, leading us to a new integrable SU(3)-structure on S×ℝ^+ associated to any flat M.