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 for 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 to a new integrable SU(3)-structure on S × R + associated to any flat M .