Natural SU(2)-structures on tangent sphere bundles

TL;DR

This paper explores natural SU(2)-structures on tangent sphere bundles of 3-manifolds, discovering new geometric structures, metrics, and integrable SU(3)-structures, enriching the understanding of special geometries in Riemannian and hyperbolic contexts.

Contribution

It introduces and analyzes new classes of SU(2)-structures on tangent sphere bundles, including double-hypo and hypo types, and finds explicit metrics and integrable structures related to 3D geometry.

Findings

New double-hypo structures on S^3×S^2

A theorem for explicit metric determination of SU(2)-structures

A new integrable SU(3)-structure on tangent sphere bundles of flat 3-manifolds

Abstract

We define and study natural $\mathrm{SU}(2)$-structures, in the sense of Conti-Salamon, on the total space $\cal S$ 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 $S^3\times S^2$, 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 $\mathrm{SU}(2)$-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 $\mathrm{SU}(3)$-structure on ${\cal S}\times\mathbb{R}_+$ associated to any flat $M$.