Natural SU(2)-structures on tangent sphere bundles

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

Keywords

tangent bundle, SU(n)-structure, hypo structure, nearly-hypo structure, evolution equations

2010 Mathematics Subject Classification

Primary 53C15, 53C25, 53C44. Secondary 53C38, 53D18, 58A15, 58A32.

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The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no PIEF-GA-2012-332209.

Received 21 September 2017

Accepted 18 September 2019

Published 9 October 2020