Asian Journal of Mathematics

Volume 24 (2020)

Number 3

Natural SU(2)-structures on tangent sphere bundles

Pages: 457 – 482



R. Albuquerque (Departamento de Matemática da Universidade de Évora, Centro de Investigação em Matemática e Aplicações, Évora, Portugal)


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 $M$.


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.

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