Taut contact circles and bi-contact metric structures on three-manifolds

Abstract

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), ${\widetilde{SL}}(2,{\mathbb {R}})$, ${\widetilde{E}}(2)$, E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure $(\eta _1,\eta _2)$ with the so-called Webster function ${\mathcal {W}}$ constant along the flow of $\xi _1$ (equivalently $\xi _2$). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).