Taut contact hyperbolas on three-manifolds

In this paper, we introduce the notion of taut contact hyperbola on three-manifolds. It is the hyperbolic analogue of the taut contact circle notion introduced by Geiges and Gonzalo (Invent. Math., 121: 147-209, 1995), (J. Differ. Geom., 46: 236-286, 1997). Then, we characterize and study this notion, exhibiting several examples, and emphasizing differences and analogies between taut contact hyperbolas and taut contact circles. Moreover, we show that taut contact hyperbolas are related to some classic notions existing in the literature. In particular, it is related to the notion of conformally Anosov flow, to the critical point condition for the Chern-Hamilton energy functional and to the generalized Finsler structures introduced by R. Bryant. Moreover, taut contact hyperbolas are related to the bi-contact metric structures introduced in D. Perrone (Ann. Global Anal. Geom., 52: 213-235, 2017).