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).

在本文中，我们介绍了三流形上的紧接触双曲线的概念。它是 Geiges 和 Gonzalo (Invent. Math., 121: 147–209, 1995), (J. Differ. Geom., 46: 236–286, 1997) 引入的紧接触圆概念的双曲线类比。然后，我们描述和研究了这个概念，展示了几个例子，并强调了拉紧接触双曲线和拉紧接触圆之间的差异和类比。此外，我们表明拉紧接触双曲线与文献中存在的一些经典概念有关。特别是，它与共形 Anosov 流的概念、Chern-Hamilton 能量泛函的临界点条件以及 R. Bryant 引入的广义 Finsler 结构有关。此外，紧接触双曲线与 D 中引入的双接触度量结构有关。