Global Differential Geometry of Curves in Three-Dimensional Heisenberg Group and CR Sphere

In this paper, we study some global properties of curves in the Heisenberg group $$H_{1}$$. In particular, we obtain Fenchel-type theorem and Fary–Milnor type theorem, together with Bray–Jauregui type theorem. We also prove the fundamental theorem of curves in the standard CR 3-sphere $$S^{3}$$. As an application, we classify some horizontally regular curves in $$S^{3}$$.