The three-dimensional pseudohermitian manifolds with positive, zero, and negative constant pseudohermitian sectional curvature, respectively, with vanishing pseudohermitian torsion are the Heisenberg group $$H_1$$H 1 , the CR sphere $$S^3$$S 3 , and the circle bundle $$B^1\times \mathbb {R}$$B 1 × R . While the differential geometry of curves in $$H_1$$H 1 and $$S^3$$S 3 has been studied before, there has not been any study on the differential geometry of curves in $$B^1\times \mathbb {R}$$B 1 × R . In this paper, we take the first step in this direction. Among other things, we find the explicit formulas of the p-curvature and T-variation for curves in $$B^1\times \mathbb {R}$$B 1 × R . We also prove the fundamental theorem of curves in $$B^1\times \mathbb {R}$$B 1 × R .