Bertrand curves in the three-dimensional sphere

Abstract A curve α immersed in the three-dimensional sphere S 3 is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. The curves α and β are said to be a pair of Bertrand curves in S 3 . One of our main results is a sort of theorem for Bertrand curves in S 3 which formally agrees with the classical one: “Bertrand curves in S 3 correspond to curves for which there exist two constants λ ≠ 0 and μ such that λ κ + μ τ = 1 ”, where κ and τ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S 3 as the only twisted curves in S 3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S 3 and (1,3)-Bertrand curves in R 4 .