Bertrand curves in the three-dimensional sphere

A curve alpha immersed in the three-dimensional sphere S-3 is said to be a Bertrand curve if there exists another curve beta and a one-to-one correspondence between alpha and beta such that both curves have common principal normal geodesics at corresponding points. The curves alpha and beta 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 lambda not equal 0 and mu such that lambda kappa + mu tau = 1", where kappa and tau 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 (C) 2012 Elsevier B.V. All rights reserved.