Bertrand curves in the three-dimensional sphere

A curve αα immersed in the three-dimensional sphere S3S3 is said to be a Bertrand curve if there exists another curve ββ and a one-to-one correspondence between αα and ββ such that both curves have common principal normal geodesics at corresponding points. The curves αα and ββ are said to be a pair of Bertrand curves in S3S3. One of our main results is a sort of theorem for Bertrand curves in S3S3 which formally agrees with the classical one: “Bertrand curves in S3S3 correspond to curves for which there exist two constants λ≠0λ≠0 and μμ such that λκ+μτ=1λκ+μτ=1”, where κκ and ττ 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 S3S3 as the only twisted curves in S3S3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S3S3 and (1,3)-Bertrand curves in R4R4.