Bertrand curves in the three-dimensional sphere

摘要

A curve α immersed in the three-dimensional sphere. S3 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. S3. One of our main results is a sort of theorem for Bertrand curves in. S3 which formally agrees with the classical one: "Bertrand curves in. S3 correspond to curves for which there exist two constants λ. ≠. 0 and μ such that λκ. +. μτ. =. 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. S3 as the only twisted curves in. S3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in. S3 and (1,3)-Bertrand curves in. R4.