Bertrand curves in 3-dimensional space forms

摘要

We define Bertrand curves in a 3-dimensional Riemannian manifold and prove that a Frenet curve α with the curvature κ and the torsion τ in a 3-dimensional simply connected space form is a Bertrand curve if and only if it satisfies τ=0 or κ+aτ=b for constants a and b≠0.