Bertrand Curves in Non-Flat 3-Dimensional (Riemannian or Lorentzian) Space Forms

Let <TEX>M³q(c)</TEX> denote the 3-dimensional space form of index <TEX>$q=0,1$</TEX>, and constant curvature <TEX>c≠0</TEX>. A curve <TEX>α</TEX> immersed in <TEX>M³q(c)</TEX> is said to be a Bertrand curve if there exists another curve <TEX>β</TEX> and a one-to-one correspondence between <TEX>α</TEX> and <TEX>β</TEX> such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in <TEX>M³q(c)</TEX> correspond with curves for which there exist two constants <TEX>λ≠0</TEX> and <TEX>μ</TEX> such that <TEX>λκ+μτ=1</TEX>, where <TEX>κ</TEX> and <TEX>τ</TEX> stand for the curvature and torsion of the curve. As a consequence, non-null helices in <TEX>M³q(c)</TEX> are the only twisted curves in <TEX>M³q(c)</TEX> having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.