Bulletin of the
Korean Mathematical Society BKMS

Let $\M^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c\neq 0$. A curve $\alpha$ immersed in $\M^3_q(c)$ 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. 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 $\M^3_q(c)$ correspond with curves for which there exist two constants $\lambda\neq 0$ and $\mu$ such that $\lambda\kappa+\mu\tau=1$, where $\kappa$ and $\tau$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\M^3_q(c)$ are the only twisted curves in $\M^3_q(c)$ 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.

Keywords: Bertrand curve, general helix, null curve, non-null curve

MSC numbers: 53C50, 53B25, 53B30