BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS

Abstract

Let M 3(c) denote the 3-dimensional space form of index q = 0,1, and constant curvature c 6 0. A curveimmersed in M 3(c) is said to be a Bertrand curve if there exists another curveand a one-to-one correspondence betweenandsuch that both curves have common principal normal geodesics at corresponding points. We obtain charac- terizations 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 M3(c) correspond with curves for which there exist two constants � 6 0 and µ such that �� + µ� = 1, whereand � stand for the curvature and torsion of the curve. As a consequence, non-null helices in M 3(c) are the only twisted curves in M 3(c) having in- finite 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.