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

Title & Authors
BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS
Lucas, Pascual; Ortega-Yagues, Jose Antonio;

Abstract
Let $\small{\mathbb{M}^3_q(c)}$ denote the 3-dimensional space form of index $\small{q=0,1}$, and constant curvature $\small{c{\neq}0}$. A curve $\small{{\alpha}}$ immersed in $\small{\mathbb{M}^3_q(c)}$ is said to be a Bertrand curve if there exists another curve $\small{{\beta}}$ and a one-to-one correspondence between $\small{{\alpha}}$ and $\small{{\beta}}$ 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 $\small{\mathbb{M}^3_q(c)}$ correspond with curves for which there exist two constants $\small{{\lambda}{\neq}0}$ and $\small{{\mu}}$ such that $\small{{\lambda}{\kappa}+{\mu}{\tau}=1}$, where $\small{{\kappa}}$ and $\small{{\tau}}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\small{\mathbb{M}^3_q(c)}$ are the only twisted curves in $\small{\mathbb{M}^3_q(c)}$ 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;
Language
English
Cited by
1.
Curves in Three Dimensional Riemannian Space Forms, Results in Mathematics, 2014, 66, 3-4, 469
2.
Mannheim Curves in Nonflat 3-Dimensional Space Forms, Advances in Mathematical Physics, 2015, 2015, 1
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