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Bertrand curves in 3-dimensional space forms. (English) Zbl 1287.53011

Summary: We define Bertrand curves in a 3-dimensional Riemannian manifold and prove that a Frenet curve \(\alpha\) with curvature \(\kappa\) and torsion \(\tau\) in a 3-dimensional simply connected space form is a Bertrand curve if and only if it satisfies \(\tau =0\) or \(\kappa +a\tau =b\) for constants \(a\) and \(b\neq 0\).

MSC:

53B25 Local submanifolds
53A04 Curves in Euclidean and related spaces
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