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

Abstract

Abstract. Let M 3 q (c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c = 0. A curve α immersed in M 3 q (c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β 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: nonnull Bertrand curves in M 3 q (c) correspond with curves for which there exist two constants λ = 0 and µ such that λκ + µτ = 1, where κ and τ 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.

Mannheim Curves in Nonflat 3-Dimensional Space Forms

“…In addition, the definition of angle in Euclidean is well known by us. However, the concept of general angle in semi-Euclidean space has been drawing our attention [9]. On this basis, we show and proof some properties of the angle between tangent vectors or binormal vectors of Mannheim curves and their partner curves at corresponding points.…”

“…The other kind of curve which has been focusing a lot of researchers' attention since the beginning is Bertrand curve [8][9][10][11]. In particular, Lucas and Ortega-Yagües have devoted their work to the research of properties of Bertrand curves which include nonnull Bertrand curves and null Bertrand curves and they obtained many perfect characterizations of Bertrand curves in nonflat 3-dimensional space forms [9].…”

Some characterizations of timelike rectifying curves in de sitter 3 space

“…We give the fundamental notions for motivation to differential geometry of timelike curves and timelike surface in De Sitter 3-space and Minkowski 4-space. For more detail and background, see [11][12][13][14].…”

“…Thus, we get that the rectifying condition (22), and so  is a timelike rectifying curve. Now, we give a characterization with respect to the ratio of geodesic torsion and geodesic curvature for timelike rectifying curves in 3 be a unit speed timelike curve in a timelike conical surface which is given by the parametrization (13) such that some differentiable functions () u u s = and () v v s = . By using ( 14) and (15), we obtain that .…”

“…be a geodesic in timelike conical surface M which is given by the parametrization(13) such that some statement (v) by using (29) and (32).…”