No abstract

B. Y. Chen introduced rectifying curves in R 3 as space curves whose position vector always lies in its rectifying plane. Recently, the authors have extended this definition (as well as several results about rectifying curves) to curves in the three-dimensional sphere. In this paper, we study rectifying curves in the three-dimensional hyperbolic space, and obtain some results of characterization and classification for such kind of curves. Our results give interesting and significant differences between hyperbolic, spherical and Euclidean geometries.Mathematics Subject Classification. 53A04, 53A05.

a b s t r a c tA curve α immersed in the three-dimensional sphere S 3 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. The curves α and β are said to be a pair of Bertrand curves in S 3 . One of our main results is a sort of theorem for Bertrand curves in S 3 which formally agrees with the classical one: ''Bertrand curves in S 3 correspond to 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; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S 3 as the only twisted curves in S 3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S 3 and (1, 3)-Bertrand curves in R 4 .

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.

In this paper, we study the surfaces whose geodesics are slant curves. We show that a unit speed curve γ in the 3-dimensional Euclidean space is a slant helix if and only if it is a geodesic of a helix surface. We prove that the striction line of a helix surface is a general helix; as a consequence, slant helices are characterized as geodesics of the tangent surface of a general helix. Finally, we provide two methods for constructing slant helices in helix surfaces.