Rectifying Curves in the Three-Dimensional Hyperbolic Space

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.

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

“…It is also possible to introduce Frenet frames in Riemannian manifolds [19,30], see also [4,19,23,24,31]. Analogously, one can also define RM frames [14,15].…”

“…This definition makes sense due to the double nature of E m+1 as both a manifold and a tangent space. To extend it to a Riemannian manifold M m+1 , we should replace α − p by a geodesic connecting p to a point α(s) on the curve, as done in [23,24] for the study of rectifying curves:…”

Some characterizations of rectifying curves in the 3-dimensional hyperbolic space $\mathbb H^{3}(-r)$

“…To know more about the characterization of rectifying curve we refer the reader to see [1,2,6]. In [7], P. Lucas and J.A.O. Yagues, studied rectifying curves in the three-dimensional hyperbolic space, and obtain some results of characterization and classification for such kind of curves.…”

Some characterizations of timelike rectifying curves in de sitter 3 space

“…Ns". In this sense, Lucas and Yagües give the concept of rectifying curves in three dimensional spherical and hyperbolic space from the viewpoint of Riemannian Space Forms by using this idea in [9,10].…”

“…In this study, as inspiration from [9,10], we introduce the notions of timelike rectifying curve with respect to curve-hypersurface frame and timelike conical surface in De Sitter 3-space as non-flat Lorentzian space form viewpoint. After, we give relationship between timelike rectifying curve and geodesic of timelike conical surface in De Sitter 3-space.…”