Mannheim Curves in 3-Dimensional Space Forms

Abstract

Abstract. We define the Mannheim curve in a 3-dimensional Riemannian manifold, which is a generalization of the Mannheim curve in Euclidean space. In particular, we study the Mannheim curves and their partner curves in 3-dimensional space forms.

Mannheim partner curves in Cartan-Vranceanu 3-space

“…Mannheim mate of a biharmonic curve in the Heisenberg group Heis 3 , which is a special case of Cartan-Vranceanu metrics, are studied in [12,13]. Choi and his colleagues characterized Mannheim curves and their mate curves in 3-dimensional space forms, making a generalization of the results obtained by Liu and Wang in [5].…”

On generalized partially null Mannheim curves in Minkowski space-time

“…In Minkowski spacetime, generalized spacelike Mannheim curves whose Frenet frame contains only non-null vectors, are defined in [6]. Mannheim curves lying in 3-dimensional space forms E 3 and S 3 in E 4 , as well as in H 3 in E 4 1 , are studied in [3]. In this paper, we define generalized partially null and pseudo null Mannheim curves in Minkowski space-time.…”

On generalized null Mannheim curves in Minkowski space-time

“…In particular, Mannheim curves in E 3 are defined by the property that their principal normal lines coincide with the binormal lines of their mate curves at the corresponding points [4,7,11]. The parameter equation of a Mannheim curve α in E 3 is given in [4] by α(t) = ( h(t) sin(t)dt, h(t) cos(t)dt, h(t)g(t)dt), where g : I → R is any smooth function and the function h : I → R is given by h = (1 + g 2 + g ′2 ) 3 + (1 + g 2 ) 3 (g + g ′′ ) 2 (1 + g 2 ) 3/2 (1 + g 2 + g ′2 ) 5/2 .…”