Mannheim B-Curves in the Euclidean 3-Space E-3

Abstract

In 1878, Mannheim defined the notion of a Mannheim pair of curves and in 2008 Liu and Wang gave the necessary and sufficient condition for the Mannheim curves using the Frenet frame as a key ingredient. In this article we consider Bishop frame and introduce Mannheim B-pair in the Euclidean 3-space E-3. Then a thorough examination of the relationships between curvatures of a Mannheim B-pair is given. As the results of this examination, the relations between the Bishop vectors and Frenet vectors of these curves are found. Besides the relation between the Bishop curvatures of the Mannheim B-curves is obtained as k(2)* = k(1)/1 + lambda k(1).