Mannheim Curves in the three-dimensional Sphere

TL;DR

This paper explores Mannheim curves in the 3-sphere, defining their properties via geodesics, establishing key relations, and comparing them with Bertrand curves, extending the concept to curves in higher-dimensional spaces.

Contribution

It introduces a new definition of Mannheim curves in S^3 using geodesics, derives fundamental relations, and compares them with Bertrand curves and generalized curves in E^4.

Findings

Existence of a constant and a non-constant function relating curvatures of Mannheim pairs.

Derived a key relation: Lambda * kappa_alpha + Mu * Tau_alpha = 1.

Established connections between Mannheim curves in S^3 and generalized Mannheim curves in E^4.

Abstract

Mannheim curves are defined for immersed curves in 3-dimensional sphere S^3 . The definition is given by considering the geodesics of S^3. First, two special geodesics, called principal normal geodesic and binormal geodesic, of S^3 are defined by using Frenet vectors of a curve immersed in S^3. Later, the curve alpha is called a Mannheim curve if there exits another curve beta in S^3 such that the principal normal geodesics of beta coincide with the binormal geodesics of S^3 . It is obtained that if alpha and beta form a Mannheim pair then there exist a constant lambda that is not equal 0 and a non-constant function Mu such that Lambda.(kappa_alpha)+M(Tau_alpha)=1 where kappa_alpha, Tau_alpha are the curvatures of alpha. Moreover, the relation between a Mannheim curve immersed in S^3 and a generalized Mannheim curve in E^4 is obtained and a table containing comparison of Bertrand and Mannheim curves in S^3 is introduced.