Bertrand and Mannheim curves in three-dimensional Walker manifolds

Abstract

We review the basic concepts of space curves, including curvature and torsion. We examine special curves such as Mannheim and Bertrand in a three-dimensional Euclidean space. We define Walker manifolds which are pseudo-Riemannian manifolds with a parallel null distribution. Then we compute Christoffel symbols and Levi-Civita connection components for an arbitrary three-dimensional Walker manifold. Finally, we derive the curvature and torsion of a regular curve on a three-dimensional Walker manifold. Then, we investigate necessary and sufficient conditions for Mannheim curves in a strict three-dimensional Walker manifold. Moreover, we also prove necessary and sufficient conditions of Bertrand curves in a three-dimensional Walker manifold.