Curves in three dimensional Riemannian space forms

Abstract

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.