A Bertrand curve in the 4-dimensional Euclidean space is a whose first normal line same as of another curve. On other hand, Mannheim second or third In this paper, we investigate not only and curves regular curves, but also singular curves. As consider framed We give conditions for to become It well-known that do exist under condition space. However, even

Curves as a subject of differential geometry have been intriguing for researchers throughout mathematical history and so they have been one of the interesting research fields. Regular curves play a central role in the theory of curves in differential geometry. In the theory of curves, there are some special curves such as Bertrand curves, Mannheim curves, involute and evolute curves, and pedal ...

In this paper, we give a new approach for Bertrand and Mannheim curves in $3D$ Lie groups with bi-invariant metrics. way, some conditions including the known results have been given curve to be or Euclidean space groups.

Dual representations of the Bertrand offset-surfaces are specified and several new results gained in terms their integral invariants. A description offsets developable surfaces is given. Furthermore, relationships through striction curves ruled invariants obtained.