  A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose
first normal line is the same as the first normal line of another curve. On the
other hand, a Mannheim curve in the 4-dimensional Euclidean space is a space
curve whose first normal line is the same as the second or third normal line of
another curve. By definitions, another curve is a parallel curve with respect
to the direction of the first normal vector. As smooth curves with singular
points, we consider framed curves in the Euclidean space. Then we define and
investigate Bertrand and Mannheim curves of framed curves. We give necessary
and sufficient conditions of Bertrand and Mannheim curves of both regular and
framed curves. It is well-known that the Bertrand curves of regular curves do
not exist under a condition. However, even if regular curves, Bertrand curves
exist as framed curves.
