@article{oai:muroran-it.repo.nii.ac.jp:02000170,
 author = {Honda, Shun’ichi and Takahashi, Masatomo and Yu, Haiou},
 issue = {2},
 journal = {Journal of Geometry},
 month = {},
 note = {A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose rst
normal line is the same as the rst normal line of another curve. On the other hand, a
Mannheim curve in the 4-dimensional Euclidean space is a space curve whose rst normal
line is the same as the second or third normal line of another curve. In this paper, we
investigate not only Bertrand and Mannheim curves of regular space curves, but also
Bertrand and Mannheim curves of singular space curves. As singular curves, we consider
framed curves. We give conditions for framed curves to become Bertrand and Mannheim
curves. It is well-known that the Bertrand curves of regular curves do not exist under a
condition in the 4-dimensional Euclidean space. However, even if regular curves, Bertrand
curves exist as framed curves.},
 title = {Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space},
 volume = {114},
 year = {2023}
}