在n維的歐氏空間，我們證明如果一個C^{infinity}-special Frenet曲線是(i_1,...,i_m)-Bertrand曲線的話，那麼在這曲線上每一點的Frenet (i_1,...,i_m)-normal平面必定包含Frenet 1-normal直線。另外,我們還證明在4維的歐氏空間如果(1,3)-Bertrand曲線有超過一個(1,3)-Bertrand mate,那麼它就會有無窮多個(1,3)-Bertrand mates。這個情況的發生若且為若k_1和k_2/k_3是常數,其中k_1、k_2和k_3分別是這曲線的曲率函數。

In an n-dimentional Euclidean space R^n, we prove that if a C^{infinity}-special Frenet curve
C is a (i_1,...,i_m)-Bertrand curve then its Frenet (i_1,...,i_m)-normal plane at c(s) must
contain the Frenet 1-normal line. In addition, we prove that if a (1,3)-Bertrand curve
in R^4 has more than one (1,3)-Bertrand mate, then it has infinitely many Bertrand
mates. This case occurs if and only if its curvature function k1 and the ratio of its
curvature functions k2 and k3 are constant.

[dC76] Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall Inc.,
Englewood Cliffs, N.J., 1976, Translated from the Portuguese.
[MY03] Hiroo Matsuda and Shinsuke Yorozu, Notes on Bertrand curves, Yokohama Math. J. 50
(2003), no. 1-2, 41–58.
[WL67] Yung-chow Wong and Hon-fei Lai, A critical examination of the theory of curves in three
dimensional differential geometry, Tˆohoku Math. J. (2) 19 (1967), 1–31.
[Won41] Yung-Chow Wong, On the generalized helices of Hayden and Sypt´ak in an N-space, Proc.
Cambridge Philos. Soc. 37 (1941), 229–243.