# The Curve in $R^n$

Let $r:(a,b)\rightarrow{R^n}$ with $|r^{'}|=1$ is a natural parameter curve in $R^n$.
If $e_1(s)=r'(s),e_2(s),...,e_n(s)$ form an orthonormal frame, then we have Frenet formulae: $e_{i}^{'}=-k_{i-1}e_{i-1}+k_{i}e_{i+1}$ and $k_{0}=k_{n+1}=0,e_{0}=e_{n+1}=0$ ($k_1$ is its curvature).
Above is from the book Geometry 1 written by R.V.Gamkrelidze.

Theorem 1 When $n=2$,
if $k_1=0$, then $r$ is a straight line.
if $k_1=c\not=0$, then $r$ is a sphere.

Theorem 2 When $n=3$,
if $k_1=0$, then $r$ is a straight line.
if $k_1=c\not=0, k_2=0$, then $r$ is a sphere.
if $k_1=c\not=0, k_2=d\not=0$, then $r$ is a spiral line.

Can these theorems be generalized to higher dimension?

• Maybe we can generalize spiral line. In $R^3$, a spiral is $(acost,asint,bt)$ and its $k_1=\frac{a}{a^2+b^2}$,$k_2=\frac{b}{a^2+b^2}$. From The Uniqueness of Curve, we can get the conclusion. I guess that in $R^n$, we can introduce a "spiral line" like this $(acost,asint,b_1t,b_2t,...,b_{n-2}t)$. I compute its $k_1=\frac{a}{a^2+b_1^2+...+b_{n-2}^2}$. Other $k$, I still have on idea. – gaoxinge Jan 2 '14 at 1:38
• $d$ and $c$ are constants, right? – Robert Lewis Jan 5 '14 at 5:40
• Yes.@RobertLewis – gaoxinge Jan 5 '14 at 5:41
• thanks and thanks for responding so rapidly. Nice question, by the way, +1! – Robert Lewis Jan 5 '14 at 5:41

• I would upvote this, but you have $8008$ rep... – The Chaz 2.0 Jan 7 '14 at 4:05