Definition of a Frenet curve Recently I followed a course in Differential Geometry. The book we use is Differential Geometry: Curves - Surfaces - Manifolds by Wolfgang Kühnel. My question is about the definition of a Frenet curve.

2.4. Definition (Frenet curve). Let $c(s)$ be a regular curve in $\mathbb{R}^n$ which is parametrized by arc length and $n$-times continuously differentiable. Then $c$ is called a Frenet curve, if at every point the vectors $c',c'',\dots,c^{(n-1)}$ are linearly independent. The Frenet $n$-frame $e_1,e_2,\dots,e_n$ is then uniquely determined by the following conditions:
(i) $e_1,\dots,e_n$ are orthonormal and positively oriented.
(ii) For every $k=1,\dots,n-1$ one has $\text{Span}(e_1,\dots,e_k)=\text{Span}(c',\dots,c^{(k)})$.
(iii) $\langle c^{(k)},e_k\rangle>0$ for $k=1,\dots,n-1$.

My thought: In this definition, I think (i) and (ii) are enough to uniquely determined $e_1,\dots,e_n$. Indeed, one obtains $e_1,\dots,e_{n-1}$ from $c',\dots,c^{(n-1)}$ by means of the Gram-Schmidt orthogonalization procedure, and the missing $e_n$ is then uniquely determined by condition (i). Why condition (iii) is necessary?
Any help would be appreciated.

Condition (iii) is required in order to ensure that the initial $n-1$ vectors have the same directions.
If you don’t impose it you could have for example $e_1=-\frac{c^\prime}{\Vert c^\prime\Vert}$ and therefore the Frenet $n$-frame « running » in the opposite direction of the one of the tangent of the curve.
