A curve $\gamma : I \rightarrow \mathbb{R}^n $ is called a curve of general type in $\mathbb{R}^n$ if the first $n-1$ derivatives are linearly independent $\forall t \in I$
A moving (orthonormal) frame along a curve $\gamma : I \rightarrow \mathbb{R}^n $ is a collection of >vector fields $t_1,\dots,t_n$ along $\gamma$ such that $< t_i(t), t_j(t)> = \delta_{ij}$
A moving frame $t_1,\dots,t_n$ along a curve $\gamma$ is called a Frenet Frame if $ \forall _{k=1,\dots,n} $, $\gamma^{(k)}(t)$ is contained in the linear span of $t_1(t),\dots,t_k(t)$.
The definitions above are from Balasz Csikos' textbook. We have previously shown that if from the first $k\leq n$ derivatives we get k linearly independent vectors, then those vectors span an osculating $k$-plane. My questions are:
- What exactly is a curve of general type? A curve that has a unique osculating $n$-$1$-plane everywhere?
- This question discussed a question from the textbook. As a follow up, how to construct a curve with infinitely many frenet frames?
- Furthermore, why does a curve of a general type has exactly $2^n$ frenet frames?
For the second question, I think it is not asking about a curve with infinitely many linearly independent derivatives, because in $\mathbb{R}^n$ there are a maximum $n$ linearly independent derivatives.
For the third question, since a curve of general type has $n$-$1$ linearly independent derivatives, and we need n linearly independent derivatives, that means we may have a 0 vector in the moving frame? I noticed that $2^n$ suggests a binary yes/no for $n$ things, perhaps $n$ linearly independent vectors, and yes means not a zero vector, and no means a zero vector.
A curve in $\Bbb R^n$ is of general type if it has a well-defined osculating $(n-1)$-plane at each point. I would prefer to say that it is not contained in any hyperplane, but that is not the definition here. However, if a curve is not of general type — in particular, if it is contained in a codimension-2 plane — then there will be infinitely many such frames (rotate the vectors normal to the codimension-2 subspace).
For your last question, think about sign changes on your frame.