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Let $\gamma: I \rightarrow \mathbb{R}^n$ be a smooth curve. We define a Frenet frame to be an orthonormal frame $X_1, \ldots X_n$ such that for all $1 \leq k \leq n$, $\gamma^{(k)}(t)$ is contained in the linear span of $X_1(t), \ldots, X_k(t)$, $t \in I$.

What is an example of a smooth curve $\gamma$ with no Frenet frame?

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    Just for reference, this is an exercise in Unit 2 of Balázs Csikós's "Budapest Semesters in Mathematics" differential geometry notes, page 7 of the following pdf: http://www.cs.elte.hu/geometry/csikos/dif/dif2.pdf2011-02-22
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    Are you sure that this can be done with a smooth curve? Also, I think it might be important to mention that $X_1, \ldots, X_n$ are smooth vector fields along $\gamma$...2011-02-26

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