# A curve is contained in a hyperplane if and only if $\kappa_{n-1} = 0$

I'm trying to prove that a curve $\gamma$ in $\mathbb{R}^n$ is contained in a hyperplane if and only if $\kappa_{n-1} = 0$, where $$\kappa_{i} = \frac{\langle e_{i}',e_{i+1}\rangle}{\| \gamma' \|}$$ is the $i^{th}$ curvature function of $\gamma$ (here $e_{i}$ is the $i$th vector in the Frenet Frame). My strategy is to show that $\kappa_{n-1} = 0$ if and only if the unit normal vector of the osculating hyperplane and the distance of the osculating hyperplane from the origin are constant. However, I'm not sure how to proceed. I can see why this result is true in 3 dimensions but am struggling to reframe it in a way that is easy to deal with in $n$ dimensions.

• what is the definition of the i-th curvature function? – Xipan Xiao Oct 3 '13 at 20:07