Last generalized curvature zero for a hyper-planar curve in $\mathbb{R}^n$?

Question

Serret-Frenet frame has been generalized to curves in $\mathbb{R}^n$. I would like to enquire about what happens to the generalized curvatures if the curve belongs to one or more planes in $\mathbb{R}^n$. Is the last (the (n-1)th) generalized curvature zero when the curve lies only on the (n-1)-dimensional plane (also called the hyper-plane)? If it lies on an (n-2)-dimensional plane, then the (n-2)th curvature zero and (n-1)th becomes undefined and so on? This is what happens in $\mathbb{R}^3$.