Parallel Normal Fields

For curves \(\gamma \colon [a,b] \to \mathbb {R}^n\) there is an analog \(\kappa \colon [a,b] \to \mathbb {R}^{n-1}\) of the curvature function of a plane curve. In the context of unit speed curves, this function \(\kappa \) determines \(\gamma \) up to an orientation-preserving rigid motion of \(\mathbb {R}^n\) . Before we can define \(\kappa \) , we have to study parallel normal vector fields along a curve in \(\mathbb {R}^n\) .