Curves of generalized \(AW(k)\)-type in Euclidean spaces (Q2927993)

A sufficiently smooth unit speed curve \(\gamma\) in \(\mathbb{R}^n\) is said to be of generalized \(AW(k)\)-type if the normal component of its fifth derivative vector depends linearly on the normal components of at most two of the preceding derivative vectors. There are seven possible types of \(AW(k)\) curves. The main theorem provides a characterization of each type by conditions on the curvatures of \(\gamma\). Since a generalized \(AW(k)\)-type curve spans a space of dimension \(d < 5\), it is natural to refine this classification by a discussion of all cases with \(1 \leq d \leq 4\). For each sub-type, the authors provide characteristic differential equations that have to be or must not be fulfilled by the curvatures.