Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in ${\mathit{R}}^N$

First, this paper presents a systematic procedure for constructing criteria for singularities of curves of finite multiplicities in ${\mathit{R}}^N$. Based on this method, we provide explicit criteria for singularities of multiplicities two, three, and four, including specific cusps appearing only in dimensions three or higher. Furthermore, we generalize the normalized curvature functions and the cuspidal curvature to singular curves in ${\mathit{R}}^N$. Using these generalized curvatures, we reinterpret the existence and uniqueness theorem given by Fukui for curves in ${\mathit{R}}^N$ of finite multiplicities.