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

TL;DR

This paper develops a systematic method to identify singularities of finite multiplicity curves in higher-dimensional spaces, generalizes curvature concepts, and reinterprets existing theorems.

Contribution

It introduces explicit criteria for singularities of multiplicities two, three, and four, and extends curvature notions to singular curves in $oldsymbol{R}^N$.

Findings

Explicit criteria for singularities of multiplicities two, three, and four.

Generalized curvature functions for singular curves in $oldsymbol{R}^N$.

Reinterpreted Fukui's existence and uniqueness theorem using these generalized curvatures.

Abstract

First, this paper presents a systematic procedure for constructing criteria for singularities of curves of finite multiplicities in $\boldsymbol{R}^N$. 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 $\boldsymbol{R}^N$. Using these generalized curvatures, we reinterpret the existence and uniqueness theorem given by Fukui for curves in $\boldsymbol{R}^N$ of finite multiplicities.