Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\boldsymbol{R}^N$
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| Format: | Preprint |
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2026
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| _version_ | 1866914558340235264 |
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| author | Matsumoto, Jun Sano, Shuki Yanagida, Kiyoto |
| author_facet | Matsumoto, Jun Sano, Shuki Yanagida, Kiyoto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12052 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\boldsymbol{R}^N$ Matsumoto, Jun Sano, Shuki Yanagida, Kiyoto Differential Geometry Primary 53A04, Secondary 58K40 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. |
| title | Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\boldsymbol{R}^N$ |
| topic | Differential Geometry Primary 53A04, Secondary 58K40 |
| url | https://arxiv.org/abs/2605.12052 |