The total absolute curvature of submanifolds with singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916033903722496 |
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| author | Yamauchi, Yuta |
| author_facet | Yamauchi, Yuta |
| contents | In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean space $\boldsymbol{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-dimensional subspace of $\boldsymbol{R}^{n+r}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The total absolute curvature of submanifolds with singularities Yamauchi, Yuta Differential Geometry Primary 53C40, Secondary 53C65, 53C42, 57R45 In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean space $\boldsymbol{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-dimensional subspace of $\boldsymbol{R}^{n+r}$. |
| title | The total absolute curvature of submanifolds with singularities |
| topic | Differential Geometry Primary 53C40, Secondary 53C65, 53C42, 57R45 |
| url | https://arxiv.org/abs/2412.19147 |