Curvature surfaces in generic conformally flat hypersurfaces arising from Poincaré metric -- Extension and Approximation
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| Format: | Preprint |
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2023
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| _version_ | 1866909366985162752 |
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| author | Matsuura, Nozomu Suyama, Yoshihiko |
| author_facet | Matsuura, Nozomu Suyama, Yoshihiko |
| contents | We study generic conformally flat (analytic-)hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$. Such a local-hypersurface is obtained as an evolution of surfaces issuing from a certain surface in $\mathbb{R}^4$, and then, in consequence, the original surface is a (principal-)curvature surface of the hypersurface. The Poincaré metric ${\check g}_H$ of the upper half plane leads to a $6$-dimensional set of rational Riemannian metrics $g_0$ of $\mathbb{R}^2$: on a simply connected open set in the regular domain of $g_0$, a curvature surface $f^0$ with the metric $g_0$ is determined, which we denote by $(f^0,g_0)$. In this paper, we choose a suitable metric $g_0$ of $\mathbb{R}^2$ determined by ${\check g}_H$ to get nice curvature surfaces (but it also has degenerate and divergent points in $\mathbb{R}^2$), and clarify the structure of the curvature surfaces $(f^0,g_0)$: the curvature surfaces $(f^0,g_0)$ extend analytically to what kind of set in $\mathbb{R}^2$ beyond the regular set of $g_0$, and then the extended surface $(f^0,g_0)$ is defined on a certain open set of $\mathbb{R}^2$ and bounded in $\mathbb{R}^4$; for the extended surface $(f^0,g_0)$, we explicitly catch the set of degenerate points and the limits in $\mathbb{R}^4$ of both ends of every principal curvature line, and then the two limits of every line for one principal curvature are parallel small circles in a standard $2$-sphere $\mathbb{S}^2$. Then, every principal curvature line in the extended surface $(f^0,g_0)$ is expressed by a frame field of $\mathbb{R}^4$ induced on the surface from a hypersurface and it lies on a standard $2$-sphere $\mathbb{S}^2$ with line-dependent radius. We also provide a general method of constructing an approximation of such frame fields, and obtain the entire pictures of those lines including degenerate points of $(f^0,g_0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12128 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature surfaces in generic conformally flat hypersurfaces arising from Poincaré metric -- Extension and Approximation Matsuura, Nozomu Suyama, Yoshihiko Differential Geometry 53A07, 53C40, 68W25 We study generic conformally flat (analytic-)hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$. Such a local-hypersurface is obtained as an evolution of surfaces issuing from a certain surface in $\mathbb{R}^4$, and then, in consequence, the original surface is a (principal-)curvature surface of the hypersurface. The Poincaré metric ${\check g}_H$ of the upper half plane leads to a $6$-dimensional set of rational Riemannian metrics $g_0$ of $\mathbb{R}^2$: on a simply connected open set in the regular domain of $g_0$, a curvature surface $f^0$ with the metric $g_0$ is determined, which we denote by $(f^0,g_0)$. In this paper, we choose a suitable metric $g_0$ of $\mathbb{R}^2$ determined by ${\check g}_H$ to get nice curvature surfaces (but it also has degenerate and divergent points in $\mathbb{R}^2$), and clarify the structure of the curvature surfaces $(f^0,g_0)$: the curvature surfaces $(f^0,g_0)$ extend analytically to what kind of set in $\mathbb{R}^2$ beyond the regular set of $g_0$, and then the extended surface $(f^0,g_0)$ is defined on a certain open set of $\mathbb{R}^2$ and bounded in $\mathbb{R}^4$; for the extended surface $(f^0,g_0)$, we explicitly catch the set of degenerate points and the limits in $\mathbb{R}^4$ of both ends of every principal curvature line, and then the two limits of every line for one principal curvature are parallel small circles in a standard $2$-sphere $\mathbb{S}^2$. Then, every principal curvature line in the extended surface $(f^0,g_0)$ is expressed by a frame field of $\mathbb{R}^4$ induced on the surface from a hypersurface and it lies on a standard $2$-sphere $\mathbb{S}^2$ with line-dependent radius. We also provide a general method of constructing an approximation of such frame fields, and obtain the entire pictures of those lines including degenerate points of $(f^0,g_0)$. |
| title | Curvature surfaces in generic conformally flat hypersurfaces arising from Poincaré metric -- Extension and Approximation |
| topic | Differential Geometry 53A07, 53C40, 68W25 |
| url | https://arxiv.org/abs/2301.12128 |