Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$
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| Format: | Preprint |
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2021
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| _version_ | 1866911801220792320 |
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| author | Güler, Erhan Turgay, Nurettin Cenk |
| author_facet | Güler, Erhan Turgay, Nurettin Cenk |
| contents | In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying the Gauss map $\mathcal{G}$ of this family with respect to the operator $\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. The operator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known by various names, including the Cheng--Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying the operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the equation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_03915 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$ Güler, Erhan Turgay, Nurettin Cenk Differential Geometry 53A07, 53C42 In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying the Gauss map $\mathcal{G}$ of this family with respect to the operator $\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. The operator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known by various names, including the Cheng--Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying the operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the equation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$. |
| title | Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$ |
| topic | Differential Geometry 53A07, 53C42 |
| url | https://arxiv.org/abs/2104.03915 |