Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map
In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ fo...
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| Format: | Article |
| Language: | English |
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Sociedade Brasileira de Matemática
2018-07-01
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| Series: | Boletim da Sociedade Paranaense de Matemática |
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| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/31263 |
| _version_ | 1797633469588701184 |
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| author | Akram Mohammadpouri |
| author_facet | Akram Mohammadpouri |
| author_sort | Akram Mohammadpouri |
| collection | DOAJ |
| description |
In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.
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| first_indexed | 2024-03-11T11:54:27Z |
| format | Article |
| id | doaj.art-0c7ece7c92fa49a2b5f291415c3f6b6a |
| institution | Directory Open Access Journal |
| issn | 0037-8712 2175-1188 |
| language | English |
| last_indexed | 2024-03-11T11:54:27Z |
| publishDate | 2018-07-01 |
| publisher | Sociedade Brasileira de Matemática |
| record_format | Article |
| series | Boletim da Sociedade Paranaense de Matemática |
| spelling | doaj.art-0c7ece7c92fa49a2b5f291415c3f6b6a2023-11-08T20:10:17ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882018-07-0136310.5269/bspm.v36i3.3126315682Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss mapAkram Mohammadpouri0Tabriz University In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone. https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/31263Linearized operators $L_r$$L_r$-pointwise 1-type Gauss map$r$-minimalRotational hypersurfaces |
| spellingShingle | Akram Mohammadpouri Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map Boletim da Sociedade Paranaense de Matemática Linearized operators $L_r$ $L_r$-pointwise 1-type Gauss map $r$-minimal Rotational hypersurfaces |
| title | Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map |
| title_full | Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map |
| title_fullStr | Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map |
| title_full_unstemmed | Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map |
| title_short | Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map |
| title_sort | rotational hypersurfaces with l r pointwise 1 type gauss map |
| topic | Linearized operators $L_r$ $L_r$-pointwise 1-type Gauss map $r$-minimal Rotational hypersurfaces |
| url | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/31263 |
| work_keys_str_mv | AT akrammohammadpouri rotationalhypersurfaceswithlrpointwise1typegaussmap |