Hypersurfaces in a Euclidean space with a Killing vector field
An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterization...
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://aimspress.com/article/doi/10.3934/math.2024093?viewType=HTML |
| _version_ | 1797355364255006720 |
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| author | Mohammed Guediri Sharief Deshmukh |
| author_facet | Mohammed Guediri Sharief Deshmukh |
| author_sort | Mohammed Guediri |
| collection | DOAJ |
| description | An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface. |
| first_indexed | 2024-03-08T14:10:03Z |
| format | Article |
| id | doaj.art-812e59684d8a4e35845f4cf8e14eea0d |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-08T14:10:03Z |
| publishDate | 2024-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-812e59684d8a4e35845f4cf8e14eea0d2024-01-15T01:26:07ZengAIMS PressAIMS Mathematics2473-69882024-01-01911899191010.3934/math.2024093Hypersurfaces in a Euclidean space with a Killing vector fieldMohammed Guediri0Sharief Deshmukh1Department of Mathematics, College of Science, King Saud University, P. O. Box-2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P. O. Box-2455, Riyadh 11451, Saudi ArabiaAn odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface.https://aimspress.com/article/doi/10.3934/math.2024093?viewType=HTMLeuclidean spacehypersurfacekilling vector field |
| spellingShingle | Mohammed Guediri Sharief Deshmukh Hypersurfaces in a Euclidean space with a Killing vector field AIMS Mathematics euclidean space hypersurface killing vector field |
| title | Hypersurfaces in a Euclidean space with a Killing vector field |
| title_full | Hypersurfaces in a Euclidean space with a Killing vector field |
| title_fullStr | Hypersurfaces in a Euclidean space with a Killing vector field |
| title_full_unstemmed | Hypersurfaces in a Euclidean space with a Killing vector field |
| title_short | Hypersurfaces in a Euclidean space with a Killing vector field |
| title_sort | hypersurfaces in a euclidean space with a killing vector field |
| topic | euclidean space hypersurface killing vector field |
| url | https://aimspress.com/article/doi/10.3934/math.2024093?viewType=HTML |
| work_keys_str_mv | AT mohammedguediri hypersurfacesinaeuclideanspacewithakillingvectorfield AT shariefdeshmukh hypersurfacesinaeuclideanspacewithakillingvectorfield |