Characterizing non-totally geodesic spheres in a unit sphere
A concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functi...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-07-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231088?viewType=HTML |
| _version_ | 1797778016541081600 |
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| author | Ibrahim Al-Dayel Sharief Deshmukh Olga Belova |
| author_facet | Ibrahim Al-Dayel Sharief Deshmukh Olga Belova |
| author_sort | Ibrahim Al-Dayel |
| collection | DOAJ |
| description | A concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functions, $ f $ and $ \sigma $, defined on the hypersurface $ M $, where $ f $ is the restriction of the potential function $ \overline{f} $ of the concircural vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ to $ M $ and $ \sigma $ is defined as $ g\left(\mathbf{u}, N\right) $, where $ N $ is the unit normal to the hypersurface. In this paper, we show that if function $ f $ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $ \mathbf{w} $ has a certain lower bound, then a characterization of a small sphere in the unit sphere $ \mathbf{S}^{n+1} $ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $ M $ in the direction of the vector field $ \mathbf{w} $ with a non-zero function $ \sigma $. |
| first_indexed | 2024-03-12T23:11:17Z |
| format | Article |
| id | doaj.art-259868814f0441939ca9181a0319addb |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-12T23:11:17Z |
| publishDate | 2023-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-259868814f0441939ca9181a0319addb2023-07-18T01:26:33ZengAIMS PressAIMS Mathematics2473-69882023-07-0189213592137010.3934/math.20231088Characterizing non-totally geodesic spheres in a unit sphereIbrahim Al-Dayel0Sharief Deshmukh 1Olga Belova21. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia2. Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh-11451, Saudi Arabia3. Educational Scientific Cluster "Institute of High Technologies", Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016, Kaliningrad, RussiaA concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functions, $ f $ and $ \sigma $, defined on the hypersurface $ M $, where $ f $ is the restriction of the potential function $ \overline{f} $ of the concircural vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ to $ M $ and $ \sigma $ is defined as $ g\left(\mathbf{u}, N\right) $, where $ N $ is the unit normal to the hypersurface. In this paper, we show that if function $ f $ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $ \mathbf{w} $ has a certain lower bound, then a characterization of a small sphere in the unit sphere $ \mathbf{S}^{n+1} $ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $ M $ in the direction of the vector field $ \mathbf{w} $ with a non-zero function $ \sigma $.https://www.aimspress.com/article/doi/10.3934/math.20231088?viewType=HTMLsmall sphereconcircular vector fieldthe fischer–marsden equationthe ricci curvature |
| spellingShingle | Ibrahim Al-Dayel Sharief Deshmukh Olga Belova Characterizing non-totally geodesic spheres in a unit sphere AIMS Mathematics small sphere concircular vector field the fischer–marsden equation the ricci curvature |
| title | Characterizing non-totally geodesic spheres in a unit sphere |
| title_full | Characterizing non-totally geodesic spheres in a unit sphere |
| title_fullStr | Characterizing non-totally geodesic spheres in a unit sphere |
| title_full_unstemmed | Characterizing non-totally geodesic spheres in a unit sphere |
| title_short | Characterizing non-totally geodesic spheres in a unit sphere |
| title_sort | characterizing non totally geodesic spheres in a unit sphere |
| topic | small sphere concircular vector field the fischer–marsden equation the ricci curvature |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231088?viewType=HTML |
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