On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold
We recall classical themes such as 'on hearing the shape of a drum' or 'can one hear the shape of a drum?', and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under wh...
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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241633 |
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| author | Sharief Deshmukh Amira Ishan Olga Belova |
| author_facet | Sharief Deshmukh Amira Ishan Olga Belova |
| author_sort | Sharief Deshmukh |
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| description | We recall classical themes such as 'on hearing the shape of a drum' or 'can one hear the shape of a drum?', and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an $ n $-dimensional closed Riemannian manifold $ \left(M^{n}, g\right) $ having a non-zero eigenvalue $ nc $ for a positive constant $ c $ (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere $ S^{n}(c) $), is isometric to $ S^{n}(c) $. We address this issue in two situations. First, we consider the compact $ \left(M^{n}, g\right) $ as the hypersurface of the Euclidean space $ \left(R^{n+1}, \langle, \rangle \right) $ with isometric immersion $ f:\left(M^{n}, g\right) \rightarrow $ $ \left(R^{n+1}, \langle, \rangle \right) $ and a constant unit vector $ \overrightarrow{a} $ such that the function $ \rho = \langle f, \overrightarrow{a}\rangle $ satisfying $ \Delta \rho = -nc\rho $ for a positive constant $ c $ is isometric to $ S^{n}(c) $ if and only if $ \left(M^{n}, g\right) $ is isometric to $ S^{n}(c) $ provided the integral of Ricci curvature $ Ric\left(\nabla \rho, \nabla \rho \right) $ has an appropriate lower bound. In the second situation, we consider that the compact $ \left(M^{n}, g\right) $ admits a non-trivial concircular vector field $ \xi $ with potential function $ \sigma $ satisfying $ \Delta \sigma = -nc\sigma $ for a positive constant $ c $ and a specific function $ f $ related to $ \xi $ (called circular function) is constant along the integral curves of $ \xi $ if and only if $ \left(M^{n}, g\right) $ is isometric to $ S^{n}(c) $. |
| format | Article |
| id | doaj-art-b1cfc0a6fb584da49f304d943edc0626 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj-art-b1cfc0a6fb584da49f304d943edc06262025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912342723428810.3934/math.20241633On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifoldSharief Deshmukh0Amira Ishan1Olga Belova2Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh-11451, Saudi Arabia; shariefd@ksu.edu.saDepartment of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia; a.ishan@tu.edu.saEducational Scientific Cluster "Institute of High Technologies", Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016, Kaliningrad, Russia; olgaobelova@mail.ruWe recall classical themes such as 'on hearing the shape of a drum' or 'can one hear the shape of a drum?', and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an $ n $-dimensional closed Riemannian manifold $ \left(M^{n}, g\right) $ having a non-zero eigenvalue $ nc $ for a positive constant $ c $ (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere $ S^{n}(c) $), is isometric to $ S^{n}(c) $. We address this issue in two situations. First, we consider the compact $ \left(M^{n}, g\right) $ as the hypersurface of the Euclidean space $ \left(R^{n+1}, \langle, \rangle \right) $ with isometric immersion $ f:\left(M^{n}, g\right) \rightarrow $ $ \left(R^{n+1}, \langle, \rangle \right) $ and a constant unit vector $ \overrightarrow{a} $ such that the function $ \rho = \langle f, \overrightarrow{a}\rangle $ satisfying $ \Delta \rho = -nc\rho $ for a positive constant $ c $ is isometric to $ S^{n}(c) $ if and only if $ \left(M^{n}, g\right) $ is isometric to $ S^{n}(c) $ provided the integral of Ricci curvature $ Ric\left(\nabla \rho, \nabla \rho \right) $ has an appropriate lower bound. In the second situation, we consider that the compact $ \left(M^{n}, g\right) $ admits a non-trivial concircular vector field $ \xi $ with potential function $ \sigma $ satisfying $ \Delta \sigma = -nc\sigma $ for a positive constant $ c $ and a specific function $ f $ related to $ \xi $ (called circular function) is constant along the integral curves of $ \xi $ if and only if $ \left(M^{n}, g\right) $ is isometric to $ S^{n}(c) $.https://www.aimspress.com/article/doi/10.3934/math.20241633first non-zero eigenvalueeigenfunctionlaplace operatorricci curvatureisometric to sphere |
| spellingShingle | Sharief Deshmukh Amira Ishan Olga Belova On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold AIMS Mathematics first non-zero eigenvalue eigenfunction laplace operator ricci curvature isometric to sphere |
| title | On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold |
| title_full | On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold |
| title_fullStr | On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold |
| title_full_unstemmed | On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold |
| title_short | On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $ S^n(c) $ on a Riemannian manifold |
| title_sort | on eigenfunctions corresponding to first non zero eigenvalue of the sphere s n c on a riemannian manifold |
| topic | first non-zero eigenvalue eigenfunction laplace operator ricci curvature isometric to sphere |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241633 |
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