Characterizing small spheres in a unit sphere by Fischer–Marsden equation
Abstract We use a nontrivial concircular vector field u on the unit sphere S n + 1 $\mathbf{S}^{n+1}$ in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere S n + 1 $\mathbf{S}^{n+1}$ naturally inherits a vector field v and a smooth function ρ. We use the conditio...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2022-09-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-022-02855-4 |
| Summary: | Abstract We use a nontrivial concircular vector field u on the unit sphere S n + 1 $\mathbf{S}^{n+1}$ in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere S n + 1 $\mathbf{S}^{n+1}$ naturally inherits a vector field v and a smooth function ρ. We use the condition that the vector field v is an eigenvector of the de-Rham Laplace operator together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find a characterization of small spheres in the unit sphere S n + 1 $\mathbf{S}^{n+1}$ . We also use the condition that the function ρ is a nontrivial solution of the Fischer–Marsden equation together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find another characterization of small spheres in the unit sphere S n + 1 $\mathbf{S}^{n+1}$ . |
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| ISSN: | 1029-242X |