Characterizing non-totally geodesic spheres in a unit sphere

Ibrahim Al-Dayel; Sharief Deshmukh; Olga Belova; Ibrahim Al-Dayel; Sharief Deshmukh; Olga Belova

doi:10.3934/math.20231088

AIMS Mathematics

2023, Volume 8, Issue 9: 21359-21370. doi: 10.3934/math.20231088

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Research article

Characterizing non-totally geodesic spheres in a unit sphere

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia
2.
Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh-11451, Saudi Arabia
3.
Educational Scientific Cluster "Institute of High Technologies", Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016, Kaliningrad, Russia

Received: 22 May 2023 Revised: 15 June 2023 Accepted: 15 June 2023 Published: 05 July 2023
MSC : 53C20, 53C99, 58J99

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 $.
- small sphere,
- concircular vector field,
- the Fischer–Marsden equation,
- the Ricci curvature
Citation: Ibrahim Al-Dayel, Sharief Deshmukh, Olga Belova. Characterizing non-totally geodesic spheres in a unit sphere[J]. AIMS Mathematics, 2023, 8(9): 21359-21370. doi: 10.3934/math.20231088

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Abstract

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 $.

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