$$\zeta $$-Ricci Soliton on Real Hypersurfaces of Nearly Kaehler 6-Sphere with SSMC

We use a nontrivial concircular vector field u on the unit sphere $\mathbf{S}^{n+1}$ S n + 1 in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere $\mathbf{S}^{n+1}$ 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 $\mathbf{S}^{n+1}$ 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 $\mathbf{S}^{n+1}$ S n + 1 .

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Characterizing small spheres in a unit sphere by Fischer–Marsden equation

Turki

Deshmukh

Vîlcu

2022

J Inequal Appl

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Characterizing non-totally geodesic spheres in a unit sphere

Al-Dayel

Deshmukh

Belova

2023

MATH

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<abstract><p>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 $.</p></abstract>

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$$\zeta $$-Ricci Soliton on Real Hypersurfaces of Nearly Kaehler 6-Sphere with SSMC

Cited by 2 publications

References 20 publications

Characterizing small spheres in a unit sphere by Fischer–Marsden equation

Characterizing small spheres in a unit sphere by Fischer–Marsden equation

Characterizing non-totally geodesic spheres in a unit sphere

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