Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms

Lee, Jae Won; Lee, Chul Woo; Vîlcu, Gabriel-Eduard

doi:10.1016/j.geomphys.2020.103768

Cited by 16 publications

(8 citation statements)

References 25 publications

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“…[25,31]) and, therefore, M is isometric to the unit sphere S 2n . Recall that an odd dimensional unit sphere S 2n+1 as a real hypersurface of the complex space C n+1 has the standard Sasakian structure (ϕ, ξ , η, g) [32]. As a particular case of the above theorem, we have the following consequence.…”

Section: Vector Fields U and V As Eigenvectors Of Laplace Operatormentioning

confidence: 87%

Hypersurfaces of a Sasakian manifold - revisited

et al. 2021

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We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

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Section: Vector Fields U and V As Eigenvectors Of Laplace Operatormentioning

confidence: 87%

Hypersurfaces of a Sasakian manifold - revisited

et al. 2021

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“…For Legendrian submanifolds, Lee et al proved the following. Theorem 14.2 [80] Let M n be a Legendrian submanifold of a Sasakian space formM 2n+1 (c) . Then:…”

Section: δ -Casorati Curvatures For Legendrian Submanifolds In Sasakimentioning

confidence: 99%

Recent developments in δ-Casorati curvature invariants

Chen¹

2021

Turk J Math

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“…Of course, the notations are well known: ζ is a structure vector, the (1, 1)-type tensor field is denoted by ψ, and η is the dual one-form. Moreover, the tensorial equation for a Sasakian manifold [22][23][24] with the structure (ψ, ζ, η) is given by…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

Mofarreh

Ali

Alluhaibi

et al. 2021

Journal of Mathematics

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In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

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Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms

Cited by 16 publications

References 25 publications

Hypersurfaces of a Sasakian manifold - revisited

Hypersurfaces of a Sasakian manifold - revisited

Recent developments in δ-Casorati curvature invariants

Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

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