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Article

Hypersurfaces of a Sasakian Manifold

1
Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
2
Department of Cybernetics, Economic Informatics, Finance and Accountancy, Petroleum-Gas University of Ploieşti, Bd. Bucureşti 39, 100680 Ploieşti, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(6), 877; https://doi.org/10.3390/math8060877
Received: 3 May 2020 / Revised: 25 May 2020 / Accepted: 25 May 2020 / Published: 1 June 2020
(This article belongs to the Special Issue Complex and Contact Manifolds)
We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ρ has a certain lower bound. View Full-Text
Keywords: hypersurface; Sasakian manifold; Laplace operator; eigenvalue hypersurface; Sasakian manifold; Laplace operator; eigenvalue
MDPI and ACS Style

Alodan, H.; Deshmukh, S.; Turki, N.B.; Vîlcu, G.-E. Hypersurfaces of a Sasakian Manifold. Mathematics 2020, 8, 877. https://doi.org/10.3390/math8060877

AMA Style

Alodan H, Deshmukh S, Turki NB, Vîlcu G-E. Hypersurfaces of a Sasakian Manifold. Mathematics. 2020; 8(6):877. https://doi.org/10.3390/math8060877

Chicago/Turabian Style

Alodan, Haila, Sharief Deshmukh, Nasser B. Turki, and Gabriel-Eduard Vîlcu. 2020. "Hypersurfaces of a Sasakian Manifold" Mathematics 8, no. 6: 877. https://doi.org/10.3390/math8060877

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