Hypersurfaces of a Sasakian Manifold

Haila Alodan, Sharief Deshmukh, Nasser Bin Turki and Gabriel-Eduard Vîlcu
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Haila Alodan: Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
Nasser Bin Turki: Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
Gabriel-Eduard Vîlcu: Department of Cybernetics, Economic Informatics, Finance and Accountancy, Petroleum-Gas University of Ploieşti, Bd. Bucureşti 39, 100680 Ploieşti, Romania

Mathematics, 2020, vol. 8, issue 6, 1-17

Abstract: 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.

Keywords: hypersurface; Sasakian manifold; Laplace operator; eigenvalue (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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