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• W3165813545 abstract "Abstract 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 $mathbf{u}=-varphi (N)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:math> . 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 $mathbf{S}^{2n+1}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . 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} ) $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>R</mml:mi> <mml:mi>i</mml:mi> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> 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 $mathbf{S}^{2n+1}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> ." @default.
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• W3165813545 date "2021-03-19" @default.
• W3165813545 modified "2023-09-23" @default.
• W3165813545 title "Hypersurfaces of a Sasakian manifold - revisited" @default.
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• W3165813545 doi "https://doi.org/10.1186/s13660-021-02584-0" @default.
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