Hypersurfaces of a Sasakian Manifold

• Main Authors: Haila Alodan, Sharief Deshmukh, Nasser Bin Turki, Gabriel-Eduard Vîlcu
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Mathematics
• Subjects: hypersurface; Sasakian manifold; Laplace operator; eigenvalue
• Online Access: https://www.mdpi.com/2227-7390/8/6/877

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> of the Sasakian manifold induces a vector field <inline-formula> <math display="inline"> <semantics> <msup> <mi>ξ</mi> <mi>T</mi> </msup> </semantics> </math> </inline-formula> on the hypersurface, namely the tangential component of <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> to hypersurface, and it also gives a smooth function <inline-formula> <math display="inline"> <semantics> <mi>ρ</mi> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∇</mo> <mi>ρ</mi> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <msup> <mi>ξ</mi> <mi>T</mi> </msup> </semantics> </math> </inline-formula>) 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∇</mo> <mi>ρ</mi> </mrow> </semantics> </math> </inline-formula> has a certain lower bound.