Hypersurfaces of a Sasakian Manifold

Haila Alodan,Nasser Bin Turki,Sharief Deshmukh,Gabriel-Eduard Vîlcu

doi:10.3390/math8060877

Haila Alodan, Nasser Bin Turki + Show 2 more

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https://doi.org/10.3390/math8060877

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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.

Highlights

It is well known that Sasakian manifolds are considered the odd dimensional analogue of Kählerian manifolds and Sasakian space forms can be regarded as the counterpart of complex space forms [1]
The differential geometry of hypersurfaces in a complex space form has been widely studied over the years, but though Sasakian manifolds are very important due to their elegant geometry as well as their important applications in theoretical physics, not as many studies have been realized for hypersurfaces in a Sasakian ambient space
A well known result is that of Watanabe, who used the Obata’s differential Equation in order to prove that a complete and connected totally umbilical hypersurface of a (2n + 1)-dimensional Sasakian manifold of constant mean curvature H is isometric with a sphere of radius √ 1 2 in the Euclidean space

Summary

Introduction

It is well known that Sasakian manifolds are considered the odd dimensional analogue of Kählerian manifolds and Sasakian space forms can be regarded as the counterpart of complex space forms [1].

Preliminaries

Volume and First Eigenvalue Estimates

A Bound on Energy of a Vector Field

CMC-Hypersurfaces