We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field $\xi$ of the Sasakian manifold induces a vector field ${\xi }^T$ on the hypersurface, namely the tangential component of $\xi$ to hypersurface, and it also gives a smooth function $\rho$ 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 $\nabla \rho$ 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 ${\xi }^T$ ) 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 $\nabla \rho$ has a certain lower bound.

中文翻译：

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Sasakian流形的超曲面

我们扩展了渡边发起的Sasakian流形中可定向超曲面的研究。锐步矢量场 $\xi$ 萨萨克族流形的感应矢量场 ${\xi }^T$ 在超曲面上，即 $\xi$ 到超表面，并且还具有光滑的功能 $\rho$ 在超曲面上，这是Reeb矢量场在单位法线上的投影。首先，我们找到一个紧凑的可定向超曲面的体积估计值，然后使用它们来找到超曲面上Laplace算子的第一个非零特征值的上限，这表明如果等式成立，那么超曲面对于某个球面是等轴测的。另外，我们对向量场的能量使用一个界限 $\nabla \rho$ 在Sasakian流形中的紧致可定向超曲面上寻找另一个几何条件（根据平均曲率和积分曲线 ${\xi }^T$ ），超曲面与球体等距。最后，我们研究了在Sasakian流形中具有恒定平均曲率的紧致可定向超曲面，并在超曲面上的Laplace算子的第一个非零特征值上找到了一个尖锐的上限。特别是，我们证明只要且仅当超曲面与球体等距时，才能达到此上限，前提是超曲面的Ricci曲率沿 $\nabla \rho$ 有一定的下限。