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 $\mathbf{u}=-\varphi (N)$ . 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 $\mathbf{S}^{2n+1}$ . 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} ) $ 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 $\mathbf{S}^{2n+1}$ .

中文翻译：

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Sasakian流形的超曲面-复习

我们研究Sasakian流形中的可定向超曲面。Sasakian流形的结构矢量场ξ确定超曲面上的矢量场v，该向量场是与该超曲面相切的Reeb矢量场ξ的分量，并且还引起超曲面上的平滑函数σ，即单位法向矢量场N上的ξ。此外，我们还有一个与超曲面相切的第二矢量场，由$ \ mathbf {u} =-\ varphi（N）$给出。在本文中，我们首先找到使可定向紧致的超曲面成为完全脐带的必要和充分条件。然后，假设向量场u是Laplace算符的特征向量，我们找到了一个紧凑的可定向超曲面与球等距的必要条件。证明了这个结果的反面成立，假设Sasakian流形是奇数维球面$ \ mathbf {S} ^ {2n + 1} $。在向量场v是拉普拉斯算子的特征向量的假设下，获得了相似的结果。同样，我们使用紧致超曲面的Ricci曲率$ Ric（\ mathbf {u}，\ mathbf {u}）$的积分上的界来找到使超曲面与球等距的必要条件。我们证明，如果Sasakian流形为$ \ mathbf {S} ^ {2n + 1} $，则该条件也足够。紧致超曲面的\ mathbf {u}）$，以找到使超曲面与球等距的必要条件。我们证明，如果Sasakian流形为$ \ mathbf {S} ^ {2n + 1} $，则该条件也足够。紧致超曲面的\ mathbf {u}）$，以找到使超曲面与球等距的必要条件。我们证明，如果Sasakian流形为$ \ mathbf {S} ^ {2n + 1} $，则该条件也足够。