Articles written in Proceedings – Mathematical Sciences

    • Hypersurfaces Satisfying $L_rx = Rx$ in Sphere $\mathbb{S}^{n+1}$ or Hyperbolic Space $\mathbb{H}^{n+1}$

      Biaogui Yang Ximin Liu

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      In this paper, using the method of moving frames, we consider hypersurfaces in Euclidean sphere $\mathbb{S}^{n+1}$ or hyperbolic space $\mathbb{H}^{n+1}$ whose position vector 𝑥 satisfies $L_r x=Rx$, where $L_r$ is the linearized operator of the $(r+1)$-th mean curvature of the hypersurfaces for a fixed $r=0,\ldots,n-1,R\in \mathbb{R}^{(n+2)\times(n+2)}$. If the 𝑟-th mean curvature $H_r$ is constant, we prove that the only hypersurfaces satisfying that condition are 𝑟-minimal $(H_{r+1}\equiv 0)$ or isoparametric. In particular, we locally classify such hypersurfaces which are not 𝑟-minimal.

    • Hypersurfaces in nearly Kaehler manifold $\mathbb{S}^3\times \mathbb{S}^3$


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      In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold $\mathbb{S}^3\times \mathbb{S}^3$ with a conformal vector field. There are three almost contact metric structures on a hypersurface of $\mathbb{S}^3\times \mathbb{S}^3$, and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in $\mathbb{S}^3\times \mathbb{S}^3$.

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