Para-Blaschke isoparametric hypersurfaces in a unit sphere \(S^{n+1}(1)\) (Q2902679)

Let \(\mathbf{A}=\rho^2\sum_{i,j}A_{ij}\theta_i\otimes\theta_j\) and \(\mathbf{B}=\rho^2\sum_{i,j}B_{ij}\theta_i\otimes\theta_j\) be the Blaschke tensor and the Möbius second fundamental form of the immersion \(\mathbf{x}:M^n\to S^{n+1}(1)\). Let \(\mathbf{D}=\mathbf{A}+\lambda\mathbf{B}\) be the para-Blaschke tensor of \(\mathbf{x}\), where \(\lambda\) is a constant. A hypersurface \(\mathbf{x}:M^n\to S^{n+1}(1)\) without umbilical points is called a para-Blaschke isoparametric hypersurface if the Möbius form \(\Phi\equiv 0\) and the para-Blaschke eigenvalues of the immersion \(\mathbf{x}\) are constant.NEWLINENEWLINEHere are the main results of this paper.NEWLINENEWLINETheorem 1.2. Let \(\mathbf{x}:M^n\to S^{n+1}(1)\) be an \(n\;(n\geq 4)\)-dimensional immersed para-Blaschke isoparametric hypersurface in a unit sphere \(S^{n+1}(1)\) and \(\mathbf{D}=\mathbf{A}+\lambda\mathbf{B}\;(\lambda\neq 0)\) be the para-Blaschke tensor of \(\mathbf{x}\). If \(\mathbf{x}\) has three distinct Blaschke eigenvalues, one of which is simple, then \(\mathbf{x}\) is locally Möbius equivalent to one of the following objects: {\parindent=6mm \begin{itemize}\item[(1)] a hypersurface with constant mean curvature and constant scalar curvature in \(S^{n+1}(1)\); \item[(2)] the image of \(\sigma\) of a hypersurface with constant mean curvature and constant scalar curvature in \(\mathbb{R}^{n+1}\); \item[(3)] the image of \(\tau\) of a hypersurface with constant mean curvature and constant scalar curvature in \(\mathbb{H}^{n+1}\); \item[(4)] an embedding of \(S^p(r)\times S^q \left ( \sqrt {1-r^2} \right ) \times \mathbb R^+ \times \mathbb R^{m-p-q-1}\) in \(S^{m+1}\) (denoted by \(CSS(p,q,r)\)) for some constants \(p,q,r,p\neq q\) and \(r\neq 1/\sqrt{2}\); \item[(5)] one of the hypersurfaces as indicated in Example 3.4 of this paper, where \(k=3\) and \(\tilde y_1:M_1\to S^4(1)\) is one of Cartan's non-minimal isoparametric hypersurfaces with three principal curvatures satisfying \(\lambda\mu_i=1/r^2\) for some \(i\in\{1,2,3\}\). NEWLINENEWLINE\end{itemize}} For a related paper, see [\textit{X. Li} and \textit{Y. Peng}, Result. Math. 58, No. 1--2, 145--172 (201; Zbl 1202.53016)].NEWLINENEWLINETheorem 1.4. Let \(\mathbf{x}:M^n\to S^{n+1}(1)\) be an \(n\;(n\geq 4)\)-dimensional immersed para-Blaschke isoparametric hypersurface in a unit sphere \(S^{n+1}(1)\). If \(\mathbf{x}\) has three distinct Möbius eigenvalues, one of which is simple, then \(\mathbf{x}\) is locally Möbius equivalent to one of the following objects:{\parindent=6mm \begin{itemize}\item[(1)] a hypersurface with constant mean curvature and constant scalar curvature in \(S^{n+1}(1)\); \item[(2)] the image of \(\sigma\) of a hypersurface with constant mean curvature and constant scalar curvature in \(\mathbb{R}^{n+1}\); \item[(3)] the image of \(\tau\) of a hypersurface with constant mean curvature and constant scalar curvature in \(\mathbb{H}^{n+1}\); \item[(4)] \(CSS(p,q,r)\) for some constants \(p,q,r\); \item[(5)] an open part of the image under \(\sigma\) of the cone \(\bar{x}:N^3\times \mathbb{R}^+\to \mathbb{R}^5\) defined by \(\bar{x}(\varphi,t)=t\varphi\), where \(t\in\mathbb{R}^+\) and \(\varphi:N^3\to S^4\hookrightarrow\mathbb{R}^5\) is a minimal isoparametric immersion in \(S^4\) with three principal curvatures; \item[(6)] one of the hypersurfaces as indicated in Example 3.4 of this paper, where \(k=3\), \(r=\sqrt{\frac{6n}{n-1}},\;\lambda=0\) and \(\tilde y_1:M_1\to S^4(1)\) is one of Cartan's minimal isoparametric hypersurfaces with vanishing scalar curvature and three principal curvatures of values \(\pm\sqrt{\frac{n-1}{2n}}\), \(0\). NEWLINENEWLINE\end{itemize}} For related papers. see [\textit{Z. Hu} and \textit{D. Li}, Pac. J. Math. 232, No. 2, 289--311 (2007; Zbl 1154.53011); \textit{Z. Hu, H. Li} and \textit{C. Wang}, Monatsh. Math. 151, No. 3, 201--222 (2007; Zbl 1144.53021)].