Li, Haizhong Rigidity theorems of hypersurfaces in a sphere. (English) Zbl 0951.53037 Publ. Inst. Math., Nouv. Sér. 67(81), 112-120 (2000). Let \(M\) be an \(n\)-dimensional \((n\geq 3)\) compact hypersurface in an \((n+1)\)-dimensional unit sphere \(S^{n+1}\). Studying Cheng-Yau’s self-adjoint operator the author gives conditions such that \(M\) is one of the following: (1) a totally umbilical hypersurface; (2) \(M=S^1(r_1)\times S^{n-1}(r_2)\), where \(r_1^2=\frac{1}{1+\sqrt{n-1}},\) \(r_2^2=\frac{\sqrt{n-1}} {1+\sqrt{n-1}}\); (3) \(M=S^m(r_1)\times S^{n-m}(r_2)\), for some \(m\) with \(1\leq m\leq n-1\), where \(r_1^2=\frac{m-1}{n},\) \(r_2^2=\frac{n-m-1}{n}\). Reviewer: Neda Bokan (Novi Beograd) Cited in 2 Documents MSC: 53C40 Global submanifolds 53C24 Rigidity results Keywords:selfadjoint operator; compact hypersurface; unit sphere PDFBibTeX XMLCite \textit{H. Li}, Publ. Inst. Math., Nouv. Sér. 67(81), 112--120 (2000; Zbl 0951.53037) Full Text: EuDML