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Rigidity theorems of hypersurfaces in a sphere. (English) Zbl 0951.53037

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 \(r_1^2=\frac{m-1}{n},\) \(r_2^2=\frac{n-m-1}{n}\).

MSC:

53C40 Global submanifolds
53C24 Rigidity results
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