## 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

### Keywords:

selfadjoint operator; compact hypersurface; unit sphere
Full Text: