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Compact hypersurfaces with constant higher order mean curvatures. (English) Zbl 0673.53003
Let $k\sb i$, $i=1,...,n$ be the principal curvatures of an orientable hypersurface $M\sp n$ of the Euclidean space $R\sp{n+1}$, with some fixed orientation. The mean curvature of order r, $H\sb r$ is defined by the identity $$ (1+tk\sb 1)...(1+tk\sb n)=1+\left( \matrix n\\ 1\endmatrix \right)H\sb 1T+\left( \matrix n\\ 2\endmatrix \right)H\sb 2t\sp 2+...\quad +\left( \matrix n\\ n\endmatrix \right)H\sb nt\sp n $$ for any real number t. In this paper it is proved that the sphere is the only embedded compact hypersurface in the Euclidean space with $H\sb r$ constant for some $r=1,...,n$.
Reviewer: R.Tribuzy

53A07Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
53C40Global submanifolds (differential geometry)
Full Text: DOI EuDML