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Compact hypersurfaces with constant higher order mean curvatures. (English) Zbl 0673.53003

Let k i , i=1,···,n be the principal curvatures of an orientable hypersurface M n of the Euclidean space R n+1 , with some fixed orientation. The mean curvature of order r, H r is defined by the identity

(1+tk 1 )···(1+tk n )=1+n1H 1 T+n2H 2 t 2 +···+nnH n t 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 r constant for some r=1,···,n.

Reviewer: R.Tribuzy

MSC:
53A07Higher-dimensional and -codimensional surfaces in Euclidean n-space
53C40Global submanifolds (differential geometry)