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On the \(Hp\)-theorem for hypersurfaces. (English) Zbl 0678.53004

Consider a smooth oriented hypersurface \(M^n\) of Euclidean space \(R^{n+1}\). If we choose an origin \(0\ne M^n\), we can define on \(M^n\) two smooth functions, the distance function \(r=| x|\) and the support function \(p=-<x,n>\), where \(x\) is the position vector and \(n\) is the unit normal of \(M^n\). The constancy of \(r\) implies, even locally, that \(M^n\) is part of a hypersphere centered at 0. The present article shows that:
Let \(M\) be a closed, connected and oriented hypersurface in \(R^{n+1}\), with mean curvature \(H\); assume that \(Hp=1\). Then \(M\) is a hypersphere.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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