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

Rotondaro, Giovanni

Commentat. Math. Univ. Carol. 30, No. 2, 385-387 (1989).

Consider a smooth oriented hypersurface \(M^ n\) of Euclidean space \(R^{n+1}\). If we choose an origin \(0\neq M^ n\), 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.

Reviewer: T.Hasanis

MSC:

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

Keywords:

smooth oriented hypersurface; support function; hypersphere; mean curvature