Rotondaro, Giovanni On the \(Hp\)-theorem for hypersurfaces. (English) Zbl 0678.53004 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 \(0\ne M^n\), 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. Reviewer: Thomas Hasanis (Ioannina) 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 PDF BibTeX XML Cite \textit{G. Rotondaro}, Commentat. Math. Univ. Carol. 30, No. 2, 385--387 (1989; Zbl 0678.53004) Full Text: EuDML OpenURL