zbMATH — the first resource for mathematics

Constant-ratio hypersurfaces. (English) Zbl 1007.53006
The author introduces the notion of constant-ratio submanifolds of a Euclidean space as those for which the ratio of the lengths of the tangential and normal components of its position vector function is constant, and then he proves the following classification theorem: Let \(x: M\to E^{n+1} \) be an isometric immersion of a Riemannian \(n\)-manifold into Euclidean \((n+1)\)-space. Then \(M\) is a constant-ratio hypersurface if and only if one of the following three statements holds: (a) \(M\) is an open portion of a hypersphere \(S^n(r)\) of \(E^{n+1}\) centered at the origin, (b) \(M\) is an open portion of a cone with vertex at the origin, (c) there exist local coordinate systems \(\{s,u_2,\dots,u_n\}\) on \(M\) such that the immersion \(x\) is given by \( x(s,u_2,\dots,u_n)=(cs)Y(s,u_2,\dots,u_n),\) where \(c\) is a constant and \(Y(s,u_2,\dots,u_n)\) is a parametrization of the unit hypersphere \(S^n(1)\) centered at the origin which satisfies the following two conditions: (c.1) \(Y_s\) is perpendicular to \(Y_{u_2},\dots,Y_{u_n}\) and (c.2) \(|Y_s|=\sqrt{1-c^2}/(cs)\).

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C40 Global submanifolds