An extension of Takahashi's theorem (Q917058)

Let M be a connected hypersurface of a Euclidean space \(E^{n+1}\). The author proves that the coordinate functions of \(E^{n+1}\) restricted to M are proper functions of the Laplacian of M if and only if M is an open portion of a minimal hypersurface, an open portion of a hypersphere, or an open portion of a generalized circular cylinder. This result generalizes a result of \textit{T. Takahashi} [J. Math. Soc. Japan 18, 380- 385 (1966; Zbl 0145.186)].