Surfaces of coordinate finite II-type

Abstract

In this article, we study the class of surfaces of revolution in the 3-dimensional Euclidean space $E^3$ with nonvanishing Gauss curvature whose position vector $\mathbf{x}$ satisfies the condition $\hat{I}{”}^{II}\mathbf{x}=A\mathbf{x}$, where $A$ is a square matrix of order 3 and $\hat{I}{”}^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.