Characterization of quadric surfaces in terms of coordinate finite type Gauss map

Abstract

In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space ${\mathbb{E}}^3$. We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map $\mathbf{G}$ satisfies a relation of the form ${\Delta }^I\mathbf{G}=M\mathbf{G}$, where $M$ is a square matrix of order 3 and ${\Delta }^I$ is the Laplace-Beltrami operator corresponding to the first fundamental form $I$ of the surface.