Ruled and quadric surfaces in the 3-dimensional Euclidean space satisfying ${\Delta }^{III}\mathbf{x}=\Lambda \mathbf{x}$

Abstract

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\mathbf{x}$ satisfies the relation ${\Delta }^{III}\mathbf{x}=\Lambda \mathbf{x}$ where $\Lambda$ is a square matrix of order 3. We show that helicoids and spheres are the only surfaces in $E^3$ satisfying the preceding relation.

11 pages