Tubular Surfaces Whose Gauss Map N Satisfies $\Delta^{II}N = \Lambda N$

In this paper, we consider tubes in the Euclidean 3-space whose Gauss map N is of coordinate finite II-type, i.e., the position vector N satisfies the relation $\Delta^{II}N = \Lambda N$, where $\Delta^{II}$ is the Laplace operator with respect to the second fundamental form I of the surface and $\Lambda$ is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.