Quadric surfaces of coordinate finite type Gauss map

Abstract

We study quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type Gauss map with respect to the second fundamental form $II$, i.e., their Gauss map vector $\boldsymbol{n}$ satisfies the relation $\Delta ^{II}\boldsymbol{n}=\varLambda \boldsymbol{n}$, where $\Delta ^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface and $\varLambda$ is a square matrix of order 3. We show that helicoids and spheres are the only class of surfaces mentioned above satisfying $\Delta ^{II}\boldsymbol{n}=\varLambda \boldsymbol{n}$.