Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN

In the 3-dimensional Euclidean space (Formula presented.), a quadric surface is either ruled or of one of the following two kinds (Formula presented.) or (Formula presented.). In the present paper, we investigate these three kinds of surfaces whose Gauss map (Formula presented.) satisfies the property (Formula presented.), where (Formula presented.) is a square symmetric matrix of order 3, and (Formula presented.) denotes the Laplace operator of the second fundamental form (Formula presented.) of the surface. We prove that spheres with the nonzero symmetric matrix (Formula presented.), and helicoids with (Formula presented.) as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.