Abstract
In the 3-dimensional Euclidean space E-3, a quadric surface is either ruled or of one of the following two kinds z(2) = as(2) + bt(2) + c, abc &NOTEQUexpressionL; 0 or z = a/2 s(2) + b/2 t(2), a > 0, b > 0. In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property delta N-II = lambda N, where lambda is a square symmetric matrix of order 3, and AII denotes the Laplace operator of the second fundamental form II of the surface. We prove that spheres with the nonzero symmetric matrix lambda, and helicoids with A as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.