Ruled and Quadric Surfaces Satisfying ΔIIN = ΛN

Abstract

In the 3-dimensional Euclidean space E3, a quadric surface is either ruled or of one of the following two kinds z2=as2+bt2+c,abc≠0 or z=a2s2+b2t2,a>0,b>0. In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property ΔIIN=ΛN, where Λ is a square symmetric matrix of order 3, and ΔII denotes the Laplace operator of the second fundamental form II of the surface. We prove that spheres with the nonzero symmetric matrix Λ, and helicoids with Λ as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.