Abstract. We study the Gauss map G of ruled surfaces in the 3-dim-ensional Euclidean space E 3 with respect to the so called Cheng-Yauoperator acting on the functions defined on the surfaces. As a result,we establish the classification theorem that the only ruled surfaces withGauss map G satisfying G= AG for some 3 ×3 matrix A are the flatones. Furthermore, we show that the only ruled surfaces with Gauss mapGsatisfying G= AGfor some nonzero 3×3 matrix Aare the cylindricalsurfaces. 1. IntroductionThe theory of Gauss map of surfaces in the n-dimensional Euclidean spaceE n or in the n-dimensional Lorentz-Minkowski space L n is always one of inter-esting topics and it has been investigated from various viewpoints by a lot ofdifferential geometers ([2, 4, 7, 9, 10, 11, 14, 15, 16, 17, 18, 21, 24, 25]).We denote by M a surface of the Euclidean 3-space E 3 . The map G : M →S 2 ⊂ E 3 which sends each point p of M to the unit normal vector G(p) to M atp is called the Gauss map of the surface M, where S