RULED SURFACES AND GAUSS MAP

We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space E-3 with respect to the so called Cheng-Yau operator El acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only ruled surfaces with Gauss map G satisfying square G = AG for some 3 x 3 matrix A are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map G satisfying square G = AG for some nonzero 3x 3 matrix A are the cylindrical surfaces.