L_1 operator and Gauss map of quadric surfaces

Abstract

The quadrics are all surfaces that can be expressed as a second degree polynomial<br />in x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is the<br />1-th Newton transformation associated to the second fundamental form of<br />the surface and hessf denotes the self-adjoint linear operator metrically equivalent to the Hessian of, L_1G=(L_1G_1, L_1G_2, L_1G_3), G=(G_1, G_2, G_3). As a result, we establish the classification theorem that the only quadric surfaces with Gauss map G satisfying L_1G=AG for some 3×3 matrix A are the spheres and flat ones. Furthermore, the spheres are the only compact quadric surfaces with Gauss map G satisfying L_1G=AG for some 3×3 matrix A.