L_1 operator and Gauss map of quadric surfaces

Subject Areas : Statistics

A. Mohammadpouri ¹ , L. Kafili ² , R. Hosseinoghli ³

1 - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2 - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
3 - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

Received: 2017-12-26 Accepted : 2019-07-06 Published : 2019-11-01

Keywords: رویه های خطی, عملگر L_1, نگاشت گاوس, رویه های درجه دوم,

Abstract :

The quadrics are all surfaces that can be expressed as a second degree polynomialin 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 the1-th Newton transformation associated to the second fundamental form ofthe 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.

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