L_1 operator and Gauss map of quadric surfaces

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Mohammadpouri A.; Kafili L.; Hosseinoghli R.

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ISSN: 2588-4824

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Journal: JOURNAL OF NEW RESEARCHES IN MATHEMATICS Year:2019 | Volume:5 | Issue:20 Page(s): 81-90

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Title

L_1 operator and Gauss map of quadric surfaces

Author(s)

Mohammadpouri A. | Kafili L. | Hosseinoghli R. | Issue Writer Certificate

Keywords

Gauss map

L_1 operator quadric surfaces

Ruled surfaces

Abstract

The quadrics are all surfaces that can be expressed as a second degree polynomial 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 1-th Newton transformation associated to the second fundamental form of 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.

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APA: Copy

Mohammadpouri, A., Kafili, L., & Hosseinoghli, R.. (2019). L_1 operator and Gauss map of quadric surfaces. JOURNAL OF NEW RESEARCHES IN MATHEMATICS, 5(20 ), 81-90. SID. https://sid.ir/paper/740925/en

Vancouver: Copy

Mohammadpouri A., Kafili L., Hosseinoghli R.. L_1 operator and Gauss map of quadric surfaces. JOURNAL OF NEW RESEARCHES IN MATHEMATICS[Internet]. 2019;5(20 ):81-90. Available from: https://sid.ir/paper/740925/en

IEEE: Copy

A. Mohammadpouri, L. Kafili, and R. Hosseinoghli, “L_1 operator and Gauss map of quadric surfaces,” JOURNAL OF NEW RESEARCHES IN MATHEMATICS, vol. 5, no. 20 , pp. 81–90, 2019, [Online]. Available: https://sid.ir/paper/740925/en

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