The Conformal Surfaces In 3-dimensional Lorentz Space

The surface theory in the space form, especially, the construction and the classification of a special surface is an important and interesting problem. In this paper the conformal surface in the 3-dimension Lorentz space will be studied in detail. Mainly by studying surfaces in Q³, the space of compactification of the Lorentz space form R₁³, S₁³, H₁³, in a method of constructing a moving frame, the fundamental equations and structural equations can be obtained. Finally with the fundamental equations and structural equations the isotropic surface and the surface of C_i ≡ 0 in Q³ are classified. Thus the two important theorem of classification can be obtained. This paper is organized as follows:In section 1, Q³ the space of compactification of the 3-dimension Lorentz space form R₁³, S₁³, H₁³ is introduced. The conformal group of Lorentz space form is obtained after calculating.In section 2, The conformal surface in Q³ is studied. The main result is the fundamental equations and structural equations.In section 3, The isotropic surface in Q³ is classified, and the first theorem of classification is obtained.In section 4, The surface of C_i ≡ 0 is classified, and the second classification theorem is obtained.