The Umbilic Pionts Index Sum Theory Of 3-dimensional Lorentz Spaces

In this paper the classification of totally umbilic surfaces in three-dimensional Lorentz spaces R₁³,S₁³,H₁³ is obtained first, next by the method of inbedding R₁³,S₁³,H₁³ into their common comformal compactification Q³, the classification of totally umbilic surfaces in Q³ is obtained and the relation of those totally umbilic surfaces under the inbed is studied. At last by studying the property of isolated umbilic point in Q³, the umbilic points index sum theory of positive definite oriented closed surfaces in Q³ is discovered. This paper includes five sections.The first section is introduction.The conformal groups act on R₁³,S₁³, and H₁³ are introduced in the second section. The totally umbilic surfaces of R₁³,S₁³, and H₁³ are classified in the third section. The totally umbilic surfaces of Q³ are classified, and the relation between totally umbilic surfaces in Q³ and those in R₁³,S₁³, and H₁³ is discovered in the forth section. The index of umbilic point in Q³ is studied in the last section.This paper shows serveral invariant properties of R₁³,S₁³, and H₁³ can be derived from the conformal geometry of Q³ ,especally from the invariant property of Q³. Moreover, properties of 3-dimensional general Lorentz spaces of constant se-cional curvature can be derived from the invariant property of the 3-dimensional Lorentz space forms R₁³,S₁³, and H₁³ with constant secional curvature 0,1,-1 respectively.