Generalized Biconservative Surfaces In 3-dimensional Lorentz Space

The classification of surfaces in Lorentz space forms is an important subject in sub-manifold geometry,which has attracted much attention in recent years.Due to the un-certainty of the Lorentz space metric,the surface geometric properties and classification results are more complex and interesting compared to the surface in Riemann space.In this paper,we mainly study a class of surfaces satisfying the equation A（?）H=k H（?）H in three-dimensional Lorentz space,where A is the shape operator,H is the mean curvature and k is an arbitrary constant.Such surfaces are called generalized biconservative surfaces（simply GB surface）.GB surfaces in Minkowski 3-space have been fully classified,this paper will continue to study the complete classification of space-like and time-like GB surfaces in 3-dimensional de Sitter space S₁³and anti-de Sitter space H₁³.Firstly,we review some basic formulas and basic concepts in Lorentz space and give the definition of GB surface.Secondly,space-like and time-like GB surfaces in 3-dimensional de Sitter space S₁³are discussed separately.By choosing the local orthostandard field,using the Gauss for-mula and Weingarten formula and the integrability condition to obtain the differential equation satisfying the surface,and then solve the differential equation by parameter transformation to determine the solution to satisfy the differential equation gives the explicit parameter form of the null GB surface,then solve the differential equation by parameter transformation to determine the structure satisfying the solutions of the dif-ferential equations and give the explicit parametric form of space-like GB surface,a total of 9 empty GB surfaces were obtained;Because the metric of the time-like surface is indefiniteness,the geometric structure becomes more complex,and through the indepth discussion of the time-like GB surface,the obtained classification results are more rich,a total of 12 types of GB surfaces were obtained.Finally,we fully classify are space-like and time-like GB surfaces in 3-dimensional anti-de Sitter space H₁³.In this paper,we give the explicit parameter form of space-like GB surfaces,and get 7 kinds of space-like GB surfaces;Also because the metric of the time-like surface is indefiniteness,16 kinds of time-like GB surfaces are obtained.When k=-1,the GB surface is a Biconservative surface,all the Biconservative surfaces are included in the results of this paper.