The Classification Of1-type Surfaces Of Revolution Based On Laplace Operator

Professor CHEN put forward the fnite type theory, which has been widely used instudy of manifolds, the theory of fnite type Gauss map will be used the fnite type theoryinto Gauss map on manifolds.1-type Gauss map is a special and useful tool, which has theadvantages of simple and intuitive. With the deepening of the study, evolution conditionscan also play an important role in the classifcation of manifold. Pointwise1-type Gaussmap and Weak1-type Gauss map both are evolution conditions of1-type Gauss map.They have obvious efect on the classifcation of the manifold than1-type Gauss maptheory.Minkowski Space is a pseudo Space with concepts of time and space, it had played animportant role in the development of general relativity. In this paper we studied surfaceof revolution in Minkowski3-Space by pointwise1-type and weak1-type theory and givesa more detailed classifcation theories of surface of revolution.The article structure is as follows:In the frst chapter, we expounded the development situation of fnite diferential mapto everyone to have a general understanding of fnite type theory. Secondly, introducedgeneral situation for full text.In the second chapter, The frst section introduces the basic content of the three di-mensional Minkowski space, Including the defnition of pseudo inner product and pseudovector product, diferent type of vectors、curves,、surfaces and the structure of the sur-face of revolution in Minkowski space, and constructed four types of surface of revolutionin the space. Further, introduced Gauss map G and Laplace operator. In the secondquarter to the fourth quarter, the surface of revolution in turn into time like, space like,light like three kinds of situations were calculated specifc form with pointwise1-typeGauss map. The second section, we get some surfaces with pointwise1-type Gauss mapin time like, respectively, Euclidean plane R2, one index cylinder R111×S、non light cone surface. The third section, we get some surfaces with pointwise1-type Gauss map in spacelike, respectively, Lorentz plane R21、Lorentz cylinderS11×R1、The frst and second kindscircular cone with space axis. The fourth section, we get some surfaces with pointwise1-type Gauss map in light like, respectively, the second kind of Enneper surface、de Sitterpseudosphere、hyperbolic pseudosphere.In the third chapter, the surface of revolution also discussed in three kinds of situa-tions with weak1-type Gauss map The frst section, we get four kinds of surfaces with weak1-type Gauss map in time like, respectively, the frst and third kinds of catenoid、de Sitterpseudosphere、Hyperbolic pseudosphere. The second section, we get fve kinds of surfaceswith weak1-type Gauss map in space like, respectively, the second kind of catenoid、theforth kind of catenoid、the ffth kind of catenoid、de Sitter pseudosphere、Hyperbolicpseudosphere. The third section, we get four kinds of surfaces with weak1-type Gaussmap in space like, respectively, the second kind of Enneper surface、the third kind ofEnneper surface、de Sitter pseudosphere、Hyperbolic pseudosphere. And also gives therelationship between operators and h. The forth section, Proved Laplace operatorinduced by third fundamental form of surface, unable to classify surface of revolution.