The Differential Geometry Of Smooth Curves And Surfaces

In this thesis, we focus on the differential geometry in the neiborhood of a singular point of smooth curves and surfaces in semi-Euclidean space. In 2009, Saji, Umehara, Ya-mada proposed the definition of curvature at cuspidal edges, and characterized the Gaussian curvature at cuspidal edge and swallowtail in the article they published in the journal of Annals of Mathematics. This article is a milestone working on the differential geom-etry in the neiborhood of a singular point of submanifolds. During this period, many mathematicians devoted themselves to the research of geometrical properties at singular points. In this thesis, we first focus on the smooth curves with singular points on hyperbolic plane, give the definitions of curvature and evolute at singu-lar points, characterize the evolutes in the three pseudo spheres, and further research the relationship between singular points and geodesic vertexes. Second, we discuss the pseudo null curves and partially null curves in semi-Euclidean four space with index two, study the geometrical properties of their nullcone Gaussian surface, point the relationship between the singularities of null hypersurfaces and some geometrical invariants. At last, we investigate the recognition problem of inflection points and H singularities of spacelike surfaces in Anti de Sitter four space from the view point of lightlike geometry, and explore the relationship between inflection points and H singularities.There are four parts in this thesis.In Chapter 1, we introduce the outline of development of this subject in recent years and review briefly the background of this thesis. Moreover, we introduce the structure of the full thesis and describe the main content of this thesis.In Chapter 2, we present the basic notations and results in both differential geometry and singularity theory of some submanifolds related to smooth curves and surfaces.In Chapter 3, we research the differential geometry and singularity of some submanifolds related to curves. First, we briefly introduce curves with singular points in Euclidean plane. Further, we study curves with singular points in hyperbolic plane. We give the definition of curvature at singular points and study the evolutes with multiple degree and the four-vertex theorem. Moreover, with the help of Legendrian singularity theory, we reveal the singularity classifications of null hypersurfaces of pseudo and partially null curves.In Chapter 4, we discuss the recognition problem of inflection points of spacelike sur-faces in Anti de Sitter four space. As we all know, every point of a surface corresponds a curvature ellipse. The point is called an inflection point, when the curvature ellipse degener-ates into a radial line. We can classify inflection points into three kinds, real type, imaginary type and flat type, depending on the position relationship between the degenerated curva-ture ellipse and the inflection points. First, we give the the methods of judging inflection points through the traditional approach in Euclidean case. Second, we find the equivalent conditions of inflection points of real type, imaginary type and flat type in terms of lightlike geometry. At last, we give the differential equations of the mean direction curves, and reveal that H singularities are definitely inflection points and stationary points.