Geometry Of Singular Submanifolds In Non-flat Space

It is well known that the hyperbolic space is one of pseudo-spheres in Minkowski space with negative constant sectional curvature.The sphere space is a subspace of Euclidean space with positive constant sectional curvature.They are all non-flat Riemannian space.In the thesis,we focus on the topology and differential geometry of singular submanifolds in these two kinds of non-flat spaces from the viewpoint of singularity theory.We investigate a special class of smooth curves,namely framed curves,in hyperbolic3-space.By using a moving frame along the curve,we define evolutes and focal surfaces related to the framed curve.Moreover,taking advantage of distance square function family and bifurcation theory as basic tools,we discuss the relationship between the evolutes and the focal surfaces from the point of view of singularity theory.As results,the bifurcation set of the distance square function family is the focal surface,and the singular set of the bifurcation set is the evolute.On the other hand,in sphere 3-space,we consider the differential geometric properties of two kinds of singular surfaces.Both of the singular sets of these two surfaces are nondegenerate curves,we call them the cuspidal edge and the swallowtail,respectively.Firstly,we define the developable surfaces in sphere 3-space along the non-degenerate curves and call them the extrinsic flat great circular surfaces along the cuspidal edge and the swallowtail,respectively.For the classification of singularities of such flat surfaces,we introduce new invariants.Finally,the relationship between singularities and those invariants is well revealed.