Differential Geometry Of Singular Submanifolds And Its Application

We mainly study the differential geometry of some singular geometrical objects,including the local differential geometry and the global differential geometry.In the first part,we study the local differential geometry of real analytic curves in Euclidean 3-space,which may have singularities.We classify all non-flat real analytic curves by the notion of(n,m)-cusp curves.Most of classical differential geometrical results about regular curves are generalized,especially the fundamental theorem and the local shape.In order to calculate conveniently,we define the modified Frenet-Serret type frame and obtain the corresponding Frenet-Serret type formula as well as some invariants.As an application,we study some special curves in the classical differential geometry,such as general helices,evolutes and involutes.They may have singularities.As a result,we obtain the duality theorem between evolutes and involutes.Furthermore,we apply our results to solve some explicit problems in optics and mechanics.In a classical optical system,we give the one-to-one correspondence among(n,m)-cusps on planar curvilinear mirrors,caustics and wavefronts.We also obtain the type of singularities on the locus of rolling balls and then describe their behavior at singularities in mechanics.In the second part,we study the global differential geometry of submanifolds in different pseudo-Riemannian spaces with respect to the lightlike geometry.We give the corresponding Gauss-Bonnet type theorem for spacelike submanifolds in Minkowski space with respect to the lightlike geometry.This result enlarges the research scope of Gauss-Bonnet type formulas and promotes the research on the global property of spacelike submanifolds in Minkowski space.Furthermore,we study the lightlike geometry of fronts which may have singularities in de Sitter 3-space and hyperbolic 3-space.We define the corresponding lightcone Gauss map and obtain some lightlike geometrical invariants.As the main result,we give the corresponding Gauss-Bonnet type theorem.It connects the lightlike geometry of singular submanifolds with their topology.This result can be regarded as a supplement and improvement of the global property of singular objects satisfying the Einstein field equation.There are five chapters in this thesis.In Chapter 1,we introduce the background,the history,the motivation as well as the meaning of our research and then give the explicit arrange of this thesis.In Chapter 2,we review some basic notions involved in this thesis.In Chapter 3,we study the local differential geometry of real analytic curves in Euclidean3-space.The notion of(n,m)-cusp curves is introduced and the fundamental theorem as well as the local shape are given.Using the modified Frenet-Serret type frame,we study the general helices,evolutes as well as involutes of(n,m)-cusp curves and give the duality theorem between evolutes and involutes.In Chapter 4,we apply the duality theorem between evolutes and involutes to solve some explicit problems in optics and mechanics.We firstly study the one-to-one correspondence among(n,m)-cusps on planar curvilinear mirrors,caustics and wavefronts in a classical optical system.We then give the type of singularities on the locus of rolling balls and describe their behavior at singularities.In Chapter 5,we study the global lightlike geometrical property of submanifolds in different pseudo-Riemannian spaces.We give the corresponding Gauss-Bonnet type theorem of spacelike submanifolds in Minkowski space.Furthermore,we study the lightlike geometry of fronts in de Sitter 3-space and hyperbolic 3-space.As the main result,we give the corresponding Gauss-Bonnet type theorem.