| This thesis is mainly concerned with the singularities of some geometric objects asso-ciated to lightlike submanfolds. In the last few decades, the singularity theory has enjoyedrapid development, it is an important tool for us to study the physics systems. From Eu-clidean space to semi-Euclidean space, the research range of the geometric objects that itinvolves is wider and wider also. For the study of the submanifolds in Euclidean space,regarding singularity, most of them have focused on the spacelike and timelike subman-ifolds, whether nonlightlike curves, surfaces, or high dimensional spacelike and timelikehypersurfaces, most of the classification problems of the singularities have been solved,however there are few studies on the singularities of lightlike submanifolds in the litera-ture. Few papers which study the singularities of lightlike submanifolds also merely staysin the research of the lightlike hypersurface with index1(i.e., lightlike submanifolds withcodimension1in Minkowski space), the research methods is limited by the degeneracyof the lightlike submanifold. Thus, we focus our attention on the study of singularity oflightlike submanifolds. We use some fundamental results of lightlike diferential geometryestablished by Bonner[1]and Duggal[45] as the basic tools in researching the singularitiesof lightlike submanifolds.This paper is organized as follows. In Chapter2, we review basic concepts of semi-Euclidean spaces. In Chapter3, we investigate the classification of the singularities ofnull Darboux developables, Gaussian surfaces, pseudo-spherical Darboux images and focalsurfaces associated with a null Cartan curve in Minkowski3-space. In Chapter4, weshow the classification of the singularities of null ruled surface of null Cartan curve inde Sitter3-space. In Chapter5, we give the classification of the singularities of the nulldevelopables of timelike curves that lie on nullcone in3-dimensional semi-Euclidean spacewith index2. In Chapter6, Geometrical properties associated with the contact of anisotropic submanifolds with a pseudospherical hyperhorosphere of a pseudosphere are studied. In Chapter7, we investigate the diferential geometry of1-lightlike submanifoldsof in anti de Sitter n space as an application of the theory of Legendrian singularities.Based on some theory of lightlike sunmanifold, we also introduce the notion of1-lightlikehorospherical Gauss curvature which is important for us to study the singularities of1-lightlike horospherical hypersurfaces. Moreover, we discuss the related geometric propertyof1-lightlike horospherical hypersurfaces in anti de Sitter n-space.. |
