Advances in Differential Geometry and Singularity Theory, 2nd Edition

Dear Colleagues,

The geometry of curves and surfaces is a subject that has long fascinated many mathematicians and related scholars. In recent years, a new approach has been applied to studying this classical subject from the point of view of singularity theory. For example, robust geometric features on a surface in the Euclidean 3-space, some of which are detectable by the naked eye, can be captured by certain types of singularities for some functions and mappings on the surface. The mappings in question are in fact members of some natural families of mappings on the surface. The singularities of the individual members of these families of mappings measure the contact of the surface with model objects, such as lines, circles, planes, and spheres. It is interesting to investigate how to apply singularity theory to the study of the extrinsic geometry of surfaces, as well as how such methods are applied to any smooth submanifolds of higher-dimensional Euclidean space and to other settings, such as affine, hyperbolic or Minkowski spaces. Singularities arise naturally in a huge number of different areas of mathematics and science. In recent years, there has been fast-growing interest in developing theories and tools for studying singular submanifolds because singular submanifolds are produced in physics, mechanics and other fields of application, and they are the breakthrough point in the discovery of new problems. Therefore, it is of great scientific significance to study the geometric and topological properties of singular submanifolds. However, due to the existence of singular sets, the traditional analysis methods and geometric mathematical tools are no longer applicable, which makes the study of singular submanifolds difficult. Currently, there is a growing and justified interest in the study of the differential geometry of singular submanifolds (such as caustics, wavefronts, images of singular mappings, etc.) of Euclidean or Minkowski spaces, as well as of submanifolds with induced (pseudo) metrics that change signature on some subsets of submanifolds. We hope that this Special Issue can bring together experts within the field and those from adjacent areas where singularity theory has existing or potential applications. One of the aims of this Special Issue is to provide a platform for papers focused on differential geometry and singularity theory, devoted to surveying the remarkable insights derived from any related fields, and exploring promising new developments.

We look forward to receiving your contributions.

Prof. Dr. Zhigang Wang
Dr. Yanlin Li
Prof. Dr. Juan De Dios Pérez
Guest Editors

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• singularity theory
• morse theory
• singularities
• singular submanifolds
• light-like submanifolds
• differentiable manifolds
• submanifold theory
• legendrian duality
• front and frontal