Differential Geometry of Curves and Surfaces with Singularities

This book offers a novel and accessible approach to singularity theory through the lens of differential geometry of curves and surfaces, written by three renowned experts in the field. It introduces singularities, recognition theorems, and their applications in geometry and topology, focusing on plane curves and surfaces in Euclidean 3-space. The book presents the singular curvature and generalizes the Gauss–Bonnet theorem for surfaces with singularities, highlighting its intrinsic nature for 2-dimensional Riemannian manifolds. It also explores Riemannian manifolds with singularities and provides explicit examples, models, and exercise problems to aid in understanding the material.

Format: Hardback
Length: 380 pages
Publication date: 30 August 2021
Publisher: World Scientific Publishing Co Pte Ltd

This groundbreaking book offers a unique and highly accessible approach to singularity theory, drawing from the field of differential geometry of curves and surfaces. Written by three esteemed experts in the interplay between singularity theory and differential geometry, it provides a comprehensive exploration of singularities and their recognition theorems.

The book begins by introducing the concept of singularities and their theorems, which play a crucial role in understanding the behavior of curves and surfaces. It then delves into their applications to geometry and topology, focusing specifically on singularities of plane curves and surfaces in the Euclidean 3-space. By presenting the singular curvature, a concept developed through the authors' research, the Gauss-Bonnet theorem for surfaces is generalized to encompass those with singularities. This theorem holds intrinsic significance, as it extends to 2-dimensional Riemannian manifolds as well.

Furthermore, the book elucidates the notion of Riemannian manifolds with singularities, shedding light on their mathematical properties and applications. Explicit examples and models are provided in abundance, accompanied by insightful explanations of the underlying theory. Numerous figures and exercise problems are included to aid in developing a deep understanding of the material.

This text serves as an exceptional introduction to singularities of curves and surfaces from the perspective of differential geometry. It is invaluable for students and researchers interested in this field, offering a comprehensive and up-to-date resource that goes beyond the scope of other books. By providing a clear and concise explanation of complex concepts, this book empowers readers to grasp the intricacies of singularity theory and its applications in modern mathematics.

ISBN-13: 9789811237133