Differential Geometry From A Singularity Theory Viewpoint

Shyuichi Izumiya, Maria del Carmen Romero Fuster, Maria Aparecida Soares Ruas, Farid Tari

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814590464
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.

Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.

Readership: Advanced undergraduates and post-graduate students, and researchers in the fields of differential geometry and singularity theory.
Key Features:

• The book is unique in its nature. It provides a coherent approach for studying the geometry of sub-manifolds of various ambient spaces from the singularity theory point of view
• The book informs the reader about the progress in the field of extrinsic differential geometry and singularity theory. The information is new and has not been treated in previous textbooks
• The book gathers scattered work from various research articles, most of which are recent, and describes techniques that could be used to tackle problems in other areas of mathematics

Table of contents

• Cover Page
• Title Page
• Copyright
• Preface
• Contents
• 1. The case for the singularity theory approach
• 2. Submanifolds of the Euclidean space
• 3. Singularities of germs of smooth mappings
• 4. Contact between submanifolds of ℝn
• 5. Lagrangian and Legendrian Singularities
• 6. Surfaces in the Euclidean 3-space
• 7. Surfaces in the Euclidean 4-space
• 8. Surfaces in the Euclidean 5-space
• 9. Spacelike surfaces in the Minkowski space-time
• 10. Global viewpoint
• Bibliography
• Index