Differential geometry from singularity theory viewpoint / Shyuichi Izumiya (Hokkaido University, Japan) [and three others].

Format:

Book

Author/Creator:

Izumiya, Shūichi, author.

Contributor:

Izumiya, Shūichi.

Language:

English

Subjects (All):

Surfaces--Areas and volumes.

Surfaces.

Singularities (Mathematics).

Geometry, Differential.

Curvature.

Physical Description:

1 online resource (383 p.)

Place of Publication:

New Jersey : World Scientific, 2015.

Language Note:

English

Summary:

"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."-- Provided by publisher.

Contents:

Contents; Preface; 1. The case for the singularity theory approach; 1.1 Plane curves; 1.1.1 The evolute of a plane curve; 1.1.2 Parallels of a plane curve; 1.1.3 The evolute from the singularity theory viewpoint; 1.1.4 Parallels from the singularity theory viewpoint; 1.2 Surfaces in the Euclidean 3-space; 1.2.1 The focal set; 1.3 Special surfaces in the Euclidean 3-space; 1.3.1 Ruled surfaces; 1.3.2 Developable surfaces; 1.4 Notes; 2. Submanifolds of the Euclidean space; 2.1 Hypersurfaces in Rn+1; 2.1.1 The first fundamental form; 2.1.2 The shape operator; 2.1.3 Totally umbilic hypersurfaces

2.1.4 Parabolic and umbilic points2.2 Higher codimension submanifolds of Rn+r; 2.2.1 Totally -umbilic submanifolds; 2.2.2 -parabolic and -umbilic points; 2.2.3 The canal hypersurface; 3. Singularities of germs of smooth mappings; 3.1 Germs of smooth mappings; 3.2 Multi-germs of smooth mappings; 3.3 Singularities of germs of smooth mappings; 3.4 The Thom-Boardman symbols; 3.5 Mather's groups; 3.6 Tangent spaces to the G-orbits; 3.7 Finite determinacy; 3.8 Versal unfoldings; 3.9 Classification of singularities; 3.9.1 Germs of functions; 3.9.2 Discriminants and bifurcation sets

3.10 Damon's geometric subgroups3.11 Notes; 4. Contact between submanifolds of Rn; 4.1 Contact between submanifolds; 4.2 Genericity; 4.3 The meaning of generic immersions; 4.4 Contact with hyperplanes; 4.5 The family of distance squared functions; 4.6 The family of projections into hyperplanes; 4.7 Notes; 5. Lagrangian and Legendrian Singularities; 5.1 Symplectic manifolds; 5.1.1 Lagrangian submanifolds and Langrangian maps; 5.1.2 Lagrangian singularities; 5.2 Contact manifolds; 5.2.1 Legendrian submanifolds and Legendrian maps; 5.2.2 Legendrian singularities

5.3 Graph-like Legendrian submanifolds5.4 Versal unfoldings and Morse families of functions; 5.5 Families of functions on hypersurfaces in Rn; 5.5.1 The family of height functions; 5.5.2 The extended family of height functions; 5.5.3 The family of distance squared functions; 5.6 Contact from the viewpoint of Lagrangian and Legendrian singularities; 5.6.1 Contact of hypersurfaces with hyperplanes; 5.6.2 Contact of hypersurfaces with hyperspheres; 5.6.3 Contact of submanifolds with hyperplanes; 6. Surfaces in the Euclidean 3-space; 6.1 First and second fundamental forms

6.2 Surfaces in Monge form6.3 Contact with planes; 6.4 Contact with lines; 6.4.1 Contour generators and apparent contours; 6.4.2 The generic singularities of orthogonal projections; 6.5 Contact with spheres; 6.6 Robust features of surfaces; 6.6.1 The parabolic curve; 6.6.2 The ecnodal curve; 6.6.3 The ridge curve; 6.6.4 The sub-parabolic curve; 6.7 Notes; 7. Surfaces in the Euclidean 4-space; 7.1 The curvature ellipse; 7.2 Second order affine properties; 7.2.1 Pencils of quadratic forms; 7.3 Asymptotic directions; 7.4 Surfaces in Monge form; 7.5 Examples of surfaces in R4

7.6 Contact with hyperplanes

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4590-45-2