![]() Readership: Advanced undergraduate and graduate students, and researchers interested in the singularity theory from the perspective of differential geometry of curves and surfaces. The Half-space Theorem for Swallowtails.The Fundamental Theorem of Frontals as Hypersurfaces.Proof of the Criterion for Whitney Cusps.The treatment is rather old-fashioned manifolds are always. Proof of the Criterion for Swallowtails This book features a rather nice collection of examples of interesting curves and surfaces.Gauss–Bonnet Type Formulas and Applications. ![]() Applications of Criteria for Singularities.Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.Ĭhapter 1: Our World is Three-Dimensional Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. The book also elucidates the notion of Riemannian manifolds with singularities. The Gauss–Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss–Bonnet theorem for surfaces is generalized to those with singularities. The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. It is written by three leading experts on the interplay between two important fields - singularity theory and differential geometry. M Do Carmo, Differential geometry of curves and surfaces, Prentice Hall.This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. Struik, Lectures on classical differential geometry Addison-Wesley 1950. To apply this understanding in specific examples.īooks: John McCleary, Geometry from a differential viewpoint Cambridge University Press 1994.ĭirk J. It eventually leads on to the very general theory of manifolds.Īims: To gain an understanding of Frenet formulae for curves, the first and second fundamental forms of surfaces in 3-space, parallel transport of vectors and gaussian curvature. This is mostly mathematics from the first half of the nineteenth century, seen from a more modern perspective. One can relate these geometric notions to topology, for example, via the so-called Gauss-Bonnet formula. The gaussian curvature of a sphere is strictly positive, which is why planar maps of the earth's surface invariably distort distances. The gaussian curvature of a cone is zero, which is why we can make a cone out of a flat piece of paper. The former describes the intrinsic geometry of the surface, whereas the latter describes how it bends in space. For example, we have two notions of curvature: the gaussian curvature and the mean curvature. The case a surface is rather more subtle. The manner in which a curve can twist in 3-space is measured by two quantities: its curvature and torsion. ![]() Leads to: The following modules have this module listed as assumed knowledge or useful background:Ĭontent: This will be an introduction to some of the ``classical'' theory of differential geometry, as illustrated by the geometry of curves and surfaces lying (mostly) in 3-dimensional space.
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