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Über dieses Buch

This book presents the classical theory of curves in the plane and three-dimensional space, and the classical theory of surfaces in three-dimensional space. It pays particular attention to the historical development of the theory and the preliminary approaches that support contemporary geometrical notions. It includes a chapter that lists a very wide scope of plane curves and their properties. The book approaches the threshold of algebraic topology, providing an integrated presentation fully accessible to undergraduate-level students.

At the end of the 17th century, Newton and Leibniz developed differential calculus, thus making available the very wide range of differentiable functions, not just those constructed from polynomials. During the 18th century, Euler applied these ideas to establish what is still today the classical theory of most general curves and surfaces, largely used in engineering. Enter this fascinating world through amazing theorems and a wide supply of surprising examples. Reach the doors of algebraic topology by discovering just how an integer (= the Euler-Poincaré characteristics) associated with a surface gives you a lot of interesting information on the shape of the surface. And penetrate the intriguing world of Riemannian geometry, the geometry that underlies the theory of relativity.

The book is of interest to all those who teach classical differential geometry up to quite an advanced level. The chapter on Riemannian geometry is of great interest to those who have to “intuitively” introduce students to the highly technical nature of this branch of mathematics, in particular when preparing students for courses on relativity.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Genesis of Differential Methods

Abstract
The study of arbitrary curves—not just conics—becomes possible during the 17th century, first via the consideration of polynomial equations. The development of differential calculus allows next the study of very general curves. We describe the first historical attempts for handling questions like the tangent to a curve, its length or its curvature. We pay a special attention to some curves, like the cycloid, which have played an important role in the development of these notions.
Francis Borceux

Chapter 2. Plane Curves

Abstract
Plane curves are studied both via their parametric equations or their Cartesian equation. We study the tangent to a curve and the related problem of the envelope of a family of curves; we exhibit some interesting applications in physics. After a careful study of the curvature of a plane curve, its intrinsic equation and the famous Umlaufsatz, we switch to the more involved question of simple closed and convex curves and we prove in particular the Hopf and the “Four vertices” theorems.
Francis Borceux

Chapter 3. A Museum of Curves

Abstract
We present a whole bunch of historically important plane curves and list their major characteristics and properties.
Francis Borceux

Chapter 4. Skew Curves

Abstract
Curves in the three dimensional real space are studied from the points of view of their equations, their tangent, their curvature and their torsion. We establish the Frenet formulas and we investigate the more involved question of the intrinsic equations of a skew curve.
Francis Borceux

Chapter 5. The Local Theory of Surfaces

Abstract
First, we study the equations and the tangent plane to a surface in the three dimensional real space. The central notion of the chapter is that of normal curvature, together with the related notions of umbilical point and principal directions. We establish the important results concerning these notions and prove in particular the famous Rodrigues formula. We conclude the chapter with the study of the Gaussian curvature and its relation with the normal curvature.
Francis Borceux

Chapter 6. Towards Riemannian Geometry

Abstract
The first purpose of this chapter is to provide a deep intuition of formal notions like the metric tensor, the Christoffel symbols, the Riemann tensor, vector fields, the covariant derivative, and so on: an intuition based on the consideration of surfaces in the three dimensional real space. We switch next to the study of geodesics, geodesic curvature and systems of geodesic coordinates. As an example of a Riemann surface, we develop the study of the Poincaré half plane and prove that it is a model of non-Euclidean geometry. We conclude with the Gauss–Codazzi–Mainardi equations and the related question of the embeddability of Riemann surfaces in the three dimensional real space.
Francis Borceux

Chapter 7. Elements of the Global Theory of Surfaces

Abstract
Global theory of surfaces is interested in those properties which refer to wide pieces of the surface, not just to the neighborhood of each point. We study surfaces of revolution, ruled surfaces, developable surfaces. We study when two surfaces are just an “isometric deformation” of each other and establish the classification of developable surfaces. We pay special attention to the surfaces with constant Gaussian curvature and prove the Liebmann characterization of the sphere. We conclude with the study of polygonal decompositions, the Gauss–Bonnet theorem and the Euler–Poincaré characteristic.
Francis Borceux

Backmatter

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