A Differential Approach to Geometry

This third volume in a trilogy of texts on geometry guides students through the development of differential geometry. It links classical surface theory with modern Riemannian geometry and prepares readers for advanced topics such as algebraic topology.

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.

Pays particular attention to historical development and preliminary approaches that support the contemporary geometrical notions Links classical surface theory in the three dimensional real space to modern Riemannian geometry Can be used to teach classical differential geometry up to an advanced level ? Includes supplementary material: sn.pub/extras

Auteur
Francis Borceux is Professor of mathematics at the University of Louvain since many years. He has developed research in algebra and essentially taught geometry, number theory and algebra courses and he has been dean of the Faculty of Sciences of his University and chairman of the Mathematical Committee of the Belgian National Scientific Research Foundation.

Contenu
Introduction.- Preface.- 1.The Genesis of Differential Methods.- 2.Plane Curves.- 3.A Museum of Curves.- 4.Skew Curves.- 5.Local Theory of Surfaces.- 6.Towards Riemannian Geometry.- 7.Elements of Global Theory of Surfaces.- Appendices: A.Topology.- B.Differential Equations.- Index.- Bibliography.