Differential geometry has developed in many directions since its beginnings with Euler and Gauss. This often poses a problem for undergraduates: which direction should be followed? What do these ideas have to do with geometry? This book is designed to make differential geometry an approachable subject for advanced undergraduates. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of the non-Euclidean plane. The book begins with the theorems of non-Euclidean geometry, then introduces the methods of differential geometry and develops them towards the goal of constructing models of the hyperbolic plane. Interesting diversions are offered, such as Huygens' pendulum clock and mathematical cartography; however, the focus of the book is on the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. Although the main use of this text is as an advanced undergraduate course book, the historical aspect of the text should appeal to most mathematicians.
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
ISBN-13
978-1-139-17392-6 (9781139173926)
Schweitzer Klassifikation
Autor*in
Vassar College, New York
John McCleary is Professor of Mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role. His papers on topology have appeared in Inventiones Mathematicae, the American Journal of Mathematics and other journals, and he has written expository papers that have appeared in American Mathematical Monthly. He is also interested in the history of mathematics, especially the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of A User's Guide to Spectral Sequences and A First Course in Topology: Continuity and Dimension and he has edited proceedings in topology and in history, as well as a volume of the collected works of John Milnor. He has been a visitor to the mathematics institutes in Goettingen, Strasbourg and Cambridge, and to MSRI in Berkeley.
Prelude and themes; Part I. Synthetic Methods and Results: 1. Spherical geometry; 2. Euclid; 3. The theory of parallels; 4. Non-Euclidean geometry I; 5. Non-Euclidean geometry II; Part II. Development: Differential Geometry: 6. Curves; 7. Curves in space; 8. Surfaces; 9. Curvature for surfaces; 10. Metric equivalence of surfaces; 11. Geodesics; 12. The Gauss-Bonnet theorem; 13. Constant curvature surfaces; Part III. Recapitulation; 14. Abstract surfaces; 15. Modelling non-Euclidean geometry; 16. Coda: generalizations.
