Synopses & Reviews
This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds.
Review
"...the author has succeeded in making differential geometry an approachable subject for advanced undergraduates." Andrej Bucki, Mathematical Reviews
Synopsis
This book offers a new treatment of differential geometry which is designed to make the subject approachable for advanced undergraduates.
Synopsis
Designed to make the subject approachable for advanced undergraduates, this new presentation of differential geometry considers its historical development and introduces its methods within a context that geometry students will be familiar with from high school.
Description
Includes bibliographical references (p. 297-302) and index.
Table of Contents
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.