Geometry from a Differentiable Viewpoint

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. Cover......Page __sk_0000.djvu Copyright......Page __sk_0002.djvu Contents......Page __sk_0005.djvu Introduction......Page __sk_0007.djvu Part A. Prelude and themes: Synthetic methods and results......Page __sk_0011.djvu 1. Spherical geometry......Page __sk_0013.djvu 2. Euclid......Page __sk_0020.djvu Euclid's theory of parallels......Page __sk_0026.djvu Appendix. The Elements: Book I......Page __sk_0031.djvu Uniqueness of parallels......Page __sk_0034.djvu Equidistance and boundedness of parallels......Page __sk_0036.djvu On the angle sum of a triangle......Page __sk_0038.djvu Similarity of triangles......Page __sk_0041.djvu The work of Saccheri......Page __sk_0044.djvu The work of Gauss, Bolyai, and Lobachevskii......Page __sk_0049.djvu 5. Non-Euclidean geometry II......Page __sk_0055.djvu The circumference of a circle......Page __sk_0066.djvu Part B. Development: Differential geometry......Page __sk_0071.djvu 6. Curves......Page __sk_0073.djvu Early work on plane curves (Huygens, Leibniz, Newton, Euler)......Page __sk_0076.djvu The tractrix......Page __sk_0079.djvu Directed curvature......Page __sk_0080.djvu Digression: Involutes and evolutes......Page __sk_0082.djvu 7. Curves in space......Page __sk_0090.djvu Appendix: On Euclidean rigid motions......Page __sk_0099.djvu 8. Surfaces......Page __sk_0105.djvu The tangent plane......Page __sk_0111.djvu The first fundamental form......Page __sk_0116.djvu Area......Page __sk_0122.djvu 8 bis. Map projections......Page __sk_0126.djvu Stereographic projection......Page __sk_0130.djvu Central projection......Page __sk_0133.djvu Mercator projection......Page __sk_0134.djvu Azimuthal projection......Page __sk_0136.djvu Sample map projections......Page __sk_0137.djvu Euler's work on surfaces......Page __sk_0141.djvu The Gauss map......Page __sk_0144.djvu 10. Metric equivalence of surfaces......Page __sk_0155.djvu Special coordinates......Page __sk_0161.djvu 11. Geodesics......Page __sk_0167.djvu Euclid revisited I: The Hopf-Rinow theorem......Page __sk_0175.djvu 12. The Gauss-Bonnet theorem......Page __sk_0181.djvu Euclid revisited II: Uniqueness of lines......Page __sk_0185.djvu Compact surfaces......Page __sk_0186.djvu A digression on curves......Page __sk_0190.djvu 13. Constant-curvature surfaces......Page __sk_0196.djvu Euclid revisited III: Congruences......Page __sk_0201.djvu The work of Minding......Page __sk_0202.djvu Part C. Recapitulation and coda......Page __sk_0209.djvu 14. Abstract surfaces......Page __sk_0211.djvu Hilbert's theorem......Page __sk_0213.djvu Abstract surfaces......Page __sk_0216.djvu 15. Modeling the non-Euclidean plane......Page __sk_0227.djvu The Beltrami disk......Page __sk_0230.djvu The Poincaré disk......Page __sk_0234.djvu The Poincaré half-plane......Page __sk_0237.djvu 16. Epilog: Where from here?......Page __sk_0252.djvu Manifolds (differential topology)......Page __sk_0253.djvu Vector and tensor fields......Page __sk_0257.djvu Metrical relations (Riemannian manifolds)......Page __sk_0259.djvu Curvature......Page __sk_0262.djvu Covariant differentiation......Page __sk_0271.djvu Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry......Page __sk_0279.djvu Appendix: Notes on selected exercises......Page __sk_0289.djvu Bibliography......Page __sk_0307.djvu Symbol index......Page __sk_0313.djvu Name index......Page __sk_0314.djvu Subject index......Page __sk_0315.djvu 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.