Geometry from a Differentiable Viewpoint

The Development Of Geometry From Euclid To Euler To Lobachevsky, Bolyai, Gauss, And Riemann Is A Story That Is Often Broken Into Parts - Axiomatic Geometry, Non-euclidean Geometry, And Differential Geometry. This Poses A Problem For Undergraduates: Which Part Is Geometry? What Is The Big Picture To Which These Parts Belong? In This Introduction To Differential Geometry, The Parts Are United With All Of Their Interrelations, Motivated By The History Of The Parallel Postulate. Beginning With The Ancient Sources, The Author First Explores Synthetic Methods In Euclidean And Non-euclidean Geometry And Then Introduces Differential Geometry In Its Classical Formulation, Leading To The Modern Formulation On Manifolds Such As Space-time. The Presentation Is Enlivened By Historical Diversions Such As Hugyens's Clock And The Mathematics Of Cartography. The Intertwined Approaches Will Help Undergraduates Understand The Role Of Elementary Ideas In The More General, Differential Setting. This Thoroughly Revised Second Edition Includes Numerous New Exercises And A New Solution Key. New Topics Include Clairaut's Relation For Geodesics, Euclid's Geometry Of Space, Further Properties Of Cycloids And Map Projections, And The Use Of Transformations Such As The Reflections Of The Beltrami Disk-- Machine Generated Contents Note: Part I. Prelude And Themes: Synthetic Methods And Results: 1. Spherical Geometry; 2. Euclid; 3. The Theory Of Parallels; 4. Non-euclidean Geometry; Part Ii. Development: Differential Geometry: 5. Curves In The Plane; 6. Curves In Space; 7. Surfaces; 8. Curvature For Surfaces; 9. Metric Equivalence Of Surfaces; 10. Geodesics; 11. The Gauss-bonnet Theorem; 12. Constant-curvature Surfaces; Part Iii. Recapitulation And Coda: 13. Abstract Surfaces; 14. Modeling The Non-euclidean Plane; 15. Epilogue: Where From Here?. John Mccleary. Includes Bibliographical References And Indexes. Cover ......Page 1 Geometry from a Differentiable Viewpoint......Page 2 Title......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Preface to the second edition......Page 10 Introduction......Page 12 How to use this book......Page 14 Acknowledgments......Page 15 PART A: Prelude and themes: Synthetic methods and results ......Page 18 1 Spherical geometry......Page 20 Exercises......Page 27 2 Euclid......Page 29 Euclid's theory of parallels......Page 36 Exercises......Page 39 Propositions......Page 41 3 The theory of parallels......Page 44 Uniqueness of parallels......Page 45 Equidistance and boundedness of parallels......Page 46 On the angle sum of a triangle......Page 48 Similarity of triangles......Page 51 The work of Saccheri......Page 54 Exercises......Page 58 The work of Gauss......Page 60 The hyperbolic plane......Page 64 Digression: Neutral space......Page 72 Hyperbolic space......Page 79 Exercises......Page 86 Propositions......Page 89 PART B: Development: Differential geometry ......Page 92 5 Curves in the plane......Page 94 Early work on plane curves (Huygens, Leibniz, Newton, and Euler)......Page 98 The tractrix......Page 101 Oriented curvature......Page 103 Involutes and evolutes......Page 106 Exercises......Page 114 6 Curves in space......Page 116 Appendix: On Euclidean rigid motions......Page 127 Exercises......Page 129 7 Surfaces......Page 133 The tangent plane......Page 141 The first fundamental form......Page 145 Lengths, angles, and areas......Page 147 Exercises......Page 153 7bis Map projections......Page 155 Stereographic projection......Page 160 Central (gnomonic) projection......Page 164 Cylindrical projections......Page 165 Sinusoidal projection......Page 169 Azimuthal projection......Page 170 Exercises......Page 171 Euler's work on surfaces......Page 173 The Gauss map......Page 176 Exercises......Page 186 9 Metric equivalence of surfaces......Page 188 Special coordinates......Page 196 Exercises......Page 200 10 Geodesics......Page 202 Euclid revisited I: The Hopf–Rinow Theorem......Page 212 Exercises......Page 217 11 The Gauss–Bonnet Theorem......Page 218 Euclid revisited II: Uniqueness of lines......Page 222 Compact surfaces......Page 224 A digression on curves......Page 228 Exercises......Page 234 12 Constant-curvature surfaces......Page 235 Euclid revisited III: Congruences......Page 240 The work of Minding......Page 241 Hilbert's Theorem......Page 248 Exercises......Page 250 PART C: Recapitulation and coda ......Page 252 13 Abstract surfaces......Page 254 Exercises......Page 267 14 Modeling the non-Euclidean plane......Page 268 The Beltrami disk......Page 272 The Poincaré disk......Page 279 The Poincaré half-plane......Page 282 15 Epilogue: Where from here?......Page 299 Manifolds (differential topology)......Page 300 Vector and tensor fields......Page 304 Metrical relations (Riemannian manifolds)......Page 306 Curvature......Page 311 Covariant differentiation......Page 320 Exercises......Page 327 I. Concept of an n-fold extended quantity......Page 330 1......Page 331 3......Page 332 1......Page 333 2......Page 334 4......Page 336 1......Page 337 2......Page 338 3......Page 339 Solutions to selected exercises......Page 342 Books......Page 358 Articles......Page 363 Symbol index......Page 368 Name index......Page 369 Subject index......Page 371 "The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss, and Riemann is a story that is often broken into parts - axiomatic geometry, non-Euclidean geometry, and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Hugyens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics, Euclid's geometry of space, further properties of cycloids and map projections, and the use of transformations such as the reflections of the Beltrami disk"-- Provided by publisher The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss and Riemann is a story that is often broken into parts – axiomatic geometry, non-Euclidean geometry and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Huygens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics and the use of transformations such as the reflections of the Beltrami disk. A thoroughly revised second edition of a textbook for a first course in differential/modern geometry that introduces methods within a historical context