Geometry with an Introduction to Cosmic Topology

by Michael P. Hitchman

Read here Read at LibreTextsPDF

Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology uses Mobius transformations to develop hyperbolic, elliptic, and Euclidean geometry - three possibilities for the global geometry of the universe. The text, written for students who have taken vector calculus, also explores the interplay between the shape of a space and the type of geometry it admits.

Description from LibreTexts.

Publisher
Linfield College
Subjects
Mathematics, Geometry
Licence
CC BY-SA 4.0, as LibreTexts states it
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Preface
  5. Table of Contents
  6. Acknowledgements
  7. Licensing
  8. Table of Contents
  9. 1: An Invitation to Geometry
  10. 1.1: Introduction
  11. 1.2: A Brief History of Geometry
  12. 1.3: Geometry on Surfaces- A First Look
  13. 2: The Complex Plane
  14. 2.1: Basic Notions
  15. 2.2: Polar Form of a Complex Number
  16. 2.3: Division and Angle Measure
  17. 2.4: Complex Expressions
  18. 3: Transformations
  19. 3.1: Basic Transformations of Complex Numbers
  20. 3.2: Inversion
  21. 3.3: The Extended Plane
  22. 3.4: Möbius Transformations
  23. 3.5: Möbius Transformations: A Closer Look
  24. 4: Geometry
  25. 4.1: The Basics
  26. 4.2: Möbius Geometry
  27. 5: Hyperbolic Geometry
  28. 5.1: The Poincaré Disk Model
  29. 5.2: Figures of Hyperbolic Geometry
  30. 5.3: Measurement in Hyperbolic Geometry
  31. 5.4: Area and Triangle Trigonometry
  32. 5.5: The Upper Half-Plane Model
  33. 6: Elliptic Geometry
  34. 6.1: Antipodal Points
  35. 6.2: Elliptic Geometry
  36. 6.3: Measurement in Elliptic Geometry
  37. 6.4: Revisiting Euclid's Postulates
  38. 7: Geometry on Surfaces
  39. 7.1: Curvature
  40. 7.2: Elliptic Geometry with Curvature k > 0
  41. 7.3: Hyperbolic Geometry with Curvature k < 0
  42. 7.4: The Family of Geometries (Xₖ, Gₖ)
  43. 7.5: Surfaces
  44. 7.6: Geometry of Surfaces
  45. 7.7: Quotient Spaces
  46. 8: Cosmic Topology
  47. 8.1: Three-Dimensional Geometry and 3-Manifolds
  48. 8.2: Cosmic Crystallography
  49. 8.3: Circles in the Sky
  50. 8.4: Our Universe
  51. Back Matter
  52. Index
  53. Appendix A: List of Symbols
  54. References
  55. Detailed Licensing