Chapters, Act I, The Nature of Space

• 2. Gaussian Curvature, Notes and Exercises
• 3. Exercises for Prologue and Act 1

Chapters, Act II, The Metric

• 4. Notes and exercises
• 5. Notes and exercises
• 6. Notes and exercises
• 7. Exercises for Act II

Chapters, Act III, Curvature

• 8. Curvature of Plane Curves
• 11. Geodesics and Geodesic Curvature
• 15. The Shape Operator
• 19. Graphics: Streamlines and Indices
• 20. Exercises for Act III
  1. Computational Proof of the Curvature Formula
  2. Binormal Cannot Tip in Direction of Motion
  3. The Darboux Vector.
  4. Variable Speed Frenet-Serret Equations
  5. The Helix
  6. The Frenet-Serret Approximation to a Curve.
  7. Series Expansion of Surfaces
  8. Variable-Speed Formulas for the Curvature of a Surface of Revolution
  9. Polar Formula for the Curvature of a Surface of Revolution
  10. Curvature of surface z = r^4
  11. Why Paper Folds into a Straight Line
  12. Visual Linear Algebra
  13. Vanishing Shape Operator <--> Flat
  14. Gaussian Curvature in terms of the Shape Operator
  15. Cartesian Formulas for the Shape Operator, Curvature, and Mean Curvature
  16. Nonprincipal Coordinates
  17. Validate Results of Ex. 7
  18. Curvature Formula in Polar Coordinates (tentative)
  19. Nonisometric Surfaces with Equal Curvatures
  20. Curvature of the n-Saddle
  21. Genus-Dependent Predictions of GGB
  22. Holy Broken Gauss-Bonnet, Batman!
  23. Hopf's Umlaufsatz
  24. There Are Only Five Regular Polyhedra
  25. There Are Only Five Topologically Regular Polyhedra
  26. Legendre's Projection of Convex Polygon
  27. Hopf's Proof That X(S_g)=2-2g
  28. Dual Polyhedra
  29. Dipole Streamlines are Circular
  30. Honey-flow on S_g
  31. Existence of the Stiefel Vector Field
  32. A Vector Field on S_g with One Singular Point
  33. Vector Fields on S_g with -X Saddle Points

Chapters, Act IV Parallel Transport, Curvature

• 23. Notes and Exercises
• 27. Notes on Metric Proof of Curvature Formula
• 28. Notes on Jacobi Equation
• 29. Notes on Riemann's Curvature

  29(a) Annotated OpenAI Notes on Riemann's Curvature

• 30. Notes on Einstein's Curved Spacetime
• 31. Exercises for Act IV
  1. Geodesic Curvature on the Sphere
  2. Geodesic Curvature on the Cone
  3. Geodesic Curvature along Touching Surfaces
  4. Geodesic Curvature via Intrinsic Differentiation
  5. Holonomy on the Cone and Sphere
  6. General Local Gauss-Bonnet Theorem
  7. Minding's Theorem
  8. Jacobi Equation on a General Surface of Revolution
  9. Gauss's Lemma via Computation
  10. Two Symmetries of the Riemann Tensor
  11. Counting the Components of the Riemann Tensor
  12. The Exponential Operator and Curvature
  13. Eclipses and Tides

  Chapters, Act V forms

• 32. Notes on 1-Forms, Dual Bases
• 33. Notes on Tensors
• 34. Notes on 2-Forms
• 35. Notes on 3-Forms
• 36. Differentiation
• 39. Exercises for Act V
  1. Dirac Delta Functions as 1-Form.
  2. Geometrical interpretation of ω(v) in ℝ³
  3. Explanation of Covariant and Contravariant
  4. Matrix Multiplication (as tensors)
  5. Tensor Contractions
  6. Tensor View of Matrix Multiplication as Contraction
  7. Factorizes in ℝ³
  8. Need not factorizein ℝ⁴
  9. Area formula for surface f = const.
  10. Factorizes in ℝ⁴
  11. Closed and Exact Forms
  12. Closed and Exact Forms
  13. Closed and Exact Forms
  14. Vector Calculus Identities as Forms
  15. Hodge Star Duality Operator (*)
  16. Conservation of Electric Charge

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