Differential Geometry

Timeline

Textbook: VDGAF by Needham

Week 1

• Prologue
  • Newtonian Reasoning
  • Ultimate Equality
• Act I
  • Three different geometries
  • Angular Excess
  • Gaussian Curvature
  • Local Gauss-Bonnet Theorem

Week 2

• Act II: Chapter 4 - Mapping Surfaces
  • Maps . Projective, Conformal Steregraphic
  • Metrics . General Surface . Metric Curvature

Week 3

• Act II: Chapter 5 - Pseudosphere & \(\mathbb{H}^2\)
  • Beltrami
  • Tractrix
  • Half-plane

Week 3

• Act II: Chapter 6 - Isometries
• Act III: Chapter 8 - Curvature of Plane Curves

Week 4

• Act III: Chapter 9 - Curves in 3-Space
• Act III: Chapter 10 - Principal Curvatures of a Surface

Week 5

• Act III: Chapter 11 - Geodesics & Geodesic Curvature
• Act III: Chapter 12 - Extrinsic Curvature of a Surface

Week 6

• Act III: Chapter 13 - Gauss’s Theorema Egregium
• Act III: Chapter 14 - Curvature of a Spike

Week 7

• Act III: Chapter 15 - The Shape Operator
• Act III: Chapter 16 - Intro to Global Gauss-Bonnet Theorem

Week 8

• Act III: Chapter 17 - Heuristic Proof of GGB
• Act III: Chapter 18 - Angular Excess Proof of GGB

Week 9

• Act III: Chapter 19 - Vector Proof of GGB
• Act IV: Chapter 22 - Parallel Transport: Extrinsic Constructions

Week 10

• Act III: Chapter 23 - Intrinsic Contructions
• Act IV: Chapter 24 - Holonomy

Week 11

• Act III: Chapter 25 - Intuitive Proof of Theorema Egregium
• Act IV: Chapter 26 - Holonomy Proof of GGB

Week 12

• Act III: Chapter 27 - Proof of Metric Curvature Formula
• Act IV: Chapter 28 - Curvature as Force b/w Geodesics

Week 13

• Act III: Chapter 29
  • Covariant Derivative
  • Riemannian Curvature Tensor
• Act III: Chapter 30
  • Tidal Forces
  • Spacetime Metric

Week 14

• Act IV: Forms
  • 1-forms
  • 2-forms
  • Tensors