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  • Differential Geometry
    • 1. Basic Information
    • 2. Preliminary course plan
    • References
  • Part 1 Curves
    • Lecture 1 - parametrized curves, vector, inner product
    • Lecture 2 - regular curves, arclength, reparametrization
    • Lecture 3 - curvature of curve, Frenet formula
    • Lecture 4 - plane curves, chain rule
    • Lecture 5 - regid motion, fundamental theorem of space curves
  • Part 2 Surfaces
    • Lecture 6 - topology, derivatives, regular surfaces
    • Lecture 7 - regualr surfaces, tangent plane
    • Lecture 8 - surface orientation, surface area
    • Lecture 9 - the first fundamental form, isometry, equiareal map
    • Lecture 10 - conformal map, stereographic/Mercator projection, Homework 1
  • Part 3 Curvature of a surface
    • Lecture 11 - Gauss map
    • Lecture 12 - self-adjoint linear transformation, normal curvature
    • Lecture 13 - normal curvature, geometric characterization of Gaussian curvature
    • Lecture 14 - the second fundamental form, SageMath demo
    • Lecture 15 - linear algebra on forms and tensor, the first and second fundamental forms revisited
  • Part 4 Geodesics
    • Lecture 16 - geodesics, Clairaut's relation
    • Lecture 17 - exercise session, Homework 2
    • Lecture 18 - exponential map, normal polar coordinates
    • Lecture 19 - Gauss's Theorema Egregium, constant Gaussian curvature surfaces, complete surfaces
    • Lecture 20 - parallel transport, holonomy, covariant derivative
    • Lecture 21 - geodesics in local coordinates, proof of Gauss's Theorema Egregium
    • Lecture 22 - Gaussian curvature measures infinitesimal holonomy
  • Part 5 Gauss-Bonnet theorem
    • Lecture 23 - the local Gauss-Bonnet theorem
    • Lecture 24 - exercise session, Homework 3 and 4
    • Lecture 25 - the global Gauss-Bonnet theorem
  • Roadmap, where to go, References
  • Homework 1
  • Homework 2
  • Homework 2 - remark
  • Homework 3
  • Homework 4
  • Oral Exam Facts
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