Math 335 - Differential Geometry
Fall 2023
Emily Proctor




Here are the videos for our course, arranged chronologically by topic.





Curves

• Section 1.1 (parametrized, smooth curves): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 1.2 (arc length): Part 1/notes and Part 2/notes.


• Section 1.3 (intuitive introduction to curvature of plane curves): Part 1/notes.


• Section 1.5 (space curves: moving frames, curvature, and torsion): Part 1/notes, Part 2/notes, Part 3/notes, Part 4/notes, and Part 5/notes.

Regular Surfaces

• Section 2.1/Background (definition of the derivative): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 2.1 (definition of a regular surface): Part 4/notes, Part 5/notes, and Part 6/notes.


• Sections 2.2-2.4 (examples of regular surfaces): Part 1/notes, Part 2/notes, and Part 3/notes.

The Tangent Plane to a Regular Surface

• Section 3.1 (the tangent plane to a regular surface): Part 1/notes, Part 2/notes, Part 3/notes, Part 4/notes, and Part 5/notes.


• Sections 3.2, 3.3, and 3.4 (the first fundamental form): Part 1/notes, Part 2/notes, and Part 3/notes.

Differentiable Functions on Surfaces

• Section 3.8 (change of parameters): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 4.1 (smooth functions on surfaces): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 4.1 (orientability and the Gauss map): Part 4/notes and Part 5/notes.


• Section 4.2 (the derivative of a differentiable function): Part 1/notes, Part 2/notes, Part 3/notes, and Part 4/notes.


• Section 4.3 (global and local isometries): Part 1/notes, Part 2/notes, and Part 3/notes.

Curvature

• Section 5.1 (the derivative of the Gauss map): Part 1/notes, Part 2/notes, Part 3/notes, and Part 4/notes.


• Sections 5.2 and 5.7 (the second fundamental form and normal curvature): Part 1/notes, Part 2/notes, Part 3/notes, and Part 4/notes.


• Sections 5.4, 5.9, and 5.10 (Gaussian and mean curvature): Part 1/notes.

Local Expressions of Curvature

• Section 5.3 (local expression of the second fundamental): Part 1/notes and Part 2/notes.


• Section 5.3 (local expression of the derivative of the Gauss map): Part 3/notes.


• Section 5.3 (local expression of Gaussian and mean curvature): Part 4/notes and Part 5/notes.


• Section 5.3 (local expression of asymptotic curves): Part 6/notes.

Gauss' Theorema Egregium

• Section 6.1 (Christoffel symbols): Part 1/notes and Part 2/notes.


• Section 6.2 (Gauss' Theorema Egregium): Part 1/notes, Part 2/notes, and Part 3/notes.

Geodesics

• Section 7.1 (geodesic curvature): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 7.2 (geodesics): Part 1/notes, Part 2/notes, and Part 3/notes.


• Sections 7.3 and 7.4 (properties of geodesic): Part 1/notes and Part 2/notes.

The Gauss-Bonnet Theorem

• Sections 8.2 and 8.3 (preliminaries and turning of tangents): Part 1/notes, Part 2/notes, Part 3/notes, and Part 4/notes.


• Section 8.3 (the local Gauss-Bonnet theorem): Part 1/notes, Part 2/notes, and Part 3/notes.


• Section 8.4 (triangulations and the Euler-Poincare characteristic): Part 1/notes and Part 2/notes.


• Section 8.5 (the global Gauss-Bonnet theorem): Part 1/notes and Part 2/notes.