Math 136: Differential Geometry (Fall 2021)

Instructor: Dori Bejleri (bejleri [at] math [dot] university [dot] edu)

Time and place: Wednesdays and Fridays at 12:00pm - 1:15pm in Science Center 507

Syllabus

Canvas

Lectures

Date	Topic	Text	Notes
09/03/2021	Introduction, parametrized curves	[Kü] 2A	Lecture 1
09/08/2021	Plane curves, space curves, Frenet curves	[Kü] 2B-2C	Lecture 2
09/10/2021	Frenet space curves and spherical curves	[Kü] 2C	Lecture 3
09/15/2021	Frenet equations, fundamental existence and uniqueness theorem	[Kü] 2D	Lecture 4
09/17/2021	Fundamental theorem continued, global theory of plane curves	[Kü] 2F	Lecture 5
09/22/2021	Regular surfaces	[Kü] 3A, [DC] 2-2	Lecture 6
09/24/2021	Regular surfaces continued, first fundamental form	[Kü] 3A, [DC] 2-2, 2-5	Lecture 7
09/29/2021	First fundamental form, length, area	[Kü] 3A, [DC] 2-5	Lecture 8
10/01/2021	Area continued, vector fields, orientation	[Kü] 3A, [DC] 2-5	Lecture 9
10/06/2021	The Gauss map and second fundamental form	[Kü] 3B, [DC] 3-2	Lecture 10
10/08/2021	Curvature of surfaces	[Kü] 3B, [DC] 3-2, 3-3	Lecture 11
10/13/2021	Curvature continued	[Kü] 3B, [DC] 3-2, 3-3	Lecture 12
10/15/2021	Surfaces of revolution, ruled surfaces	[Kü] 3C, [DC] 3-3	Lecture 13
10/20/2021	Directional and covariant derivatives	[Kü] 4A	Lecture 14
10/22/2021	Christoffel symbols, isometries	[Kü] 4A, [DC] 4-2	Lecture 15
10/27/2021	Theorem Egregium, Gauss and Codazzi-Mainardi equations	[Kü] 4C, [DC] 4-3	Lecture 16
10/29/2021	Gauss and Codazzi-Mainardi equations continued, parallel transport	[Kü] 4B	Lecture 17
11/03/2021	Geodesics	[Kü] 4B, [DC] 4-4	Lecture 18
11/05/2021	Geodesics continued	[Kü] 4B, [DC] 4-4	Lecture 19
11/10/2021	Riemannian curvature	[Kü] 4C	Lecture 20
11/12/2021	Riemannian curvature continued	[Kü] 4C	Lecture 21
11/17/2021	The fundamental theorem of local surfaces, the exponential map	[Kü] 4D	Lecture 22
11/19/2021	The Gauss-Bonnet theorem	[Kü] 4F	Lecture 23
12/01/2021	The Gauss-Bonnet theorem continued	[Kü] 4F	Lecture 24

Midterm project. Due October 29.

Problem Sets

1. Problem set 1. Due September 17.
2. Problem set 2. Due September 24.
3. Problem set 3. Due October 1.
4. Problem set 4. Due October 8.
5. Problem set 5. Due October 15.
6. Problem set 6. Due October 22.
7. Problem set 7. Due November 5.
8. Problem set 8. Due November 12.
9. Problem set 9. Due November 22.

Textbook

[Kü] Kühnel, Differential Geometry: Curves-Surfaces-Manifolds. Third Edition, 2015
[DC] Do Carmo, Differential Geometry of Curves and Surfaces. Second Edition, 2016