This is a proof-based introduction to differential geometry, beginning with curves and surfaces in Euclidean space and continuing to intrinsic geometry and Riemannian manifolds. A syllabus can be found here.

The course text is Wolfgang Kühnel, Differential Geometry: Curves–Surfaces–Manifolds, third edition.
Lecture
Time: T/Th 2:00pm – 3:15pm.
Location: CAS 214

Course schedule and notes

Approximate sections of Kühnel are indicated in yellow.

Lecture 1: Prerequisites from calculus.
§1
Lecture 2: Regular curves, arclength, and curvature.
§2A,B
Lecture 3: Torsion and Frenet curves.
§2C,D
Lecture 4: Global theory of curves.
§2F
Lecture 5: Spherical curves; fundamental theorem of curves.
§2F
Lecture 6: Immersions, surfaces
§2F
Lecture 7: Inner products and the first fundamental form
§2F
Lecture 8: Surface integrals, vector fields
§2F
Lecture 9: Coordinate changes, orientations
§2F
Lecture 10: Second and third fundamental forms
§2F
— Midterm Exam 1
Lecture 11: Hypersurfaces and the covariant derivative
§3F, §4A
Lecture 12: Covariant derivatives, brackets, and geodesics
§4A–B

Problem sets

Suggested exercises from Kühnel are listed below. Selected solutions are posted when available.

Chapter 2
1, 4, 5, 7, 9, 10, 12, 16, 18, 20, 23
Chapter 3
1, 2, 3, 5, 7, 8, 9, 18, 19

Exams and practice exams

Midterm Exam 1: Concept review and practice problems with solutions

Midterm Exam 1 — Practice Exam A
Exam Solutions
Midterm Exam 1 — Practice Exam B
Exam Solutions

Project

The group project is worth 25% of the course grade. Groups of two or three will prepare a 6–10 page mathematical expository paper and give a joint presentation. Complete expectations, suggested topics, and the grading rubric are in the project handout.

  1. Partner and topic selection
  2. Preliminary bibliography and proposal
  3. Dec 1 Written report due
  4. Dec 3, 8 Project presentations
Project resources
Sample bibliography Coming soon
Sample proposal Coming soon
Sample write-up Coming soon
Presentation resources Coming soon