Differential Geometry I
Math 2800 Fall 2010
Instructor
- Piotr Hajlasz
- Office: Thaceray Hall 622
- Office hours: MWF 10-11 am + by appointment.
- E-mail: hajlasz@pitt.edu or hajlasz@gmail.com (preferred one)
Textbook
The main material for the course will be contained in
my notes.
Differential Geometry
We will cover the following topics:
- Theory of curves in the Euclidean space (curvature, torsion, Frenet equations, global theory of curves).
- Submanifolds of R^n. Riemannian metric.
- Theory of surfaces in R^3 (first and second fundamental form, curvature, the Gauss Theorema Egregium, covariant derivative, the Gauss-Bonnet theorem, minimal surfaces, surfaces of constant curvature, the Liouville theorem on conformal mappings in R^n, n>2).
If time permits I will cover the following topics (if I run out of time the topics will be moved to Spring). It is likely that I will not be able to cover all of the
topics listed below in the Fall, but I hope to cover at least some of them:
- Abstract manifolds, the Sard theorem, the Whitney embedding theorem, degree and the Hopf theorem on homotopic mappings into spheres.
- Vector fields, commutators, the Frobenius theorem.
- Tensors and differential forms, Lie derivative.
- Integration of differential forms, the Stokes theorem.
Grades
Homework
The homework with due dates will be posted online.
HW#1 Due day: November 8
HW#2