Math 519, Spring 2010

Course:Differentiable Manifolds II, Math 519, Section M1
Time: TR 9:30--10:50am.
Location: 149 Henry Administration Bldg
Instructor: Chris Leininger
Phone: 265-6763
Email: clein (at) math.uiuc.edu
Office: 265 Altgeld Hall
Office hours: TBA
Text(s):

Riemannian Geometry, Manfredo Perdigao Do Carmo (Required text),

A Comprehensive Introduction to Differential Geometry, Michael Spivak,

Metric Spaces of Non-Positive Curvature, Martin R. Bridson and André Haefliger,

Foundations of Differentiable Manifolds and Lie Groups, Frank Warner.

The books will be put on reserve in the library. We will primarily cover chapter 1--9 from Do Carmo's book, but will supplement from the other three books as needed.

Homework: I will make 5 to 7 assignments throughout the course of the semester. These will not be collected, but to get the most from the class, I strongly suggest doing these assignments. I will have problem sessions periodically (scheduled throughout the semester) in which we can discuss problems from any of the problem sets.

Grading: Grades are based entirely on class participation.

Syllabus: Riemannian geometry: Riemannian metrics, connections, geodesics, parallel transport, curvature. Some generalizations of these notions to other bundles. Theorems of Hopf--Rinow, Bonnet--Myers, Hadamard, Cartan; Comparison Geometry; Flat Torus Theorem; Spaces of constant curvature and space forms.

Homework:

Problem set 1.

Problem set 2 (revised with some typos fixed).

Lectures:

corrected sectional curvature of SO(3) (lecture 14, p.65),

corrected solution to Ricci flow on Einstein manifold (lecture 15, p.67).

Class notes: I will be gone a few times throughout the semester. The problem sessions will be considered as the make-up classes for these.

Back to Prof. Leininger's homepage