**Course:** Math 520, Differential Geometry, Section B1

**Time:** MWF 9:00--9:50am

**Location:** 445 Atlgeld Hall

**Instructor:** Chris Leininger

**Phone:** 265-6763

**Email:** clein (at) math.uiuc.edu

**Office:** 324 Illini Hall

**Office hours:** MONDAY 10:00--12:00, 2:00--3:00, and by appointment

**Text:** *An Introduction to Differentiable Manifolds and Riemannian Geometry* by W. M. Boothby and *Differential Topology* by V. Guillemin and A. Pollack.

**Homework:** This will be assigned on Wednesday and collected the following Wednesday. Everyone should turn in their own solutions each Wednesday, but I also expect you to work together as well as independently.

An optimal strategy is to: (1) try each problem yourself first; (2) get together with others to discuss your solutions and questions; (3) write up the solutions yourself. There is a balance which must be achieved to optimize the benefits of homework: Struggling with a problem leads to a much deeper understanding of the concepts. On the other hand, if you are really stuck, talking to other can help you get ``unstuck'', and can help you move on.

I may give more problems than you can reasonably do, and some may be difficult. Please be sure you can do the more routine problems before tackling the harder ones. Turn in only polished solutions---don't hand in your scratch work!

**Tests:** There will be both a midterm and a final exam. The final is scheduled for 8:00--11:00 AM, Tuesday, December 12. Parts of either or both may be take-home exams.

**Grading:** The grades for the course will be determined by homework 20%, midterm 30%, and the final 50%. I will not assign any letter grades until the end of the semester.

**Syllabus:** Differential calculus of smooth manifolds: definitions, (co)tangent spaces, submanifolds, regular values, immersions and submersions, transversality, vector fields and flows, Lie derivatives. Integral calculus of manifolds: multilinear algebra, differential forms and exterior derivatives, integration, Stokes' Theorem, Lie derivatives, DeRham cohomology. Vector bundles: definitions, connections, curvature. Riemannian geometry: definitions, Levi-Civita connection, intrinsic v. extrinsic geometry.

Homework 1 due 8/30.

Homework 2 due 9/8.

Homework 3 due 9/22.

Homework 4 due 10/6.

Midterm: take-home portion due 10/9.

Homework 5 due 10/27.

Homework 6 due
11/13 (note the change!!)

Homework 7 due
11/29.

Homework 8 due
12/8.

Final Exam due
12/12 at 3:00pm. Presentations will be from 8:00--11:00, and I'll email
you with your 1/2-hour time slot. Good luck!

Notes on point set topology

Notes on group actions

Here's a couple notes repairing a couple mistakes from class.

Notes on bases and induced maps for tensor and exterior algebras

Notes on notation

(Standard) proof of existence and uniqueness of L-C connection.

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