D216 Wells Hall
parker@math.msu.edu
Office hours: Mon 2-3, Tues. 1-2, Thurs. 2-3
and by appointment
Goals: This course introduces the intuition and techniques used to study manifolds. Manifolds are the natural setting for calculus in its most appealing and flexible form, and are the primary objects of study in much of modern geometry and topology.
The course will start with a rapid introduction (which should be a review for most students) to linear algebra, metric spaces and multi-variable calculus. Then come the main topics of the course: differentiable manifolds and tangent spaces, vector bundles, transversality, calculus on manifolds, differential forms, tensor bundles, the Frobenius Theorem, the deRham Theorem and cohomology groups. If time permits, we will cover the beginnings of Riemannian geometry. Course outline
Background: The official prerequisites are a 400-level course on Abstract Algebra and one on Real Analysis. In reality, the main prerequisite is a solid knowledge of multi-variable calculus and linear algebra and some knowledge of basic point set topology (open sets, compactness).
Text: Introduction to Smooth Manifolds by John M. Lee.
Other helpful reference books:
- A Comprhensive Introduction to Differential Geometry, Vol. I by Michael Spivak.
- An Introduction to Differential Manifolds by D. Barden and C. Thomas.
- Differential Topology by V. Guillemin and A. Pollack.
- Foundations of Differentiable Manifolds and Lie Groups by Frank Warner.
- Morse Theory by John Milnor.
- Lectures on the Geometry of Manifolds by Liviu Nicolaescu.
Homework: HW1 HW 2 HW3 HW4 HW5 HW6 HW7 HW8 HW9
Comments on HW 2 HW3 Sol'ns HW4 Sol'ns HW7 Sol'ns HW9 Sol'ns
Math 868 Student Seminar Times and location to be announced. Seminar Topics and Schedule
During this time students will give informal talks on small topics related to the class material. You have a choice of either giving a talk, or doing a written alternative project (there may not be time for everyone in the class to give a talk).