Smooth Manifolds

The notion of a smooth manifold plays an important role in many areas of mathematics and physics, including: differential geometry, algebraic and dif- ferential topology, global analysis, classical mechanics, general relativity and high energy physics. Our first goal in this course is to extend the multi- variable differential and integral calculus to the setting of smooth manifolds. This requires the introduction of tensor fields and in particular differential forms. Our final goal is to give a proof of the Hodge decomposition theo- rem for compact Riemannian manifolds. Time permitting, I shall also prove Weyl’s law regarding the asymptotic distribution of eigenvalues of the Lapla- cian of a Riemannian manifold.

Outlines and Lecture notes:

Here is a tentative list of topics I plan to cover.

  • Smooth manifolds, tangent bundles, vector fields,
  • Distributions and the Frobenius integrability theorem,
  • Tensor fields, differential forms, the Lie derivative, Differential ideals,
  • Integrations on manifolds, de Rham cohomology, de Rham theorem,
  • The Hodge theorem: elliptic regularity and finiteness theorems, the Laplace- Beltrami operator.

You can find topics discussed per day at class here

Prerequisite:

linear algebra, general topology, multivariable calculus and analysis.

Marking Policy:  50% assignments, 30% midterm exam, 20% project. Projects will be chosen in consolation with each student, and usually reelects student’s area of spe- cialization or interest. I expect students to prepare an essay on their project and present it in class in a one hour lecture towards the end of the term. Students should make sure to see me no later than 3 weeks after the start of the class regarding their projects.

Assignments:

Assignement 1: pdf Due Date:  ....

Textbook:

warner    I shall follow Frank Warner’s Foundations of Differentiable Manifolds and Lie Groups and plan to cover Chapters 1, 2, 4, 6. Complementary examples and materials will be presented from other sources.