Math 518: Differentiable Manifolds I
Fall 2011

### Basic Information

• Instructor: Eugene Lerman
• e-mail: lerman at math dot uiuc dot edu
• Homepage: `https://math.uiuc.edu/~lerman`
• Course page: `https://math.uiuc.edu/~lerman/518/f11/syl.html`
• Office: 336 Illini Hall
• Office Hours: see my calendar
• Phone: 244-9510
• Class meets: MWF 11 am    in 441 Altgeld Hall

### Prerequisites

Point set topology and linear algebra will be very useful.

## Course outline

1. Manifolds: Definitions and examples including projective spaces and Lie groups; smooth functions and mappings; submanifolds; Inverse Function Theorem and its applications including transversality; (co)tangent vectors and bundles; Whitney Embedding Theorem; manifolds with boundary; orientations.

2. Calculus on Manifolds: Vector fields, flows, and Lie derivative/bracket; differential forms and the exterior algebra of forms; orientations again; exterior derivative, contraction, and Lie derivative of forms; integration and Stokes Theorem.

3. Other topics (depending on how much time we'll have): Sard's Theorem, Distributions and the Frobenius Theorem; Lie groups; DeRham cohomology.

## Texts

Introduction to Smooth Manifolds by John M. Lee may be useful. It is not required.

I will mostly follow these notes. Their state is not final and they will be edited over the course of the semester. So don't print out the whole file.
If your point set topology is rusty, have a look at this short file.

The course grade will be based on weekly homework and a final exam. If there is an overwhelming popular demand, a midterm may be arranged.

## Homework assignments

• homework assignment 1 (due 8/31/2011 in class)
• homework assignment 2 (due 9/7/2011 in class) and a sketch of solutions
• homework assignment 3 (due 9/14/2011 in class). Sketch of solutions
• homework assignment 4 (corrected and expanded; due 9/21/2011 in class). Sketch of solutions
• homework assignment 5 (due 9/28/2011 in class). Solutions
• homework assignment 6 (due 10/5/2011 in class). One typo reported. Solutions
• homework assignment 7 (due 10/12/2011 in class). In problem 1 assume that the time dependent vector field is complete. That is, assume that the vector field in the hint is complete. Please report any further issues! Solutions
• homework assignment 8 (due 10/19/2011 in class) Solutions.
• homework assignment 9 (due 10/26/2011 in class) Solutions.
• homework assignment 10 (due 11/02/2011 in class) Solutions.
• homework assignment 11 (due 11/09/2011 in class) Solutions
• homework assignment 12 (due 11/16/2011 in class) Solutions
• homework assignment 13 (due 11/30/2011 in class)[small correction 11/27] Solutions
• homework assignment 14 (due 12/07/2011 in class) Last one!Solutions

If you want to learn more geometry, next semester you could take 519, where we'll be doing connections, curvature and characteristic classes. And you could take 524 .

## Notes of lectures

• lecture 19 (10/05/2011)
• lecture 20 (10/07/2011)
• lecture 21 (10/10/2011)
• lecture 22 (10/12/2011)
• lecture 23 (10/14/2011)
• lecture 24 (10/17/2011) [integration of differential forms]
• lecture 25 (10/19/2011) [definition of a vector bundle]
• lecture 26 (10/21/2011) [examples of vector bundles, transition maps]
• lecture 27 [dual bundle, exterior powers of bundles, Whitney sum, categories]
• lecture 28 (10/26/2011) [smooth functors and construction of vector bundles]
• lecture 29 (10/28/2011) [uniqueness and existence of the exterior derivative d]