# Math 518 – Differentiable Manifolds I – Fall 2013

The notion of differentiable manifold makes precise the concept of a space which locally looks like the usual euclidean space Rn. Hence, it generalizes the usual notions of curve (locally looks like R1) and surface (locally looks like R2). This course consists of a precise study of this fundamental concept of Mathematics and some of the constructions associated with it: for example, much of the infinitesimal analysis (i.e., calculus) extends from euclidean space to smooth manifolds. On the other hand, the global analysis of smooth manifolds requires new techniques and even the most elementary questions quickly lead to open questions. If you would like to have a preview of this course and experience a taste of it, you may wish to watch an old recording of 3 lectures by Fields medalist John Milnor.

Lecturer:Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: Wed 11:00 AM (or by appointment);
Class meets: MWF 12:00-12:50 PM, 447 Altgeld Hall;

## Syllabus:

• Foundations of Differentiable Manifolds. Differentiable manifolds and differentiable maps. Tangent space and differential. Immersions and submersions. Embeddings and Whitney’s Theorem. Foliations. Quotients.
• Lie Theory. Vector fields and flows. Lie derivatives and Lie brackets. Distributions and Frobenius’ Theorem. Lie groups and Lie algebras. The Exponential map. Transformation groups.
• Differential Forms. Differential forms and Tensor fields. Differential and Cartan Calculus. Integration on manifolds and Stokes Formula.

## Textbooks:

You can find my lecture notes here, but the following two textbooks are highly recommended.

### Recommended Textbooks:

• John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, GTM vol 218, 2003. See the contents and first chapter here.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, (3rd edition) Publish or Perish, 1999.

There will be weekly homework, 1 midterm and a final exam. All exams/midterms will be closed book.

• Homework and in class participation (40% of the grade): Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Friday class. No late homework will be accepted. The two worst homework grades will be dropped. On Fridays, one homework problem will be discussed in class, with the participation of the students and this will be taken into account for the grade.
• Midterm (20% of the grade): The midterm will take place on Friday, October 11, in the regular classroom (the date is subject to change).
• Final Exam (40% of the grade): You have to pass the final to pass the course. According to the non-combined final examination schedule it will take place 7:00-10:00PM, Tuesday, December 17, in the regular classroom.

## Homework Assignments and Sections covered so far:

• Homework #0: Watch the video Differential Topology by J. Milnor.
• Homework #1 (PDF): Read first two lectures of the Lecture Notes and solve exercises 0.2, 0.4, 0.5, 1.2, 1.3, 1.5.
• Homework #2 (PDF): Read Lecture 2 and solve exercises 1.8, 1.9, 1.10, 2.1, 2.2, 2.4.
• Homework #3 (PDF): Read Lecture 3 and solve exercises 3.1, 3.3, 3.5, 3.6, 4.2, 4.4.
• Homework #4 (PDF): Read Lecture 4 and solve exercises 4.5, 4.6, 5.1, 5.2, 5.3, 5.4.
• Homework #5 (PDF): Read Lecture 5 and solve exercises 5.5, 5.6, 5.7, 5.8, 5.9, 5.10.
• Homework #6 Practice for the Midterm by attempting to solve problems 1-7 in Lecture 6 and the following Midterm Sample Exam. You can check one solution of the Midterm Sample Exam.
• Homework #7 (PDF): Read Lecture 7 and solve exercises 6.5, 6.6, 7.1, 7.2, 7.3.
• Homework #8 (PDF): Read Lecture 8 and solve exercises 7.4, 7.5, 7.6, 8.1, 8.2, 8.3.
• Homework #9 (PDF): Read Lecture 9 and solve exercises 8.5, 8.7, 9.2, 9.3, 9.6, 9.9.
• Homework #10 (PDF): Read Lecture 10 and solve exercises 10.1, 10.2, 10.4, 10.5 and 10.6.
• Homework #11 (PDF): Read Lecture 11 and solve exercises 11.1, 11.2, 11.3, 11.4, 11.5 and 11.6.
• Homework #12 (PDF): Read Lecture 12 and solve exercises 12.2, 12.3, 12.4, 12.5, 12.7 and 12.9.
• Homework #13 (PDF): Read Lectures 15 and 16 and solve exercises 15.5, 15.8, 15.10, 16.1 and 16.2.
• Homework #14 (PDF): Read Lecture 17 and solve exercises 16.5, 16.6, 16.7, 17.2, 17.3 and 17.4.
• Homework #15 Practice for the Final by attempting to solve the following Final Sample Exam. You can check one solution of the Sample Final Exam.
• Final Exam and its Solutions.

Sections of the Lecture Notes covered: Lectures 0-12, 15-17 (Lectures 13-14 will not be covered).