The notion of differentiable manifold makes precise the concept of a space which locally looks like the usual euclidean space R^{n}. Hence, it generalizes the usual notions of curve (locally looks like R^{1}) and surface (locally looks like R^{2}). 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;**Prerequisites:** Restricted to Graduate Students; Undergraduate students may register with approval.

**In this page:**

- Announcements
- Syllabus
- Textbooks
- Grading Policy and Exams
- Homework Assignments
- Emergency information for students in Mathematics courses

**Announcements:**

- I have made available my lecture notes.
- I have made available the Midterm and a Solution.
- I have made available the Sample Final Exam and its solution.
- You can view the scores of homeworks and midterms on-line (please note that you will have to log in).
- Solutions to the Final Exam have been posted here.
- The grades of the Final Exam as well as the overall grades for the course are available here. You can come by my office Wednesday, December 18, at 5 pm, to look at your Final Exam.

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

**Grading Policy and Exams**

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).

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

**Emergency information for students in Mathematics courses**

For important emergency information related to fires, tornados or active threats, please look at the following leaflet.

*Last updated December 17, 2013*