Differentiable Manifolds I, Math 518, Fall 2022

Basic information

• Lectures: MWF 9:00–9:50am in 141 Altgeld Hall.
• Instructor: James Pascaleff
• Email: jpascale@illinois.edu; Office: 357 Altgeld Hall; Phone: (217) 244-7277.
• Office hours: Wednesdays 12:00–2:00pm.
• Course website: https://pascaleff.github.io/518fa22/
• Prerequisites: Multivariable calculus. (Officially: MATH 423 or MATH 481, or consent of instructor.)
• Texts: We shall follow the lecture notes on Differential Geometry by Rui Loja Fernandes. Two recommended textbooks are
• John M. Lee, Introduction to Smooth Manifolds, Springer Graduate Texts in Mathematics volume 218. A full-text PDF is available through the UIUC Library. Homework problems may be assigned from this book.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1.
• Final Exam: Monday, December 12, 2022, 8:00–11:00am in 141 Altgeld Hall.

Course outline

• Description: Differentiable manifolds are a class of spaces that includes Euclidean spaces, smooth curves and surfaces in 3-space, higher-dimensional generalizations such as the $$n$$ dimensional spheres, and infinitely much more. Manifolds locally look like open sets in Euclidean space, but they may have a nontrivial global topology. A differentiable manifold is one where, at least locally, we can “do multivariable calculus” in a meaningful way. This local structure has consequences for the global topology. Even for Euclidean spaces (whose global topology is trivial) the theory of manifolds provides a new perspective on geometry that vastly generalizes traditional Euclidean geometry in multiple directions.

The foundations of differentiable manifolds are not particularly simple, but they support a theory that has a lot of intuitive appeal. One goal of this course is to convey both the technical foundations and the intuitive picture.

• Lectures: Lectures will be held in person.
• Assessment: Grades will be based on homework (25%), a midterm exams (25%), and the final exam (50%). The two lowest homework scores will be dropped.
• Homework: There will be weekly homework assignments whose due dates are listed in the schedule. Homework should be submitted on paper in class. If you are unable to come to class, you may turn in your homework to James Pascaleff’s mailbox in the mailroom, 250 Altgeld Hall.
• Exams: There will be one midterm exam and a final exam. The midterm exam will be held in class on 5 October. The final exam will be held Monday, December 12, 2022, 8:00–11:00am in 141 Altgeld Hall.

Policies

• Collaboration and Academic Integrity: For homework assignments, collaboration is permitted and expected, but you must write up your solutions individually and understand them completely. On exams, no collaboration is permitted.
• Disability accommodations: Students who require special accommodations should contact the instructor as soon as possible. Any accommodations on exams must be requested at least one week in advance and will require a letter from DRES.

Homework assignments

• Grades may be viewed on the Canvas website.
• Optional homework: Watch this series of lectures on Differential Topology by John Milnor. Many of the ideas presented in these lectures will be covered in this course.
• Homework 1 due Friday, September 2
• Fernandes, p. 11: problems 4, 5.
• Fernandes, p. 18-19: problems 1, 4, 5, 7, 9.
• Homework 2 due Friday, September 9
• Fernandes, p. 23: 2, 3.
• Fernandes, p. 26-27: 1, 2, 3, 4, 5, 6.
• Homework 3 due Friday, September 16
• Fernandes, p. 35: 1, 2, 3, 4, 5 (all problems).
• Homework 4 due Friday, September 23
• Fernandes, p. 38: 1, 2, 3, 4, 5.
• Fernandes, p. 46: 1, 3, 5.
• Homework 5 due Friday, September 30
• Fernandes, p. 47: 6, 7, 8, 9.
• Fernandes, p. 54: 1, 3, 4, 5.
• There is no homework due on Friday, October 7.
• Homework 6 due Monday, October 17
• Fernandes, p. 55: 7, 8 (some ideas towards this were discussed in lecture).
• Fernandes, p. 82: 2, 3, 4, 5, 6.
• Homework 7 due Monday, October 24
• Fernandes, p. 87: 1, 3, 4, 5, 6.
• Fernandes, p. 63: 4, 5.
• Homework 8 due Wednesday, November 2
• Fernandes, p. 92: 1, 3, 4, 5.
• Fernandes, p. 98: 2, 3, 5.
• Homework 9 due Friday, November 11
• Fernandes, p. 99: 6, 7, 8, 10.
• Fernandes, p. 109: 1, 3, 6.
• Homework 10 due Wednesday, December 7
• Fernandes, p. 131: 1, 2, 4, 5, 6, 7, 8.
• Homework 11 due Wednesday, December 7
• Fernandes, p. 136: 2, 3, 5, 8 parts (a) and (b).

Schedule

This schedule is a tentative outline of the topics that we will cover. Section numbers refer to Fernandes’ notes. As the semester progresses, this schedule will mutate into a log of what was covered in each lecture.

Week 1 [2022-08-22 Mon] Welcome, historical intro, diffeomorphic subsets of Euclidean space
[2022-08-24 Wed] Examples of smooth manifolds in Euclidean space
[2022-08-26 Fri] Abstract topological manifolds
Week 2 [2022-08-29 Mon] Abstract smooth manifolds
[2022-08-31 Wed] Manifolds with boundary
[2022-09-02 Fri] Partitions of unity
Week 3 [2022-09-05 Mon] Labor day
[2022-09-07 Wed] Partitions of unity, cont’d.
[2022-09-09 Fri] Tangent vectors
Week 4 [2022-09-12 Mon] Tangent vectors as derivations
[2022-09-14 Wed] Tangent spaces
[2022-09-16 Fri] The differential of a map
Week 5 [2022-09-19 Mon] The differential, cont’d.
[2022-09-21 Wed] Immersions, submersions, and submanifolds
[2022-09-23 Fri] Submanifolds, cont’d.
Week 6 [2022-09-26 Mon] Submanifolds; critical points
[2022-09-28 Wed] Regular value theorem
[2022-09-30 Fri] Transversal intersections
Week 7 [2022-10-03 Mon] Whitney’s theorem
[2022-10-05 Wed] Midterm exam covering Sections 0–6.
[2022-10-07 Fri] Vector fields
Week 8 [2022-10-10 Mon] Ordinary differential equations
[2022-10-12 Wed] Vector fields and Flows
[2022-10-14 Fri] Flows cont’d.
Week 9 [2022-10-17 Mon] Lie bracket and Lie derivative
[2022-10-19 Wed] Lie bracket cont’d.
[2022-10-21 Fri] Foliations and distributions
Week 10 [2022-10-24 Mon] The Frobenius theorem
[2022-10-26 Wed] Lie groups and Lie algebras
[2022-10-28 Fri] The Lie algebra of a Lie group
Week 11 [2022-10-31 Mon] Class canceled
[2022-11-02 Wed] Lie theory
[2022-11-04 Fri]
Week 12 [2022-11-07 Mon] The exponential map
[2022-11-09 Wed]
[2022-11-11 Fri]
Week 13 [2022-11-14 Mon] Differential forms and tensor fields
[2022-11-16 Wed]
[2022-11-18 Fri]
Week 14 [2022-11-21 Mon] Fall break
[2022-11-23 Wed] Fall break
[2022-11-25 Fri] Fall break
Week 15 [2022-11-28 Mon] Differential forms cont’d.
[2022-11-30 Wed] Differential and Cartan calculus
[2022-12-02 Fri]
Week 16 [2022-12-05 Mon] Integration of differential forms.
[2022-12-07 Wed]