Math 549: Differentiable Manifolds I

David Dumas

University of Illinois at Chicago
Fall 2017

Boy's surface: An immersion of the projective plane into R3.

Main Text	J. M. Lee, Introduction to smooth manifolds, 2ed, Springer GTM, 2012. (Free ebook access through UIC library)
Other Texts	W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2ed, Academic Press, 1986.
	F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer GTM, 1983.
Lectures	MWF 9:00am in Taft 305
	Optional make-up lecture: Dec 11, 9:00am, in Taft 305
Email	david@dumas.io
Office	SEO 503
Office hours	Monday 2-3 in SEO 503 Wednesday 11-12 in MSLC (SEO 430) or by appointment
CRN	38306

About the final exam

The final exam will be held on Tuesday December 12, 10:30am-12:30pm in Taft Hall 305 (our usual classroom).

The exam will have a total of six problems, of which you will be required to complete any four.

The exam is cumulative. There will be a slight emphasis on material covered after the midterm.

The exam problems will be shorter than most of the homework problems, but similar in overall difficulty.

Documents

• Course syllabus
• Topic outline (updated December 9)

Calendar & Grading

This is a condensed summary of some important dates and the weights used to compute your course grade. The course syllabus is the definitive reference for course policies.

	Date/time	Course grade fraction
Homework	Most Mondays (See list below.)	40%
Take-home midterm	Due Mon Oct 16 at 9:00am in class	20%
Final Exam	Tue Dec 12, 10:30am-12:30pm	40%

Homework

Homework should be submitted directly to the mailbox of the grader, Victor Jatoba, in the third floor mailroom of SEO by 1:00pm on the date indicated.

Note that the submission procedure for the midterm exam is different; it will be collected by the instructor at the beginning of lecture on October 16.

• Homework 0 due Wednesday August 30: Read the syllabus.
• Homework 1 due Monday September 11
• Homework 2 due Monday September 18
• Homework 3 due Monday September 25
• Homework 4 due Monday October 2
• Homework 5 due Monday October 9
• Midterm Exam due Monday October 16 at 9:00am in Taft 305
• Homework 6 due Monday October 23
• Homework 7 due Monday October 30
• Homework 8 due Monday November 6
• Homework 9 due Monday November 13
• Homework 10 due Monday November 20
• Homework 12 due Monday December 4

Typeset solutions are encouraged but not required. If writing solutions by hand, please make sure they are legible. Staple homework if it spans several sheets of paper.

Many of the homework problems are assigned directly from the primary textbook. Some other problems are modified versions of exercises in the primary or secondary textbooks (as indicated by a tag with the author's name) or are adapted from the following sources:

[ND]   Homework problems assigned in Nathan Dunfield's course

Extra problems

• Additional suggested problems from the main textbook (Lee):
  • Chapter 1:   2,6,8,11
  • Chapter 2:   3,9,11; after reading Theorem 2.29, also problem 14
  • Chapter 5:   1,4,6,10,19
  • Chapter 7:   1,3,6,23
  • Chapter 8:   20,24,30
  • Chapter 9:   7
  • Chapter 11:   1,4,9,10
  • Chapter 13:   10, 11, 21
  • Chapter 15:   3
  • Chapter 16:   4, 5, 11, 13
  • Chapter 17:   2
  • Chapter 20:   6

• All problems from the textbook that have been assigned so far:
  • Chapter 1:  7,9
  • Chapter 2:  8
  • Chapter 3:  4,5,8
  • Chapter 4:  6,10
  • Chapter 5:  7
  • Chapter 7:  2,4,13,16
  • Chapter 8:  3,10,11,13,18,29
  • Chapter 9:  2,3,16
  • Chapter 11:  7
  • Chapter 12:  7
  • Chapter 13:  4
  • Chapter 14:  5,6
  • Chapter 15:  1,5
  • Chapter 16:  2,10
  • Chapter 17:  1
  • Chapter 20:  3,5

Resources

• Math 518 (Differentiable Manifolds I), Fall 2014, at UIUC.   Instructor: Nathan Dunfield
  • The lecture notes on this site are very nice, and are based on our textbook.
• Fundamental theorem of ordinary differential equations
  • P. Hartman, Ordinary Differential Equations, Wiley, 1964.
    • See chapters II and V
  • E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, 1969.
    • Includes Osgood's criterion (unique solutions with low regularity)
• Calculus by Gilbert Strang, Wellesley-Cambridge Press, 1991. (Homework 10 refers to statements from a multivariable calculus text. This is provided as an example.)
• The History of Stokes' Theorem by Victor Katz, Mathematics Magazine 52(3), 1979.

Some pathologies and counterexamples

• A function with partial derivatives of every order, at every point, but which is not continuous (hence not differentiable).
  • N. Kimura and J. Burr, Discontinuous Function with Partial Derivatives Everywhere, The American Mathematical Monthly, Vol. 67, No. 8 (Oct 1960), pp. 813-814.
  • Full text: Direct link (works on campus) - UIC proxy link (works off campus; login required)
• Space-filling curves
  • H. Sagan, Space-Filling Curves, Springer, 1994.
• Vector fields that are not uniquely integrable sometimes arise naturally in geometry, for example when considering osculating circles of plane curves (or similar osculation constructions). See e.g.
  • E. Ghys, S. Tabachnikov, V. Timorin, Osculating curves: around the Tait-Kneser Theorem

LaTeX

Students who are learning TeX / LaTeX and who want to typeset their homework may find this template helpful:

• LaTeX homework template

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