Math 518, Differentiable Manifolds I
Fall 2014
Course Description
This is a first graduate course on smooth manifolds, introducing
various aspects of their topology, geometry, and analysis. We will
start at the beginning with the definition of a smooth manifold, look
at some examples, and then explore the basic associated objects,
including submanifolds, tangent vectors, bundles, and derivatives.
We will apply the inverse function theorem to geometric issues like
transversality, and then look at vector fields, associated flows, and
the Lie derivative. Differential forms on manifolds will also be a
focus, including how to differentiate and integrate them. Time
permitting, we might look at the very basics of Lie groups,
foliations (the Frobenius theorem), Morse theory, or de Rham
cohomology. In addition to treating the foundations of the subject
carefully, this course aims to emphasize examples and geometric
intuition throughout.
Prerequisites: A good understanding of basic real analysis
in several variables (e.g. the inverse and implicit function theorems) and some
knowledge of metric spaces or point-set topology.
Texts
Here are some recommended references; the first one is the official
text, but the others are good too. For the first three, the book
title is linked to a PDF version which is free to all U of I folks.
- John M. Lee, Introduction to smooth manifolds, Springer
GTM, 2nd ed, 2012.
- William M. Boothby, An introduction to differentiable
manifolds and Riemannian geometry, 2nd ed, 1986.
- Rui Loja Fernandes, Differential
Geometry, notes from Math 518 and 519 for 2013-14.
- Victor Guillemin and Alan Pollack. Differential
topology, Pentice-Hall 1974 or AMS Chelsea 2010.
Grading
Your course grade will be based on:
- Weekly homework assignments: (30%) These will
typically be due on Wednesday. Late homework will not be
accepted; however, your lowest two homework grades will be
dropped so you are effectively allowed two infinitely late
assignments. Collaboration on homework is permitted, nay
encouraged, but you must write up your solutions
individually and understand them completely.
- Midterm Exam: (30%) The midterm exam will be
Monday, October 13. This will be a 2-hour exam, held in our usual
classroom, 443 AH, from 8-10am.
- Final Exam: (40%) This will be Monday, December 15
from 8-11am in our usual classroom.
Homework Assignments
- HW 1. Due Wed, Sept 3.
- HW 2. Due Wed, Sept 10.
- HW 3. Due Wed, Sept 17.
- HW 4. Due Wed, Sept 24.
- HW 5. Due Wed, Oct 1.
- HW 6. Due Wed, Oct 8.
- HW 7. Due Wed, Oct 22.
- HW 8. Due Wed, Oct 29.
- HW 9. Due Wed, Nov 5.
- HW 10. Due Wed, Nov 12.
- HW 11. Due Wed, Nov 19.
- HW 12. Due Wed, Dec 3.
- HW 13. Due Wed, Dec 10.
Lecture notes
Here scans of my lecture notes, in PDF format.
- August 25. Introduction. Actually
did pages 1-4.
- August 27. Smooth manifolds. Actually
did pages 1-3, page 4 up to the lemma, and the top part of page 7.
- August 29. Smooth maps and
diffeomorphisms. Did pages 1-5 with some brief discussion of 6.
- Sept 3. Tangent spaces. Did pages
1-4 with a somewhat extended discussion of how this will be helpful.
- Sept 5. Derivations as tangent
vectors. Did through first lemma on page 6.
- Sept 8. More on tangent spaces.
Did everything except the discussion of germs.
- Sept 10. Immersions, embeddings, and
covering maps. Everything with the exception of the examples
at the very bottom of page 4.
- Sept 12. Covering maps and
submersions. Did everything except the statement of the inverse
function theorem.
- Sept 15. Inverse Function Theorem.
Did everything except the proof that the inverse is smooth.
- Sept 17. Immersions and submersions
in local coordinates. Did pages 1-3.
- Sept 19. Preimages of
submersions. Did 1-3 and the high points of 4-5.
- Sept 22. Tangent bundles and vector
fields. Did 1-5 through the statement of the theorem.
- Sept 24. Lie groups. Did 1-4.
- Sept 26. More on Lie
groups. Did 1-4 and very top of 5.
- Sept 29. Lie groups: subgroups and
actions. Did everything, though last page was rushed.
- Oct 1. Vector fields integral curves,
and flows. Did everything, though last page was rushed.
- Oct 3. Lie algebras, Lie derivatives,
and Lie brackets. Did everything, though last page was rushed.
- Oct 6. Equality of the Lie bracket and
the Lie derivative. Did 1-4.
- Oct 8. Lie algebras of Lie groups.
Did 1-4, but ended 5 minutes early.
- Oct 10. 1-parameter subgroups and the
exponential map. Did everything.
- Oct 15. Prelude to integration.
Did everything, almost ran out of material.
- Oct 17. Covector fields. Did everything.
- Oct 20. Integrating covector
fields. Did 1-4.
- Oct 22. Riemannian metrics. Did
1-5.
- Oct 24. Riemannian geometry. Did
1-3 and a very condensed version of 4-5.
- Oct 27. Tensors and
k-forms. Did 1-5.
- Oct 29. Differential
forms. Through definition of orientation on page 5.
- Oct 31. Orientations of
manifolds. Did everything.
- Nov 3. Partitions of Unity. Did
everything, almost ran out of material.
- Nov 5. Integration on
Manifolds. Covered everything; the missing page 4 does not exist.
- Nov 7. Exterior
differentiation. Through very top of page 5.
- Nov 10. Lie derivatives of forms;
statement of Stokes theorem. Did everything.
- Nov 12. Proof of Stokes theorem.
Did 1-3 and 5.
- Nov 14. No class. Read the section
The Divergence Theorem in Chapter 16 of Lee (very bottom of
page 422 through the top of page 426).
- Nov 17. Calculus to cohomology. Did
1-(top of 4), plus subsequent items 2 and 3.
- Nov 19. Properties of de Rham
cohomology. Did through top half of 4.
- Nov 21. Homotopies and
cohomology. Did everything.
- Dec 1. Mayer-Vietoris by example. Did
1-4.
- Dec 3. The Mayer-Vietoris sequence.
Did 1-(very top of 4) but purposely ended 10 minutes early.
- Dec 5. Degrees of maps of
spheres. Did everything.
- Dec 8. Applications of cohomology.
Did everything with time left for ICES
forms.
- Dec 10. PoincarĂ© duality. Did 1-4
and the 5 really fast.