# Math 214: Differentiable manifolds

## Instructor

Michael Hutchings. [My last name with the last letter removed]@math.berkeley.edu Tentative office hours: Tuesday, Thursday, 1:00-2:00, 923 Evans.

## Textbooks

The official textbook for the course is John Lee, Introduction to smooth manifolds, second edition. (The first edition presents the material in a different order and omits some key topics such as Sard's theorem.) The following are some other books which you might also find useful, in order of increasing difficulty:
• Munkres, Topology, second edition. Clearly and gently explains point set topology, if you need to review this. (However we won't be going into details of point set topology very much in the course.) Also gives a nice introduction to the fundamental group and the classification of surfaces. (Familiarity with the fundamental group is useful but we will not use this much.)
• Guillemin and Pollack, Differential topology. Explains the basic s of smooth manifolds (defining them as subsets of Euclidean space instead of giving the abstract definition). More elementary than Lee's book, but gives nice explanations of transversality and differential forms (which we wil be covering).
• Spivak, A comprehensive introduction to differential geometry, vol. I, 3rd edition. This is a classic, which I considered using as the text for this course. Volumes 2-5 are also good (but go beyond this course). I learned a lot from this series when I was a student.
• Bott and Tu, Differential forms in algebraic topology. As the title suggests, it introduces various topics in algebraic topology using differential forms. We will not be doing much algebraic topology in this class, but you might still enjoy looking at this book while we are discussing differential forms.

## Syllabus

The basic plan is to cover most of the material in chapters 1-19 of Lee's book (adding a few interesting things which are not in the book, and possibly some material from chapters 20-22, as time permits). My goal is for you to understand the basic concepts listed below and to be able to work with them. This material is all essential background for graduate level geometry (except for the most algebraic kind). In class I will try to introduce the main ideas, explain where they come from, and demonstrate how to use them. I will tend to leave technical lemmas for you to read in the book (or not).
• Basic definitions: topological manifolds, smooth manifolds, smooth maps, diffeomorphisms. Also manifolds with boundary. (Lee, chapters 1-2) A bit about classification results (not in the book).
• Tangent vectors, tangent space, differential of a smooth map, tangent bundle. Calculations in coordinates. (Lee, chapter 3)
• Immersions, embeddings, and submanifolds. Submersions. (Lee, chapters 4 and 5)
• Vector fields and Lie bracket. (Lee, chapter 8)
• Lie groups and Lie algebras. (Lee, chapters 7 and 8)
• Flows and the Lie derivative. (Lee, chapter 9)
• Vector bundles, tensors. (Lee, chapters 10, 11, 12)
• Riemannian metrics. (Lee, chapter 13)
• Differential forms and Stokes' theorem. (Lee, chapters 14, 15, 16. For alternate treatments see Guilleman and Pollack or Bott and Tu.)
• de Rham cohomology. (Lee, chapters 17, 18)
• Distributions and foliations. (Lee, chapter 19)
• A bit of Morse theory (if time permits; not covered in Lee's book).

## Homework

The course grade will be based on homework, which will be due at the beginnig of class on Tuesdays. (There might be no homework on occasional weeks if we have not covered enough material for a new assignment.) We will hopefully have a homework reader, but if not, you will grade each other's assignments. Collaboration on homework is encouraged but must be acknowledged.
• HW #1, due 2/5.
• HW#2, due 2/12: Grade HW#1.
• HW#3, due 2/19.
• HW#4, due 2/26: Grade HW#3.
• HW#5, due 3/5: Lee 6.15, 7.2, 7.3, 7.13, 7.14, 7.16. Also, as usual, please give feedback on the difficulty of the assignment.
• HW#6, due 3/12.
• HW#7, due 3/19: Grade HW#6. Also do Lee 10.6, 10.10, 10.13, 11.9 (not to be graded but make sure you can do this), 11.11, 11.16, 11.17.
• HW#8, due 4/2.
• HW#9, due 4/9: Grade HW#8. Also do Lee 14.1, 15.1, 15.3 (suggestion: ignore the hint), 15.5, 16.2. (Optional: you might want to make sure you can do 14.6 and 14.7, but these will not be graded.)
• HW#10, due 4/16: Grade HW#9. Also do Lee 16.9, 16.12, 16.15, 16.18, 16.21, 16.22. (Hint: to do 16.15, first do 16.21 and 16.22 and show that the Laplacian equals d^*d.)
• HW#11, due 4/23. Grade HW#10. Also do Lee 17.1, 17.2, 17.10, 17.12, 17.13. (This should be easier than the previous assignment.)
• HW#12, due 4/30. Grade HW#11. Also prove the Five Lemma (Lee page 481) and do Lee 18.1, 18.6, and 18.7(a). Extra credit: 18.7(b).
• HW#13, due 5/7. TBA

## What we actually did in class

• (1/22) Definitions of topological and smooth manifolds, basic examples. (Lee chapter 1. I skipped manifolds with boundary; we will do this a little later. I also reviewed some classification results which are not in the book.)
• (1/24) Smooth maps and diffeomorphisms. Introduction to the tangent space of a smooth manifold. (Lee chapters 2 and 3. I skipped the existence of smooth partitions of unity from chapter 2. We will need this later but I probably won't go over the proof in class.)
• (1/29) Differential of a smooth map. See chapter 3.
• (1/31) Inverse function theorem, implicit function theorem, regular level sets. See chapter 3 and Appendix C.
• (2/5) Tangent bundle. See chapter 3.
• (2/7) Embeddings and immersions. Isotopy and regular homotopy. See Chapter 4.
• (2/12) Submersions. Submanifolds. Sard's theorem. See Chapters 4 and 5.
• (2/14) The Whitney embedding theorem, see Chapter 6.
• (2/19, 2/21) Transversality, see Chapter 6 and also Guillemin-Pollack.
• (2/26) Introduction to Lie groups, see Chapter 7. (The theorem I stated about quotients by Lie group actions is Theorem 21.10 in the book.)
• (2/28) Vector fields. Lie bracket. The Lie algebra of a Lie group. See Chapter 8.
• (3/5) Integral curves, flows, Lie derivative (Chapter 9). Exponential map on a Lie group (Chapter 20). Briefly mentioned the Poincare-Hopf index theorem (see e.g. Guillemin-Pollack).
• (3/7) Vector bundles, (Chapter 10).
• (3/12) 1-forms (Chapter 11).
• (3/14) Exterior derivative and wedge product of 1-forms.
• (3/19) Riemannian metrics (Chapter 13).
• (3/21) Differential forms on manifolds: wedge product, pullback, exterior derivative, Lie derivative (Chapter 14).
• (4/2) Orientations (Chapter 15). Integration of differential forms on oriented manifolds (beginning of Chapter 16). Manifolds with boundary (Chapter 1).
• (4/4) Stokes's theorem (Chapter 16). The volume form on a Riemannian manifold (Chapter 15) and the divergence theorem (Chapter 16).
• (4/9) Started on de Rham cohomology (Chapter 17).
• (4/11) Top degree de Rham cohomology, and the degree of a smooth map (Chapter 17).
• (4/16) Fun with degree theory. Sketched proofs of the Gauss-Bonnet theorem for surfaces in R^3 and the Hopf degree theorem. (Most of this is not in Lee's book. Some of this is in the beautiful book "Topology from the differential viewpoint" by Milnor.)
• (4/18) Computation of de Rham cohomology. Mayer-Vietoris sequence. See Lee chapter 17.
• (4/23) More about the Mayer-Vietoris sequence. Introduction to the deRham theorem. See Lee chapter 18.
• (4/25) TBA
• (4/30) TBA
• (5/2) TBA