# Math 214: Differentiable manifolds

## Instructor

Michael Hutchings. [My last name with the last letter removed]@math.berkeley.edu Tentative office hours: Thursday, 10:00-11:30, 2:00-3:30, 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 I recommend, in order of increasing difficulty. These are classics; I read all of them (except Munkres) when I was a student and really enjoyed them.
• 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.)
• Milnor, Topology from the differentiable viewpoint. A beautiful little book which introduces some of the most important ideas of the subject.
• Guillemin and Pollack, Differential topology. Explains the basics 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. I considered using this as the text for this course. Volumes 2-5 are also good (but go beyond this course).
• 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 some bits from chapters 20 and 21). 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 possibly 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 Lee's book (or not).
• Basic definitions: topological manifolds, smooth manifolds, smooth maps, diffeomorphisms. (Lee, chapters 1-2; we will discuss manifolds with boundary later.) A bit about classification of manifolds (not in the book).
• Tangent vectors, tangent space, differential of a smooth map, tangent bundle. Calculations in coordinates. Inverse function theorem, implicit function theorem, and regular level sets. (Lee, chapter 3 and Appendix C)
• Immersions, embeddings, and submanifolds. Isotopy, regular homotopy. Submersions. (Lee, chapters 4 and 5)
• Introduction to Lie groups. (Lee, chapter 7 and a bit of chapter 21)
• Vector fields and Lie bracket. The Lie algebra of a Lie group. (Lee, chapter 8)
• Flows, integral curves, Lie derivative. Exponential map on a Lie group. (Lie, chapter 9 and a bit of chapter 20) Introduction to the Poincare-Hopf index theorem. (Guillemin and Pollack)
• Vector bundles. (Lee chapter 10)
• 1-forms. Exterior derivative and wedge product of same. (Lee, chapter 11)
• Riemannian metrics. (Lee, chapter 13)
• Differential forms on manifolds: wedge product, pullback, exterior derivative, Lie derivative. (Lee, chapter 14)
• Orientations. Integration of differential forms on oriented manifolds. (Lee chapters 15 and 16)
• Stokes' theorem. The volume form on a Riemannian manifold and the divergence theorem. (Lee chapters 15 and 16; for alternate treatments of Stokes' theorem see Guillemin and Pollack or Bott and Tu.)
• de Rham cohomology. Computation using the Mayer-Vietoris sequence. (Lee chapter 17) Degree of a smooth map and applications. (Milnor)
• Introduction to the de Rham theorem. (Lee chapter 18)
• Distributions and foliations. (Lee chapter 19)
• Introduction to Morse theory, if time permits. (not in Lee's book)

## Homework

The course grade will be based on homework, which will be due at the beginning of class on Tuesdays. (There might be no homework on occasional weeks if we have not covered enough material for a new assignment.) Homework will be peer-graded. Collaboration on homework is encouraged but must be acknowledged.
• HW#1, due 2/3.
• HW#2, due 2/10: Grade HW#1. Each of the 7 questions should receive 0-3 points, where 3 points = nearly perfect, 2 points = minor flaws, 1 point = major flaws, and 0 points = no significant progress toward a solution. Readability matters. When you take off points, please write a clear explanation of why you did so. Also, please be sure to write your (the grader's) name on the paper. Thanks!
• HW#3, due 2/17.
• HW#4, due 2/24: Grade HW#3. As before, each numbered question (aside from the final feedback question) is worth 3 points, so the maximum possible score is 18.
• HW#5, due 3/3.
• HW#6, due 3/10: Grade HW#5.
• HW#7, due 3/17.
• HW#8, due 3/31: Grade HW#7.
• HW#9, due 4/7.
• HW#10, due 4/14: Grade HW#9.
• HW#11, due 4/21: Lee 16.2, 16.18, 16.22, 17.1, 17.2, 17.10, 17.13.
• HW#12 (last one), due 4/28: Grade HW#11.

## What we actually did in class

• (1/20) Definition of a topological manifold. Basic examples. Introduction to classification results. Definition of a smooth manifold.
• (1/22) Basic examples of smooth manifolds. Smooth functions on a smooth manifold. Diffeomorphisms. Introduction to classification results. Einstein summation convention. Started discussing tangent vectors.
• (1/27) Finished discussing tangent vectors and the tangent space. Derivative of a smooth map between smooth manifolds.
• (1/29) Inverse function theorem. Regular and critical values and the implicit function theorem.
• (2/3) The tangent bundle. Immersions, embeddings, and submersions. Be sure to check out the movie "Outside In" on youtube.
• (2/5) Embeddings and submanifolds. Introduction to Sard's theorem. Most of the Whitney embedding theorem.
• (2/10) Finished the Whitney embedding theorem. Introduced transversality.
• (2/12) Proved several results to the effect that submanifolds "generically" intersect transversely. Along the way, proved a special case of the tubular neighborhood theorem.
• (2/17) Orientations. Intersection numbers of compact oriented submanifolds. Introduction to the Poincare-Hopf index theorem.
• (2/19) More examples of intersection numbers, and sketch of the proof that intersection number is homotopy invariant. Along the way we introduced complex manifolds and manifolds with boundary.
• (2/24) Lie groups and basic examples. Commutators of vector fields.
• (2/26) The Lie algebra of a Lie group. Basic examples.
• (3/3) The flow of a vector field.
• (3/5) Riemannian metrics (guest lecture by Prof. Bamler).
• (3/10) The Lie derivative of a vector field. The commutator of two vector fields is indentically zero if and only if their flows commute. Nice coordinate systems for pointwise linearly independent commuting vector fields.
• (3/12) Vector bundles. Tensors. One-forms. A one-form is exact if and only if its integral over every loop is zero.
• (3/17) Two-forms. Started on k-forms.
• (3/19) Differential forms in general. Wedge product, pullback, and exterior derivative.
• (3/31) Integration of differential forms. Stokes' theorem. Volume form on a Riemannian manifold.
• (4/2) More operations on differential forms. Definition of de Rham cohomology.
• (4/7) Homotopy invariance of de Rham cohomology. Computation of top degree de Rham cohomology using compactly supported cohomology.
• (4/9) Degree of a map. Application to the Gauss-Bonnet theorem.
• (4/14) More about the Poincare-Hopf index theorem. Sketch of proof of the Hopf degree theorem. Introduction to the Mayer-Vietoris sequence.
• (4/16) The Mayer-Vietoris sequence for de Rham cohomology.
• (4/21) Review of singular homology. The de Rham theorem.
• (4/23) Distributions and foliations.
• (4/28) Connections on circle bundles.
• (4/30) Introduction to Morse theory.