Intro to Differentiable Manifolds

Dr. Jo Nelson
Math GU 4081
Spring 2018

Email: nelson [at] math [dot] columbia [dot] edu
Lectures: MW 1.10-2.25pm
Location: 520 Math
Office Hours: MW 2.30-3.30pm
Office: 624 Math

Piazza

Textbooks

The official textbook for the course is John Lee, Introduction to smooth manifolds, free through Columbia's library. We will also use Guillemin & Pollack, Differential Topology as a supplemental text. The latter book defines manifolds as subsets of Euclidean space instead of giving the abstract definition, which we will cover from Lee. It is more elementary than Lee's book, but gives nice explanations of transversality and differential forms (which we will be covering). Here are some other books which you may find helpful:

Munkres, Topology, second edition.
Clearly and gently explains point set topology, if you need to review this.

Milnor, Topology from the differentiable viewpoint.
A beautiful little book which introduces some of the most important ideas of the subject.

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.

Teaching Assistant

The teaching assistant for this course is Sara Venkatesh. She will help grade homework. She will hold office hours Tuesdays 2-3.30 in room 610 Math.

Homework

Homework will count for 30% of your final grade. Clearly print your first and last name on your assignment and indicate those students that you worked with. Staple together all of your homework that is due on a given day. You are encouraged to collaborate with your classmates on the assignments, but the write-up must be in your own words. Late homework will not be accepted. Your lowest TWO homework scores will be dropped.

Exams

There will be one take home midterm, worth 30% of your course grade, which you will have a week to do. It will be handed out in class on Wednesday February 28, 2018 and due by 5pm on Friday March 7, 2018 to my office 624 math or uploaded to gradescope. You are not permitted to work with other students and you are not permitted to consult the internet beyond the course Piazza page or wikipedia. PDFs of any textbooks you find helpful may be used.

The final exam will count for 40% of your course grade. It will be a take home exam given on Monday April 30 and you must turn it in to gradescope by 10pm on Monday May 7 (Jo has jury duty and will not be in). As with the midterm you are not permitted to work with other students and you are not permitted to consult the internet beyond the course Piazza page.

In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from the dean and doctor must be provided to me.

Help

If you find yourself confused, please seek help sooner rather than later. I will be available to answer questions during my office hours or by appointment. You should use Piazza to post questions about the course, including questions about topics covered in class or regarding the homework.

In order to receive disability-related academic accommodations, students must first be registered with the Disability Services (DS). More information on the DS registration process is available online at www.health.columbia.edu/ods. Registered students must present an accommodation letter to me in person at least one week before an exam. Students who have, or think they may have, a disability are invited to contact DS for a confidential discussion.

Outline

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).

• Topological manifolds, smooth manifolds, smooth maps, diffeomorphisms, manifolds with boundary. (Lee, chapters 1-2)
• Tangent vectors, tangent space, differential of a smooth map, tangent bundle. Calculations in coordinates. (Lee, chapter 3)
• Vector fields and Lie bracket. (Lee SECOND chapter 8, FIRST chapter 4)
• Immersions, embeddings, and submanifolds. Submersions. (Lee, SECOND chapter 4, FIRST chapters 7-8)
• Transversality and Sard's theorem. ( Guillemin and Pollack, Lee SECOND chapter 6.)
• Vector bundles and the cotangent bundle (Lee SECOND chapter 10-11, FIRST chapters 5-6)
• Tensors and Riemannian metrics. (Lee SECOND chapter 12-13, FIRST chapter 11)
• Differential forms and Stokes' theorem. (Lee SECOND chapters 14-16, FIRST chapters 12-14. See also Guilleman and Pollack or Bott and Tu.)
• Flows and the Lie derivative. (Lee, chapters 17-18)
• Distributions and foliations. (Lee, chapter 19)
• Contact structures. (will use alternate source)
• Lie groups and Lie algebras. (Lee, chapters 9)
• de Rham cohomology (just a brief sketch). (Lee, chapter 15-16)
• A bit of Morse theory (if time permits; not covered in Lee's book).

Schedule & Assignments

Date	Material Covered	Homework
1/17	Definition of topological and smooth manifold, examples.
1/22	Examples of smooth manifolds. Smooth functions. Diffeomorphisms. Einstein summation convention.
1/24	Tangent vectors and the tangent space. Derivative of a smooth map between smooth manifolds.	REVISED Homework 1 LaTeX Due 1/31 (WEDNESDAY)
1/29	Local coordinates, Vector fields and the tangent bundle.
1/31	Immersions, embeddings, and submersions. Short movie ``Outside In".	Homework 2 LaTeX Due 2/5
2/5	Embeddings and submanifolds. Introduction to Sard's theorem.
2/7	Transversality.	Homework 3 LaTeX Due 2/14
2/12	Results that submanifolds "generically" intersect transversely.
2/14	The Whitney embedding theorem. A special case of the tubular neighborhood theorem.	Homework 4 LaTeX Due MONDAY 3/5
2/19 [drop date]	Finishing Whitney Approximation. Orientations.
2/21	Class Cancelled
2/26	Intersection numbers of compact oriented submanifolds.
2/28	TAKE HOME MIDTERM handed out Poincare-Hopf index theorem	Midterm LaTeX Due FRIDAY 3/9
3/5	The flow of a vector field. Hopf Fibration and Video
3/7	TAKE HOME MIDTERM due Friday 3/9 The Lie derivative of a vector field. Commutat or of two VFs = 0 iff their flows commute. Nice coordinate systems for pointwise linearly independent commuting vector fields.
3/12	Spring Break!
3/14	Spring Break!
3/19	Tensors.	Homework 5 LaTeX Due Monday 3/26
3/21	SNOW DAY
3/26	The cotangent bundle. Symmetric Tensors. Riemannian Metrics.	Homework 6 LaTeX Due WEDNESDAY 4/4
3/28	Alternating tensors in gory detail
4/2	Differential forms in general. Wedge product, pullback, and exterior derivative.	Homework 7 LaTeX Due WEDNESDAY 4/11
4/4	Distributions of the integrable, involutive, and contact persuasion. Contact Slides sketches of topology
4/9	Differential ideals, the Frobenius theorem. Foliations.	Homework 8 LaTeX Due Wednesday 4/18
4/11	A one-form is exact iff its integral over every loop is 0. Integration of differential forms. Orientations revisted. Volume form on a Riemannian manifold.
4/16	Stokes' theorem.	Homework 9 LaTeX Due Wednesday 4/25
4/18	Overview of de Rham cohomology. proof that TM = T*M
4/23	Sketch of singular homology Isomorphism with de Rham cohomology
4/25	TAKE HOME FINAL handed out Overview of Morse Theory. Moduli spaces of gradient flows. Isomorphism between singular and Morse homologies.	Final LaTeX due by 10pm to gradescope Monday 5/7
4/30	A differential topology adventure with Jo
5/7	TAKE HOME FINAL due by 10pm to gradescope