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Smooth Manifolds Homework

440-2 - Geometry/Topology: Differentiable Manifolds

Monday, Wednesday, Friday 10:00am - 10:50am, Lunt 102
Course website:


Valentino Tosatti
Office: Lunt 225

Office hours:

Wednesdays, 2:00pm - 3:00pm
or by appointment

Teaching Assistant:

Yajnaseni Dutta -

Course Description:

This course is an introduction to smooth manifolds and basic differential geometry. See the syllabus below for more detailed content information.


J.M.Lee - Introduction to Smooth Manifolds (Second edition), Springer 2012.


There will be weekly written assignments which can be found below along with the due date and time. Problem sets are due on Mondays in class, except as marked below. The solutions will be posted below.

Grading and Final:

The class grades will be based on the weekly homework and on the final exam. The projected final exam date is Thursday, March 19 from 3:00pm to 5:00pm in Lunt 102.

Daily Schedule:

This is a tentative syllabus and it is likely to change as the course progresses.
Date Topics Covered Remarks
Jan. 5, 7, 9 Multilinear algebraHomework 1
Due January 12 in class
Jan. 12, 14, 16 Differentiable manifoldsHomework 2
Due January 21 in class
Jan. 19MLK day, no class
Jan. 21, 23 Tangent vectorsHomework 3
Due January 26 in class
Jan. 26, 28, 30 Sumbanifolds, vector fieldsHomework 4
Due February 2 in class
Feb. 2, 4, 6 Flows, Lie derivativeHomework 5
Due February 9 in class
Feb. 9, 11, 13 Frobenius theorem, Lie groups and Lie algebrasHomework 6
Due February 16 in class
Feb. 16, 18, 20Vector bundlesHomework 7
Due February 25 in class
Feb. 23, 25, 27 Differential formsHomework 8
Due March 2 in class
Mar. 2, 4, 6 Orientation, integration, Stokes' theoremHomework 9
Due March 9 in class
Mar. 9, 11, 13 Riemannian metrics, geodesicsPractice problems for the final exam


Final exam with solutions

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: TBA
Office Hours: MW 2.30-3.30pm
Office: 624 Math



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 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.


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 during the scheduled final exam time, Monday May 7, from 3-4pm to my office in person. 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.


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 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.


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

DateMaterial Covered Homework (WEDNESDAYS)      
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 1LaTeX
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
Finishing Whitney Approximation. Orientations.
2/21 Class Cancelled
2/26 Intersection numbers of compact oriented submanifolds.
2/28TAKE 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
3/7TAKE HOME MIDTERM due Friday 3/9
The Lie derivative of a vector field.
The commutator of two VFs is identically 0 iff their flows commute.
Nice coordinate systems for pointwise linearly independent
commuting vector fields.
Homework 5 TBA
Due 3/21
3/12Spring Break!
3/14Spring Break!
3/19 Tensors. Riemannian metrics One-forms.
3/21 A one-form is exact iff its integral over every loop is 0.
Homework 6 TBA
Due 3/28
3/26Differential forms in general.
Wedge product, pullback, and exterior derivative.
3/28Distributions and foliations. Homework 7 TBA
Due 4/4
4/2Contact structures and symplectic forms.
4/4Integration of differential forms. Stokes' theorem.
Volume form on a Riemannian manifold.
Homework 8 TBA
Due 4/11
4/9 More operations on differential forms.
A few words about de Rham cohomology.
4/11Overview of Morse Theory. Moduli spaces of gradient flows Homework 9 TBA
Due 4/18
4/16Morse chain complex and invariance
4/18Sketch of singular homology Homework 10 TBA
Due 4/25
4/23Isomorphism between singular and Morse homologies.
4/25TAKE HOME FINAL handed out
Lie groups and basic examples.

4/30 More Lie group fun.
5/7TAKE HOME FINAL due between 3-4pm in 624 Math