## Introduction to differentiable manifolds 1, Leiden university, fall 2018

Instructor, Roland van der Veen (office Snellius 234), email: manifolds18@gmail.com

Assistants, Kevin van Helden, Lars Koekenbier, Mathijs Kolkhuis Tanke

Time/Location: Lectures Monday 15:30-17:15, Snellius 174, Problem sessions Thursday 13:30-15:15, Snellius 402/403.

### Course description

The goal is to upgrade multivariate calulus to a modern theory of differentiation and integration where it is as easy to change coordinates as it is in linear algebra. Unlike calculus we will try to understand and prove everything, including major theorems like the implicit/inverse function theorem, and the Stokes theorem (n-dim fundamental theorem of calculus). After doing some elementary differential geometry we move on to lift everything we've learned to curved spaces called manifolds.

The final grade is a weighted average of weekly homework and the final written exam. The homework counts as the area of a disk with radius a quarter in the hyperbolic plane. Homework should be handed in or emailed. Monday before class. The homework is meant to help you prepare for the next lecture and to give you feedback. The homework grade is an average of all your homework sets, ignoring the two worst scores. Finding corrections to the lecture notes/ suggesting improvements may further improve your mark.

The retake exam will be a written exam and in this case your homework counts for only 10% and the exam for 90% of your final grade.

### Schedule

Date Class material Exercises
Mon Sep 3Introduction, implicit function theorem (Sect 2.4) linear algebra (2.1), derivative (2.2) up to proof of chain ruleHomework due Sept 10: Syllabus Exercise 2.2:1 (derivative of product) and 2.4:1 (crazy system)
Thu Sep 6Chapter 2Additional exercises for Chapter 2: 1,2,3,4,10,11,14
Mon Sep 10Finish derivative (2.2), mean value theorems (2.3), intrinsic formulation of implicit function theorem (2.4) Homework due Sept 17: Syllabus Exercise 2.2:2 (derivative of det), Exercise 2.3:1(Mean failure) and 2.3:2(Constant?)
Thu Sep 13Chapter 2Additional exercises for Chapter 2: 17,19,20,...
Mon Sep 17Implicit function theorem, inverse function theorem section 2.4Homework due Sept 24: Exercise 2.4.2 (inverse implies implicit) and 2.4.3(Line bundle)
Thu Sep 20Chapter 2Additional exercises for Chapter 2: 23,26,30,..
Mon Sep 24Definition of integration section 3.1, Determinant, intersection number and definition of k-(co)vectorsHomework due Oct 8: Syllabus Exercise 3.1:1 (change of variables), 3.2:5,6,7. This set only counts as bonus
Thu Sep 26Chapter 3
Mon Oct 8Wedge products and k-vectorsHomework due Oct 8: 3.2.8 and 3.2.9.
Thu Oct 11Chapter 3.2Additional exercises for Chapter 3
Mon Oct 15Integration of k-covector fields, meaning of dx, differentialHomework due Oct 22: 3.3.1, 3.3.2
Thu Oct 18Chapter 3.3,3.4Additional exercises for Chapter 3, 21,22,23,24a and 25
Mon Oct 22Substitution lemma, Exterior derivative, Proof of StokesHomework due Oct 29 exercises 3.3.3,3.5.1
Thu Oct 25Chapter 3Additional exercises for Chapter 3, 28-31
Mon Oct 29Change of variables theorem, Poincare lemmaExercises 3.6.1, 3.6.3, 3.7.1a,b due Nov 12.
Thu Nov 1Chapter 3Additional exercises for Chapter 3, 32-38
Mon Nov 12Chapter 4 Metric, volume form and Hodge starHomework: 4.1.1,4.2.1 due Nov 19
Thu Nov 15Chapter 47-9 from Exercises Chapter 4 by Kevin
Mon Nov 19Chapter 5: Atlasses and manifoldsHomework Exercises 5.1.1,5.1.2 due THURSDAY Nov 29 before 10am. In 5.1.1a a picture is enough to get points, providing the homeomorphisms is bonus. For the other exercises you should be more precise.
Thu Nov 22Chapter 52,3,5,6,9 from Exercises Chapter 5.
Mon Nov 26Analytic continuation and Vector bundles
Thu Nov 29Chapter 58,15,16,17 from Exercises Chapter 5. Homework: Exercise 5.5.1, due THURSDAY Dec 6 before 10am
Mon Dec 3Manifolds and vector bundles again
Thu Dec 6Ch 521-25 from Exercises Chapter 5. Homework: Exercise 5.5.2, due THURSDAY Dec 13 before 10am
Mon Dec 10Fundamental theorem of calculus and geometry on manifolds
Thu Dec 13Practice examSolutions available here
Mon Jan 7, 3pmQUESTIONSLocation Snellius 402
Tue Jan 15WRITTEN EXAM 10-13, HL204,226, now with SolutionsOne of the exam questions will be an exercise treated in the problem sessions.
Fri Mar 1RETAKE EXAM, WRITTEN 10-13, room Snellius 403Written exam.

### Literature

The lecture notes used in the course are available here. Keep in mind that the notes will be updated regularly. Suggestions and corrections are greatly appreciated. Please don't assume the proof is right and you are wrong. The reason for writing my own lecture notes is that all books I have seen are either too advanced or too calculus-like or too long and usually all of the above. Nevertheless some useful references are Calculus on Manifolds by M. Spivak and Introduction to Differentiable manifolds by Lee. At the start of the Dutch Masters program there is a 1-day course meant as a refersher on all the material you're supposed to know based on the lecture notes by E. Looijenga. We treat about half of this material.