Introduction to differentiable manifolds
Weekly outline

Introduction to Differentiable manifolds 2022
Lecturer : Francesca Carocci (francesca.carocci@epfl.ch)
Teaching Assistant: Marcos Cossarini (marcos.cossarini@epfl.ch)
Hours:
Course : Tuesday 8h1510h00 ( room MA A3 30)
Exercises : Tuesday 10h1512h00 ( room MA A3 30)
Topics and objectives: The objective of this course is to learn the basic notions in the theory of differentiable manifolds. More specifically, by the end of the classes the students are expected to: 1) know definitions and examples of differentiable manifolds and differentiable functions; 2) be familiar with the notions of tangent vector, tangent and cotangent space at a point of a differentiable manifolds; 3) know definitions and examples of submersions, immersions and embedding as well as the notion of rank for a differentiable morphism of smooth manifolds;4) understand the definition of tangent and cotangent bundle, vector fields and differential forms; 5) get familiar with the basics of tensor calculus and integration on smooth manifolds.
Suggested prerequisites: The students are expected to have standard undergraduate courses background in linear algebra, differentiable calculus in , topology
References: All the material we will cover (and in fact way more!) is contained in:
Introduction to Smooth Manifolds (Second Edition) by John M. Lee. https://link.springer.com/content/pdf/10.1007/9781441999825.pdf
A more concise version of the material we will cover can be found in the Lecture Notes of the previous editions of this course:
Marcos Cossarini (from 2021 edition) https://marcosaedro.github.io/diffman_2021/
Yash Lodha's videos (from the 2020 edition): https://mediaspace.epfl.ch/channel/MATH322+Introduction+to+differentiable+manifolds/30062
Anna Kiesenhofer's website (2019 course): https://sites.google.com/view/differentiablemanifolds/home
Homework and Exam: Every week there will be one exercise from the exercise sheet that you will be asked to hand in. The hand ins are collected every 4 weeks (so weeks 4,8,12 ). This will count for 25% of your final grade. There will be a written exam accounting for 75% of your final grade.


Lecture 1: introduction to the course; definition and examples of topological nmanifolds; definition and examples of smooth atlas and smooth manifold.
References( Lee's Book: Chapter 1, Section "Topological manifolds"+ Section "Smooth structures"; 2021 Lectures Notes: Chapter 1: Section 1.1,Section 1.2)
Exercise 1.7 is the exercise to hand in in week 4.

Lecture 2: smooth functions and smooth maps of manifolds; partition of unity, definition and proof of existence.
References: Section 1.3,1.4 of 2021 lecture notes; Chapter 2 of Lee's "Introduction to smooth manifolds"
Exercise 2.5 to hand in in week 4

Lecture 3: Existence of smooth bump functions, Extension Lemma; tangent vectors, tangent space at a point of a manifold and expression in local coordinates.
References: Lee's book: Proposition 2.25, Lemma 2.26; Chapter 3: Lemma 3.2, proposition 3.2;, Proposition 3.6(Exercise), Proposition 3.8, 3.9,3.10 and coordinate representation in local charts.
Notes from last year: tangent vectors are explained in Chapter 2, but in the notes they are first presented in a different way than what we have seen in class, and only in the second questions as derivations.
Exercise 3.8 to hand in in week 4

Lecture 4: Tangent vectors: computations in local coordinates (Lee's book chapter 3 (par 6165); Tangent bundle as a smooth manifold (Lee's Proposition 3.18/ Notes Proposition 4.1.1 + Smooth Chart Lemma, Lee's Lemma 1.3.5/Notes 1.2.8); definition and examples of maps of constant ranks, immersion, submersion, smooth embeddings; criteria for an injective immersion to be a smooth embedding (Lee's Proposition 4.22/Notes 3.1.1)
Exercise to hand in 4.8

Lecture 5: Constant Rank Theorem and proof (Lee's book Theorem 4.12/ Notes Theorem 3.2.2). Application of the CRT: existence of local slice charts and retractions for immersion and for embedded sub manifolds (Lemma 3.3.1+ Proposition 3.3.2 Notes/Theorem 5.8 Lee's book); Submanifolds from regular level sets (Theorem 3.5.2 Notes/ Theorem 5.12 Lee's book). Statement of Whitney Theorem.
Exercise to hand in 5.4

Lecture 6 (more on sub manifolds)
 slice property for immersion and sub manifolds (Notes Lemma 3.3.1, Proposition 3.3.2; Lee Theorem 5.8) Initial property of sub manifolds + corollary (sec 3.3.2 Notes) Characterisation of embedded submanifolds ( Prop 3.4.1 Notes)
Exercise 6.7 to hand in 
Lecture 7
Vector bundles: definition and Examples ( Lee's Book Chaper 10, pag 249252; Notes pag 3839, Example 4.2.5); transition functions for vector bundle and smooth chart Lemma (Lee's Book: Lemma 10.5, Lemma 10.6; (this is not in last year notes but I followed the book completely)); local and global sections of vector bundles (in particular tangent fields), local frames and relation to local trivialisations (Lee's book pag 255257 and 257259; Notes: section 4.2, 4.3)
Exercise to hand in: Exercise 7.4

Lecture 8: vector fields and their flows: vector fields as derivations (pag 180181 Lee's book); Frelated vector fields and push forward (pag 182184 Lee's book); Lie bracket (definition); integral curve of a vector field: examples, existence ( pag 206209 Lee's book); flows of a vector fields ( pag 209211 Lee's book); statement of the fundamental theorem of flows.

Lecture 9: Flows generated by a complete vector field; examples. Dual of a vector space in linear algebra; cotangent vectors and cotangent bundle; change of charts formula for cotangent vectors; convector fields (or 1forms); contraction of a one form with a vector field; differentials of a function; pullback of one form and integration along curves.
We will follow closely last year notes: Chapter 5, pag 4449.
There was no Exercise to hand in in week 8. For week 9, the hand in is Exercise 8.4 which you find in last week exercise sheet.

Lecture 10: manifolds with boundary, definition and extension of known concepts (Section 7.3 Notes/"Manifold with Boundary" Chapter 1 Lee's book +pag 5758 for tangent space to a manifold with boundary); line integrals and 1dimensional stokes Theorem (Section 5.7,5.8 Notes/"Line Integrals" Chapter 11 Lee's Book); Tensor and Alternating Tensors (6.2,6.3 Notes/Section 1 and 2 in Chapter 12 Lee's Book.)
Exercise to hand in 10.3 
Lecture 11. Alternating ktensors: definition, examples, an explicit basis, wedge product (Section 6.3 last year Notes; pag 350356 Lee's book); kdifferential forms on smooth manifolds: tensor bundle and global section, expression in local coordinates, component functions; evaluating differential forms on tangent fields; contraction of a kform to a k1form; wedge product; pullback and the particular case of top degree forms; exterior differential. (Section 6.4 last year Lecture Notes; pag 359365 Lee's book. There are a lot of explicit examples in the book, have a look!)

Lecture 12. exterior derivative of differential forms: definition, properties (Lee's book Proposition 14.23, Theorem 14.24). Orientation and volume forms: definition of orientatation on a vector space, orientable manifolds: definition, characterization in terms of consistently oriented atlas, characterization in terms of volume forms (Notes Section 7.1.1, 7.1.2; Lee's book pag 376383). Definition of integral of a differential forms on a manifold and change of coordinates (Notes, Section 7.2)

Lecture 13: Characterisation of orientable manifolds in terms of nowhere vanishing top degree forms (Proposition 5.15 Lee's book; Proposition 7.1.1 Notes). Definition of integrals for compactly supported forms on an oriented manifold (Lee's Book, Chapter 16, pages 402408; Notes Section 7.2). Boundary orientation: definition and examples (Chapter 15 Lee's book, pag 386388; Section 7.3 Notes.) Statement of Stoke's Theorem (proof in Lecture 14).

Proof of Stoke's Theorem. What can we study now that we have learnt everything on differentiable manifolds?

 The exam will be Wednesday 25.01.2023 from 15h15 to 18h15 (CM1121).
 The exam is written. You are allowed to have with you one A4 paper, handwritten.
 You do not need to know the proofs of the Theorem seen during the lectures in order to solve the exercises of the exam, but you should be able to apply them. In particular you need to know under what hypothesis a certain statement holds.
 I suggest reviewing the solution to the exercises done through the semester.