Differential geometry II  Smooth manifolds
Weekly outline

Course
 Course book: Fiche de cours
 Lecturer: Nikolaos Tsakanikas (nikolaos.tsakanikas@epfl.ch)
 Teaching Assistant: Linus Erik Rösler (linus.rosler@epfl.ch)
Time and Place
 Lecture: Tuesdays, 10:15  12:00, MA A3 30
 Exercise session: Thursdays, 16:00  17:30, INM 200
Literature
 John M. Lee: Introduction to Smooth Manifolds
 Jefrrey M. Lee: Manifolds and Differential Geometry
 Loring W. Tu: An Introduction to Manifolds
 G. Gross, E. Meinrecken: Manifolds, Vector Fields, and Differential Forms  An Introduction to Differential Geometry
Material from previous versions of the course:
 Lecture notes by Marcos Cossarini (2021)
 Videos by Yash Lodha (2020)
Further useful material:
Assessment Methods
 Exercises: There will be an exercise sheet every week, which will be discussed thoroughly in the exercise session. The solution to one designated exercise from each weakly exercise sheet must be submitted the following week (Fridays). The submissions will be corrected and graded. They will account for 25% of the final grade.
 Exam: There will be a written exam, which will account for 75% of the final grade.
 Course book: Fiche de cours

Lecture 1:
 Introduction to the course
 Definition and examples of topological manifolds
 Definition of smooth manifolds
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 1
 Section 1, pp. 17
 Section 2, pp. 1113

Lecture 2:
 Examples of smooth manifolds
 Definition and basic properties of smooth maps
Literature: John M. Lee, Introduction to Smooth Manifolds
Suggestion for selfstudy: The projective space 
Lecture 3:
 Definition and basic properties of diffeomorphisms
 Smooth partitions of unity and applications
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 2, pp. 36  47
Note: There was a question today during the break about the smooth compatibility of two smooth atlases. See Linus' answer to (essentially) the same question for the details.

Lecture 4:
 Definition and description of the (geometric) tangent space at a point of some Euclidean space
 Definition of the tangent space at a point of a smooth manifold
 Definition and basic properties of the differential of a smooth map
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 3, pp. 50  56
Note: We only proved the first half (injectivity) of Proposition 3.9; this is where Lecture 4 finished. Next time we will continue with the proof of the second half (surjectivity) of Proposition 3.9.
 During the lecture there were some questions about the proof of Proposition 3.8. Detailed answers to these questions can be found here. [Added on 10.10.2023, updated and expanded on 11.10.2023]

Lecture 5:
 Coordinate basis for the tangent space of a smooth manifold at a point
 Matrix representation of the differential of a smooth map at a point with respect to coordinate bases
 Change of coordinates
 The tangent bundle
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 3
 Section 2, pp. 5657 & 59
 Section 3, pp. 6065
 Section 4, pp. 6570
 Section 5, pp. 7172
Note:
 We completed the proof of the second half (surjectivity) of Proposition 3.9.
 We started discussing the proof of Proposition 3.12. The complete proof of this result will be given in the next lecture.
 Note that (only) the statement and the construction of the charts $\widetilde{\varphi}$ for $TM$ in the proof of Proposition 3.12 are needed to solve Exercise 4(a) from Exercise Sheet 5. Hence, the whole proof of Proposition 3.12 has already been uploaded.
Info (Indicative Student Feedback on Teaching): You are encouraged to go to ISA and to respond to the Indicative Feedback. The deadline to do so is Sunday, 22.10.2023, and your feedback is, of course, anonymous.

Lecture 6:
 The tangent bundle
 Definition and examples of smooth immersions, smooth submersions and smooth embeddings
 Criteria for an injective smooth immersion to be a smooth embedding
Literature: John M. Lee, Introduction to Smooth Manifolds
Note:
 We completed the proof of Proposition 3.12.
 See here for a minor comment on Example 4.5(2).

Lecture 7:
 The rank theorem and applications
 Properties of smooth submersions
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 4 Section 1, pp. 81  83
 Section 3, pp. 88  91
Note: Due to lack of time, the proofs of Proposition 4.11 and Theorem 4.12 were not presented in the lecture. Next time, we will thus discuss the proofs of these two results, even though they can already be found in the lecture notes. For the same reason, the proof of Theorem 4.13 was also skipped; it can also be found in the lecture notes, but since its proof is mainly topological, we will not discuss it next time.
Suggestions for selfstudy: Chapter 4, Section 4 (Smooth covering maps), pp. 91  95.
(We will not discuss smooth covering maps in the lecture. However, those who have already learnt about (continuous) covering maps in some topology course could read the aforementioned section from Lee's book in order to learn some basics about the corresponding notion in the smooth setting.) 
Lecture 8:
 Definition and characterisation of embedded submanifolds
 Slice charts for embedded submanifolds
 Level sets of smooth maps (with constant rank) and the regular level set theorem
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 5 Section 1
 Section 3
Note: The proofs of Proposition 4.11 and Theorem 4.12 were presented in the lecture.
 The proof of Theorem 5.6 was skipped due to lack of time, but it will be presented (at least partially) in the next lecture.
 Exercises 5, 6, 7 in Exercise Sheet 8 require theory that will be presented next time (Lecture 9).

Lecture 9:
 Definition and characterisation of immersed submanifolds
 Criteria for an immersed submanifold to be embedded
 The tangent space to an immersed or embedded submanifold
 Sard's theorem and Whitney's immersion/embedding theorem
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 5
 Section 2
 Section 4
Note:
 The proof of Theorem 5.6 was presented in the lecture.
 Sard's theorem and Whitney's immersion/embedding theorem were mentioned without proof. For their proof we refer to Sections 2 & 3 in Chapter 6 from Lee's book.
 See Linus' post for a detailed discussion about the (non)uniqueness of topology and smooth structure on a manifold.
 There was a small typo in the statement of Exercise 6(c) from Exercise Sheet 9. It has now been fixed, and thus the statement of this exercise has been slightly updated.

Lecture 10:
 Vector bundles
 Sections of vector bundles
 Local frames and local trivializations for vector bundles
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 10
 Section 1
 Section 2
Suggestion for selfstudy: The whole Chapter 10 from Lee's book; in particular, Section 3 (bundle homomorphisms) and Section 4 (subbundles) which will not be covered in the lecture.

Lecture 11:
 Vector fields: definition, examples, basic properties
 Pushforward of vector fields
 The Lie bracket
 Integral curves of vector fields
Literature: John M. Lee, Introduction to Smooth Manifolds
Note:
 The fundamental theorem about autonomous systems of ODEs was not discussed in the lecture.
 Example 7.11 was not covered in the lecture, but it will be discussed in detail in Lecture 12.

Lecture 12:
 Flows of vector fields
 Complete vector fields
 The cotangent bundle
Literature: John M. Lee, Introduction to Smooth Manifolds
 Chapter 9, Section 2
 Chapter 11, Section 1
Note:
 The understanding of the dual of a vector space is necessary for the understanding of differential 1forms, which will be discussed in Lecture 13, together with differential kforms.

Lecture 13:
 Covector fields = Differential 1forms
 Pullback of differential 1forms
 Differential kforms
 Pullback of differential kforms
 Exterior derivative
Literature: John M. Lee, Introduction to Smooth Manifolds
 Chapter 11, Sections 13
 Chapter 14 (except for "interior multiplication" and "Lie derivatives of differential forms")
Note:
 There were some typos on p. 125 from the lecture notes (Part II); see here for the fixes in the text. These lecture notes have now been updated and those typos have been fixed.
 The linear algebra required for the understanding of differential kforms has been summarised here. [Added on 10.12.2023; updated on 11.12.2023 and again on 16.12.2023]
 Proposition 8.18 and Corollary 8.19 were not covered in the lecture eventually.
 The "existence part" of the proof of Theorem 8.21 was not covered in the lecture, but it can be read from the lecture notes without any difficulties.
Exercise Sheets: There are two exercise sheets associated with this lecture: one about differential 1forms and one about differential kforms. One designated exercise from each should be submitted by Friday, the 22nd of December. Note that these are the last exercise sheets for this course.
Suggestions for selfstudy: Section 4 (line integrals) and Section 5 (conservative covector fields) in Chapter 11 from Lee's book.
 Section 1 (Riemannian manifolds) and Section 2 (the Riemannian distance function) in Chapter 13 from Lee's book.
Info (indepth evaluation of the course): You are encouraged to complete the survey "indepth evaluation of the course" (either via Moodle or via the EPFL Campus App) by 07.01.2024. Your feedback is anonymous. 
Lecture 14:
 Manifolds with boundary
 Orientation of manifolds
 Integration of differential forms on manifolds
 Stokes' theorem and applications
Literature: John M. Lee, Introduction to Smooth Manifolds
 Chapter 15, Sections 12
 Chapter 16, Sections 13
Note:
 The basics about orientations of vector spaces and orientations of manifolds have been summarized here. [Added on 16.12.2023; updated on 17.12.2023; updated again on 18.12.2023]
 Stokes theorem was mentioned without proof. For its proof we refer to Section 3 in Chapter 16 from Lee's book.
 The purpose of Exercise Sheet 14 is simply to practice a bit more with integrals. There is no submission for this exercise sheet and there will also be no solutions for it. Note that the definitions in this exercise sheet as well as the notions "closed form" and "exact form" (which have also appeared in Lecture 13  Part II) are considered to be examinable.
 See also this and this ED Discussion Forum posts.
Suggestion for selfstudy: Section 1 (the de Rham cohomology groups) in Chapter 17 from Lee's book.