Differential geometry II - smooth manifolds
Weekly outline
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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: Mondays, 17:15 - 19:00, MA A3 31
- 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 typed by Linley Vion (2023)
- 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 sessions. The solution to one designated exercise from each weakly exercise sheet must be submitted the following week (Thursdays). 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
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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
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Lecture 2:
- Examples of smooth manifolds
- Definition and basic properties of smooth maps
- Diffeomorphisms
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 1 and Chapter 2
Schedule change: Since Monday the 16th of September is a public holiday, the 2nd Lecture will take place instead on Wednesday the 18th of September from 16:15 to 18:00 in GC A3 31, see here. -
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Lecture 4:
- The tangent space to a smooth manifold at a point
- The differential of a smooth map
- Computations in local coordinates
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 3 - The tangent space to a smooth manifold at a point
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Lecture 5:
- The tangent bundle
- Velocity vectors of curves
- Definition and examples of smooth immersions, submersions and embeddings
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 3 and Chapter 4
Important: You are encouraged to complete the "Indicative Feedback" for the course via ISA until Sunday, the 13th of October. For further information about the process see here. - The tangent bundle
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Lecture 6:
- Further discussion of smooth immersions, submersions and embeddings
- Local diffeomorphisms
- The Rank Theorem and some applications
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 4
Suggestion for self-study: - Further discussion of smooth immersions, submersions and embeddings
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Autumn holidays: No lecture and no tutorial will take place this week!
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Lecture 8:
- Slice charts for embedded submanifolds
- Regular level set theorem
- Sard's theorem (see Chapter 6)
- Definition of immersed submanifolds
- Images of injective smooth immersions
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 5
Suggestion for self-study: John M. Lee, Introduction to Smooth Manifolds, Chapter 7, Lie Subgroups (in particular, Propositions 7.16 and 7.17) - Slice charts for embedded submanifolds
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Lecture 9:
- Immersed vs. embedded submanifolds
- Tangent space to (immersed or embedded) submanifolds
- Whitney's immersion/embedding theorems (see Chapter 6)
- Vector bundles
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 5 and Chapter 10
Suggestion for self-study: John M. Lee, Introduction to Smooth Manifolds, Chapter 10, Bundle Homomorphisms & Subbundles
Important: The second half of Thursday's tutorial (14.11.2024) will be used instead to complete the discussion about "vector bundles and their sections" that was initiated today (11.11.2024) in the second half of the lecture. -
Lecture 10:
- Sections of vector bundles
- Frames for vector bundles
- Correspondence between (smooth) local frames and (smooth) local trivializations [Exercise Sheet 10]
- Subbundles of vector bundles
- Vector fields
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 8 and Chapter 10
- Sections of vector bundles
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Lecture 11:
- Vector fields as derivations
- Vector fields and smooth maps (= pushforward of vector fields) [Exercise Sheet 11]
- Vector fields and submanifolds [Exercise Sheet 11]
- The Lie bracket [Exercise Sheet 11]
- Integral curves of vector fields
- Flows of vector fields
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 8 and Chapter 9
Suggestions for self-study: -
Lecture 12:
- Complete vector fields
- Differential 1-forms
- Topological and smooth manifolds with boundary
Literature:
- John M. Lee, Introduction to Smooth Manifolds, Chapter 11 and Chapter 12
- Loring W. Tu, Differential Geometry - Connections, Curvature, and Characteristic Classes, Chapter 4, §18 - §19
Important: Appendix C (Multilinear Algebra) from the lecture notes is essential for the understanding of the concepts of "differential forms", "orientations" and "integration on manifolds", which will be discussed during the final lectures of the course. However, due to lack of time, Appendix C will not be covered in the lectures, so you are kindly requested to study it on your own, since it is just (higher) linear algebra.
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Lecture 13:
- Differential k-forms
- The exterior derivative on R^n
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 14
Important:- There will be two lectures this week, but no tutorial. Specifically, the first lecture (= 13th lecture of the course) will take place on Monday the 9th from 17:15 to 19:00 in MA A3 31, as usual, while the second lecture (= 14th lecture of the course) will take place on Thursday the 12th from 16:00 to 18:00 in MA B1 524.
- You are strongly encouraged to complete the "In-depth Evaluation" for the course until 12.01.2025, 23:59. This can be done either on Moodle or via the EPFL Campus App. For further information about the process see here.
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Lecture 14:
- The exterior derivative on a smooth manifold
- Orientation of smooth manifolds
- Integration on smooth manifolds
- Stokes' theorem and applications
Literature: John M. Lee, Introduction to Smooth Manifolds, Chapter 14, Chapter 15 and Chapter 16
Important:- Due to lack of time, Chapter 10 (Orientations) from the lecture notes will only be briefly covered in the lecture, so you are kindly requested to study it (beforehand) on your own, especially Section 10.1, Subsection 10.2.1, Proposition 10.23 and Subsection 10.2.4, which will are required in Chapter 11 (Integration on Manifolds).
- There will be two turorials this week, but no lecture. Specifically, the first tutorial (concerning the 13th lecture of the course) will take place on Monday the 16th from 17:15 to 19:00 in MA A3 31 and line integrals will be discussed during the first hour, while the second hour will run as usual. The second tutorial (concerning the 14th lecture of the course) will take place on Thursday the 19th from 16:00 to 18:00 in IMN 200, as usual.
Suggestion for self-study:
- John M. Lee, Introduction to Smooth Manifolds, Chapter 11, Conservative Covector Fields
- John M. Lee, Introduction to Smooth Manifolds, Chapter 13 (Riemannian Metrics)
- John M. Lee, Introduction to Smooth Manifolds, Chapter 15, The Riemannian Volume Form
