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
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 session. 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 and examples of smooth manifolds
- Smooth maps between smooth manifolds
Schedule modification: The first exercise session, scheduled on Thursday the 11th of September, will be replaced by a full lecture. In other words, there will be two lectures this week: the regular one on Wednesday the 10th and an additional one on Thursday the 11th instead of the tutorial (which does not have any content yet).
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Lecture 2:
- Diffeomorphisms
- Partitions of unity and applications
- Geometric tangent space to R^n at a point
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Lecture 3:
- The tangent space to a smooth manifold at a point
- The differential of a smooth map
- Computations in local coordinates
Suggestion for self-study: Alternative definitions of the tangent space (pp. 71-73)
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Lecture 4:
- More computations in local coordinates
- Velocity vectors of smooth curves
- The tangent bundle of a smooth manifold
- Definition of smooth immersions, submersions, and embeddings
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Lecture 5:
- Smooth immersions, submersions, and embeddings: definition, examples, basic properties
- Local diffeomorphisms
- The inverse function theorem
- The rank theorem
Important information: You are strongly encouraged to complete the indicative feedback until Sunday, the 12th of October, by midnight. It is anonymous, and information about this quick procedure can be found here. - Smooth immersions, submersions, and embeddings: definition, examples, basic properties
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Lecture 6:
- Applications of the rank theorem
- Further properties of surjective smooth submersions
- Embedded submanifolds: definition, characterization, slice charts
Info: Regarding part (a) of Exercise 1 in Exercise Sheet 6, see here for some clarifications.
Suggestion for self-study: Smooth covering maps (pp. 91-95)
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Autumn holidays: No lecture and no tutorial will take place this week!
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Lecture 7:
- Slice charts for embedded submanifolds
- Level sets of smooth maps (of constant rank)
- The regular level set theorem
- Sard's theorem
- Immersed submanifolds
Info: You may use GeoGebra, for instance, to draw some level sets of various smooth functions.
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Lecture 8:
- Whitney's embedding and immersion theorems
- The tangent space to a smooth (immersed or embedded) submanifold
- Smooth vector bundles: definition and smooth local trivializations
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Lecture 9:
- Smooth sections of smooth vector bundles
- Smooth local and global frames for smooth vector bundles
- Smooth subbundles of smooth vector bundles
- Smooth vector fields (= smooth sections of the tangent bundle)
Suggestion for self-study: Bundle Homomorphisms (pp. 261-264) and Subbundles (pp. 264-267) -
Lecture 10:
- The Lie bracket of two smooth vector fields
- Integral curves of smooth vector fields
- Flows of smooth vector fields: definition and examples of global flows
Suggestion for self-study: Lie derivatives (pp. 227-231)
- The Lie bracket of two smooth vector fields
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Lecture 11:
- Flows of smooth vector fields: fundamental theorem and examples of (non)global flows
- Complete vector fields: definition and (non)examples
- The cotangent bundle of a smooth manifold
- Differential 1-forms aka smooth covector fields (= smooth sections of the cotangent bundle)
- The differential of a smooth function
Info: For various nice pictures of flows of smooth vector fields, see Integral Curves (p. 207) and Regular Points and Singular Points (p. 221).
Important info: This week's exercise session is cancelled; see Linus' email (26.11.2025, 16:13) for useful info regarding its replacement.
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Lecture 12:
- Pullback of differential 1-forms
- Differential k-forms (= smooth sections of the k-th exterior power of the cotangent bundle): definition and pullback
- Computational examples
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Lecture 13:
- The exterior derivative on $\mathbb{R}^n$: definition, properties and examples
- The exterior derivative on a smooth manifold: definition, properties and naturality
- Closed and exact differential forms on a smooth manifold
- Crash course on (smooth) manifolds with boundary
Important information: You are strongly encouraged to complete the in depth evaluation for this course until 11.01.2026 by 23:59. The feedback is anonymous, and information about the whole procedure can be found here. It is also possible to access the course evaluations via the EPFL CampusApp.
Suggestion for self-study: The de Rham cohomology groups (p. 440-443)
- The exterior derivative on $\mathbb{R}^n$: definition, properties and examples
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Lecture 14:
- Crash course on orientations of smooth manifolds
- Integration on smooth manifolds
- Stokes' theorem and applications
Info:
- See here for some comments regarding ES14 and its relevance for the final exam.
- Cheat sheet: In the final exam you can use a cheat sheet, that is, the two sides of a single A4 paper, written exclusively by yourself; see here.
Suggestion for self-study: Conservative covector fields (p. 292-298)
- Crash course on orientations of smooth manifolds
