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
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
-
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).
-
Lecture 2:
- Diffeomorphisms
- Partitions of unity and applications
- Geometric tangent space to R^n at a point
-
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)
-
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
-
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
-
Lecture 6:
- Applications of the rank theorem
- Further properties of surjective smooth submersions
- Embedded submanifolds: definition, characterization, slice charts
Suggestion for self-study: Smooth covering maps (pp. 91-95)
-
Autumn holidays: No lecture and no tutorial will take place this week!
-
Lecture 7:
- Level sets of smooth maps (of constant rank)
- The regular level set theorem
- Immersed submanifolds
- The tangent space to a smooth submanifold
- Level sets of smooth maps (of constant rank)
-
Lecture 8:
- TBC
- TBC
-
-
-
-
-
-
