Prerequisites: The standard basic notions that are taught in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a vector field, the tangent space (and bundle), differential forms, manipulations with differential forms including de Rham operator and Cartan's magic formula, the definition of de Rham cohomology, integration and Stokes theorem. See e.g. the lecture notes of M. Crainic, or the book "Introduction to smooth manifolds" by John Lee.
Aim of the course: The aim of this course is to provide an introduction to vector bundles and principal bundles and, along the way, illustrate their relevance to the study of various geometric structures on manifolds (Riemannian, foliations, etc).
The plan is to start with vector bundles and their geometry (connections, curvature, etc); when discussing metrics we will make a small digression into Riemannian geometry and the tubular neighborhood theorem. Then we move on to principal bundles, their geometry, describe their relationship with vector bundles and their relevance to the theory of G-structures.
Along the way we will also improvise a crash course on Lie groups (only what we need).
Homeworks and exercises: After each lecture you will receive some "first exercises" for you to focus on first, in order to get a first grasp on the theory. There will also be regular homeworks, which count for 40% of the final mark. The aim of the homeworks is to make the students stay in touch with the material throughout the semester and remove some of the pressure of the final exam. We do have in mind here that the weekly workload for the course is 10 hours, and the fact that both a lot more hours extra or less is not beneficial for the students. Here is a possible rough division of the 10 hours:
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2h (class): the lectures
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1h (class): the werkcollege
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2-3h (home): revise the material from the class and continue solving some of the "first exercises" that will be given to you in order to consolidate the theory
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2-3h (home): solve the homework exercises
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1-2h (home): write down the solutions of the homework
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Possible extra-time: in case you do miss some of the required background (e.g. on flows of vector fields or operations with differential forms), it doesn't mean you have to abandon the course, but it does mean you have to put in a bit of extra-time and work to fill in those gaps. Hopefully not more than 1-2h/week (but hopefully that would rarely be necessary).
(with the important note that skipping C is not a good idea, neither for the learning process nor for time management (e.g. D without C would take a lot more time)).
Extra exercises: for whoever the previous setup for the course is not stimulatin/challenging enough, or wants to invest more time and energy into this course: there are more exercises in the lecture notes and we can also propose extra-homework exercises, more challenging (and still counting for the final mark).
Exam: At the end of the course there will be a 3 hours exam. The final mark will be obtained by:
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taking the weighted average (4H+ 6E)/10 of the mark E for the exam and the mark H that results from the homeworks
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approximating it by a half integer, other than 5.5. The approximation is by the closest half integer (e.g. 6.76 is 7, but 6.74 is 6.5). When the mark is between 5 and 5.49, the approximation is by 5.
General rule for passing the ocurse: a final mark above 6, and the exam mark E at least 5 (the last rule is a general masterrmath rule).
Retakes: in case of retakes, the same rules apply.
Lecture notes/Literature: Lecture notes, made available on the website.
- Docent: Marius Crainic
- Docent: Maarten de Groot
- Docent: Sven Holtrop