Differentieerbare variëteiten

Manifolds are the main objects of differential geometry. They give a precise meaning to the more intuitive notion of “space”, when “smoothness” is important (in comparison, when interested only in “continuity”, one looks at topological spaces, and one follows the course “Inleiding Topologie”). The simplest examples are the usual embedded surfaces in R^3; in general, the underlying idea is similar to how cartographers describe the earth: there is a map, i.e., a plane representation, for every part of Earth and if two maps represent the same location or have an overlap, there is a unique (smooth) way to identify the overlapping points on both maps. Similarly, a manifold should look locally like R^n, i.e. there are maps which identify parts of the manifold with the flat space R^n and if two maps describe overlapping regions, there is a unique smooth way to identify the overlapping points. Most of the notions from calculus on R^n are local in nature and hence can be transported to manifolds. Further, some nonlocal constructions, such as integration, can be performed on manifolds using patching arguments. One interesting aspect of this pasasage from R^n to general manifolds is that various aspects of Analysis become much more geometric/intuitive- in some sense, they get a new life (in this way, a set of functions on R^n, depending on how they were used, may remain a function, or may become a vector-field, or a 1-form, etc).

This course is optional for mathematics students. The course is recommended to students interested in pure mathematics, such as differential geometry, topology, algebraic geometry, pure analysis. Please find more information about the study advisory paths in the bachelor at the https://webspace.science.uu.nl/~crain101/manifolds-2026/.

Leerdoelen:
This course will cover the following concepts:

• definition and examples of manifolds,
• smooth maps, immersions, submersions, diffeomorphisms
• special submanifolds,
• Lie groups, quotients
• tangent and cotangent spaces/bundles,
• vector fields, Lie derivatives and flows,
• differential forms, exterior derivative and de Rham cohomology,
• integration and Stoke’s theorem.

The course will also cover the following important results relating the concepts above:

• implicit and inverse function theorems,
• Cartan identities and Cartan calculus,
• Stoke’s theorem

The students should learn the contents of the course, namely

• the definition of a manifold as well as ways to obtain several examples e.g.,

• by finding parametrizations,

• as regular level sets of functions,

• as quotients of other manifolds by group actions.

• the various equivalent description of tangent vectors.

• the relationship between vector fields and curves (flows).

• Differential forms and the various interpretations/properties of the exterior (DeRham) derivatice.

• Orintations, volume forms and integration of differential forms.

• Stokes’ theorem and the very basics of DeRham cohomology.

At the end of the course, the successful student will have demonstrated their abilities to:

• Be fluent in using the regular value theorem in order to obtain (sub)manifolds and compute their tangent spaces.
• Be able to compute flows of vector fields.
• Be able to manipulate with differential forms both locally (in coordinate charts) as well as more globally (e.g. using global formulas for Lie derivarives, DeRham differential, etc). In particular, make use of the Cartan calculus.
• Be able to integrate differential forms and derive consequences of Stoke’s theorem.
• Compute DeRham cohomology of some simple spaces.

Onderwijsvormen:
Two times per week two hours of lectures and two times per week two hours of tutorials.

Toetsing:
There will be mandatory weekly homework exercises, whose solution is to be written down in the class (not at home). Each homework exercise is graded with a mark between 1 and 10. If n is the total number of homeworks (most probably n will be 7), the average of your best n-1 marks will be denoted H and called the overall homework mark, with two decimals of accuracy.

At the end of the course there will also be a 3 hour written exam, leading to a grade E with two decimals of accuracy.

There may also be occasional bonus exercises (hard ones), that add up to a certain number of points B.

The final grade F is F= min{10, (6 E+ 4 HW)/10 + B} rounded off to the closest integer if F< 6, and to a half integer otherwise. The requirements for passing the exam are: H and E have to be at least 5 (before rounding) and F has to be at least 6.

The grade rules can still change, due to AI.

Herkansing en inspanningsverplichting:
The same rules apply for the retake.

The retake rules can still change, due to AI.

Taal van het vak:
The language of instruction is English.
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Zie onder vakinhoud.

Lineaire algebra (WISB107 and WISB108), Analyse in meer variabelen (WISB212), Inleiding topologie (WISB 243) and Inleiding groepen en ringen (WISB124). Zie de cursusplanner (cursusplanner.uu.nl) voor de inhoud van deze vakken: selecteer Faculteit Betawetenschappen en vervolgens het programma van de bachelor Wiskunde van het meest recente jaar.

• Book Lee - Introduction to Smooth Manifolds, gratis verkrijgbaar via de website van Springer. (This is for the students that want a book that does more things for them, e.g. more details. But please be aware that, as good/attractive as that sounds, having (too) many things done for you is not necessarily positive ...).
• Dictation There are lecture notes for the course, which will be made available on the webpage of the course: https://webspace.science.uu.nl/~crain101/manifolds-2021/ These will be a revised version of the lecture notes from the previous year- see https://webspace.science.uu.nl/~crain101/manifolds-2020/
• Book Guillemin, Pollack - Differential Topology - (This is for the students that want to get some more geometric, intuitive insight, with some nicer stories, not all details worked out but fun to read/consult e.g. during a train ride).