Analysis on manifolds

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 student website.

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. These lead to a grade H with 1 decimal of accuracy.
At the end of the course there will be a 3 hour written exam, leading to a grade E with 1 decimal of accuracy.
The final grade F is be determined by F = max {(7E + 3H)/10, (17E + 3 H)/20} rounded off to an integral number up to 6, and to a half integer above 6 (to the closest one). 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.

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

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

Linear algebra (WISB107 and WISB108), Calculus of several variables (WISB212), Introduction to topology (WISB 243) and Introduction to Groups and Rings (WISB124). See the courseplanner (cursusplanner.uu.nl) for the contents of those courses: select Faculty of Science and then the programme of the bachelor Mathematics of the most recent year.

• 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).