Analysis on manifolds

WISB342

About this course

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 are homeworks every week graded from 1 to 10, with a final average called HW.
  • There is one final exam with a mark E between 1 and 10. To pass the course, E has to be at least $5$.
  • There will be other activities such as quizzes (with a mark Q) or bonus exercises (B).

The final mark will be obtained by combining HW, E, Q and B according to a formula that will be agreed upon during the first week of the course and posted on the web-site of the course.

Herkansing en inspanningsverplichting:
For students who failed for the course, an active participation in the course is required to participate in the retake of this course.

Taal van het vak:
The language of instruction is English.
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Learning outcomes

Zie onder vakinhoud.

Required prior knowledge

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.

Link to more information

If anything remains unclear, please check the FAQ of Utrecht University.

Offering(s)

  • Start date

    2 september 2024

    • Ends
      8 november 2024
    • Term *
      Period 1
    • Location
      Utrecht
    • Instruction language
      English
    Enrolment starts in 47 days
For guests registration, this course is handled by Utrecht University