About the subject


A manifold is an abstraction which generalizes the concept of embedded surface in R^3 and is the basic object studied in differential geometry. 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.


This course will cover the following concepts:

  1. definition and examples of manifolds

  2. quotients and Lie groups,

  3. tangent and cotangent spaces as well as vector bundles,

  4. vector fields and forms, as well as sections of vector bundles,

  5. submanifolds,

  6. diffeomorphisms,

  7. distributions,

  8. tensor and exterior algebras,

  9. exterior derivative and de Rham cohomology,

  10. integration and Stoke’s theorem.

 

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

  1. implicit and inverse function theorems,

  2. Cartan calculus,

  3. Frobenius theorem,

  4. Stoke’s theorem.

Differential geometry

Material covered until now: Cech cohomology, Chapter 1, Sections 1 and 2 of Chapter 2 and sections 1, 2 and parts of 3 from Chapter 4.


Wednesday, December 12, 2012 (week 50)


This lecture we introduced integration on manifolds and stated/proved Stoke’s theorem.


Wednesday, December 5, 2012 (week 49)


This lecture we introduced the Lie derivative of tensors and proved a number of its properties.


Wednesday, November 28, 2012 (week 48)


We introduced tensor fields on manifolds and the exterior derivative. We defined de Rham cohomology and stated the Poincare lemma.


Wednesday, November 21, 2012 (week 47)


We introduced distributions, a higher dimensional version of vector fields, and stated Frobenius theorem. Then we started studying tensors, focusing on skew symmetric tensors (forms).


Wednesday, November 14, 2012 (week 46)


We studied flows of vector fields.


Wednesday, November 7, 2012 (week 45)


Exam 1

Place: Bestuursgebouw, Van Lier en Egginkzaal (Room 1?)

Time: 13:30 to 16:30.

Material for the exam: All of Chapter 1, except for the last section (Distributions and Frobenius theorem).


Wednesday, October 31, 2012 (week 44)


This lecture finished the study of the inverse function theorem. Then we introduced the Lie bracket on vector fields.


Wednesday, October 24, 2012 (week 43)


This lecture we studied submanifolds as well as the inverse and the implicit function theorems.


Wednesday, October 17, 2012 (week 42)


In the first half of this lecture we introduced the notion of pull back of bundles. In the second half we introduced the cotangent bundle and the exterior derivative.


Wednesday, October 10, 2012 (week 41)


This lecture we introduced the tangent space  and vector fields of a manifold as well as the notion of vector bundles and sections of vector bundles.


Wednesday, October 3, 2012 (week 40)


This lecture we introduced the notions of tangent vectors as equivalence classes of paths with the same speed and then argued how one can see vectors as derivations of (germs of) functions and that TpM is the dual space of Fp/Fp2, where Fp is the set of gems of functions vanishing at p.This material is covered in Chapter 1, section 4.


Wednesday, September 26, 2012 (week 39)


This lecture we proved the existence of partitions of unity and defined Čech-cohomology with coefficients in R, Z and smooth functions. We wrote down the differential on zero, one and two chains and described the degree zero cohomology. We showed that for Čech cohomology with coefficients on smooth functions, the Čech cohomology groups vanish in degree greater than zero. Čech cohomology is treated in these notes following the same language used in lectures and in Chapter 5, paragrah 5.33 (page 200) using more advanced language.


Wednesday, September 19, 2012 (week 38)


This lecture we saw more examples of manifolds and introduced the notion of diffeomorphism. Then we covered some consequences of the fact that manifolds are Hausdorff and second countable. This material is covered in Sections 2 and 3 of Chapter 1.


Wednesday, September 12, 2012 (week 37)


We covered notation that will be used consistently throughout the course and then went on to define locally Euclidean spaces, C^k structures on such spaces and finally manifolds. For us a smooth manifold is a second countable, locally Euclidean space with  a C^∞ structure.  Variations on the definition lead to the concepts of topological, C^k, analytic and complex manifold. We went on to introduce the notion of smooth maps between manifolds. We finished the lecture with examples of manifolds: R^n, vector spaces, and the 2-dimensional sphere.

Announcements


  1. Here are some notes on Čech cohomology. As we progress in the course these will be updated to include new material.


  1. Here is a copy of the part of Hurewicz’s book on ODEs relevant to flows of vector fields.


  1. Marks for the first and second hand-in exercise are out.


  1. Exam marks are out.

  2. Exam 2 marks are out

  3. Final marks are out.

Practical information


The book we will be using as reference for this course is Warner’s “Foundation of Differentiable Manifolds”.


The lectures will take place in room 611 in the Maths building on Wednesdays from 13:15 to 15:00, starting on the 12th of September.


The exercise classes take place in room 611 in the Maths building on Wednesdays from 15:15 to 17:00, starting on the 12th of September.


There will be regular hand-in exercises and two exams for this course, in week 45 (2012) and week 3 (2013).


The the hand-in exercises contribute with 20% of the final mark, the first exam contributes with 30% and the last with 50%. If you do the re-take exam, the hand-in exercises contribute with 20% of the final mark and the retake with 80%.