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Manifolds occur throughout mathematics -- and hence other sciences -- as spaces in which something interesting happens or spaces in which something interesting is to be found. Often the set of solutions to some problem forms a manifold, or the set of possible configurations of some system forms a manifold. By studying manifolds and by developing a set of tools with which to study particular manifolds, we can gain considerable insight into those problems where manifolds occur.

The key property of a manifold is that it locally looks like ordinary Euclidean space. Therefore, things that work on small patches of Euclidean spaces can often be made to work on manifolds. The most important of these things is calculus. Indeed, one can regard manifolds as the right places to do calculus and many of the questions involving manifolds have their origin in considering a particular theorem of calculus.

This course is designed to be an introduction to the theory of manifolds, looking at particular examples and developing simple techniques for studying them.

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The original official schedule for TMA4190 was not feasible. The previously announced provisional schedule turned out to be the best possible (perhaps self-fulfilling) so the schedule is:

• Lectures:
  • Monday 13:15-15:00, B3.
  • Friday 10:15-12:00, B3.
• Øving:
  • Thursday 15:00-16:00, 656 (Sentralbygg 2, floor 6). See Message 1.

The locations may change if I find better rooms and once numbers have settled down.

Note that the øving is one hour earlier than on the official schedule. It will not take place in week 3 (15th January) or week 4 (22nd January). See Message 1.

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These are board notes from the lectures. They are essentially what I planned to write on the board each lecture. That means that they do not contain everything that I say, they may be slightly condensed compared to what I actually write, and they do not contain any extras or changes introduced in the course of the lecture. Nonetheless, they should be a useful guide as to what the lecture was about.

1. 16th January. Lecture 1: The Essence of Calculus PDF

2. 19th January. Lecture 2: Go With the Flow PDF

3. 23rd January. Lecture 3: A Point of View PDF

   Note: there are a couple of corrections/clarifications to the definitions given at the end of this lecture. These will be explained in lecture 4.

4. 26th January. Lecture 4: The Heart of the Matter PDF

5. 30th January. Lecture 5: Preparation for Pearls PDF

6. 2nd February. Lecture 6: What is ... the Derivative PDF

7. 6th February. Lecture 7: Spherical Encounters PDF

8. 9th February. Lecture 8: Off on a Tangent PDF

9. 13th February. Lecture 9: Intrinsic Tangency PDF

10. 23th February. Lecture 10: A Brief Interlude PDF

11. 27th February. Lecture 11: A Crowd, a Host, of Golden Vector Spaces PDF

12. 2rd March. Lecture 12: A Bundle of Vector Spaces PDF

13. 6th March. Lecture 13: Real Projective Space PDF

14. 9th March. Lecture 14: Real Projective Space (continued) PDF

15. 13th March. Lecture 15: Maps, Chart, Atlases: Mathematics or Geography? PDF

16. 16th March. Lecture 16: Derivative Notions PDF

17. 20th March. Lecture 17: Linear to Non-linear PDF

18. 23rd March. Lecture 18: Something for (Almost) Nothing PDF

19. 27th March. Lecture 19: Submarine Topology PDF

20. 30th March. Lecture 20: Substantial Submanifolds PDF

21. 3rd April. Lecture 21: Intrinsic Vector Bundles PDF

22. 17th April. Lecture 22: A Mapping Miscellany PDF

23. 20th April. Lecture 23: The Return of the ODE PDF

24. 24th April. Lecture 24: Ehresmann's Theorem PDF

25. 27th April. Lecture 25: Continuation of Ehresmann's Theorem (same notes as lecture 24).

26. 6th May. Lecture 26: Odds and Ends PDF

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The problem sets are not part of the assessment for this course. Nonetheless, I strongly recommend that you do them: they are designed to help you understand the material given in the lectures and the course notes.

It may not be necessary for you to do all of every problem. What I recommend is that you read each problem and spend a little time thinking about each one. If you can see clearly how to do a given problem then you should just write yourself some brief notes on how to do it (I advise you to write these down rather than just think about them - it is easy to convince yourself that you see something when you don't if you don't write it down) and move on to the next problem. Those problems that you don't see clearly how to do should be written out in more detail. In the Thursday problem session we will go through those problems that have proved most difficult for the most people.

Each week -- on the Monday -- you may hand in one question to be looked at. As well as the mathematics, I shall also look at the presentation so if you had no problems with the mathematics that week then you should choose a problem to write out in neat to get comments on the presentation aspect (make a comment on your script if you want me to focus mainly on the presentation).

I shall publish the solutions after the Monday lecture. You should go through the solutions, especially to check that those questions that you thought you understood then you really did understand them

1. Problem set one. Published 26th Janurary in PDF. Solution in PDF

   Correction: Qn3(b): p_z should be compared against 0 not 1 in both instances.

   Clarification: Qn3 deals with the n-sphere but then uses notation that suggests specifically the 2-sphere (e.g. the definition of the north pole). You may work with either (so long as you make it clear which). The simplest is to take n = 2 throughout.

2. Problem set two. Published 3rd February in PDF. Solution in PDF

3. Problem set three. Published 9th February in PDF Solution in PDF

4. Problem set four. Published 15th February in PDF Solution in PDF

5. Problem set five. Published 3rd March in PDF Solution in PDF

6. Problem set six. I consider question 2 on problem set 5 to be extremely important so I am setting it again. In addition, adapt the construction of the smooth structure on real projective space to the grassmannian of k-planes in real or complex Euclidean space. Note: although topology is not as important as the smooth structure, you should come up with a candidate for a metric.

7. Problem set seven. Published 17th March in PDF Solution in PDF

8. Problem set eight. Published 25th March in PDF Solution in PDF

9. Problem set nine. Published 31st March in PDF Solution in PDF

10. Problem set ten. Published 21st April in PDF Solution in PDF

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Exams from previous years are available:

• 2005 Solutions
• 2006 Solutions
• 2007 Solutions
• 2008 Solutions

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Lecturer: Andrew Stacey

Reference Group:

• Karin Marie Jacobsen karinmja@stud.ntnu.no
• Andreas Oppebøen andreao@stud.ntnu.no

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• Smooth Structure

  Same as a maximal smooth atlas.

• Differentiable Manifold

  Topological manifold which can be given a smooth structure.

• Differential Manifold

  Same as a smooth manifold.

• Parametrisation (Chart)

  Homeomorphism ψ: ℝⁿ ⊃U →V ⊂M.

• Coordinate (Chart)

  Homeomorphism ψ: V ⊃M →U ⊂ℝⁿ.

• Chart

  Either a parametrisation or coordinate chart depending on context or convention.

• Function

  Technically: a triple (A,B,f) where A is the domain of the function, B is the codomain of the function, and f is a rule that assigns an element of B to each element of A.

• Map

  Function or morphism, depending on convention or context.

• Morphism

  The general description would be either "things that can be composed" or "a way of getting from one object to another". When the "objects" have the form of "sets with structure" then this means "function on the underlying set that respects the structure".

• Fibre

  The preimage of a point under a map. This is usually used in contexts where all the preimages are equivalent, in some sense, so that one can talk about the _generic fibre_.

• Diffeomorphism

  A smooth bijection whose inverse is also smooth.

• Covering

  A local diffeomorphism that is surjective. "Local diffeomorphism" means that every point has a neighbourhood such that the restriction of the map to this neighbourhood is a diffeomorphism on to its image. This does not imply either injective or surjective.

• Immersion

  A smooth map whose tangent mapping is injective on tangent spaces.

• Submersion

  A smooth map whose tangent mapping is surjective on tangent spaces.

• Regular point

  For a map f : M →N, a point p ∈M for which T_q f has constant rank near p.

• Critial point

  A point in the domain that is not a regular point.

• Regular value

  A point in the codomain whose preimage only has regular points.

• Critical value

  A point in the codomain that is not a critical value.

• Section

  A section of a map f : X →Y is a map s : Y →X such that f s is the identity on Y. The map s is usually required to be of the same "type" as f; e.g. continuous or smooth. Often the map f is obvious in which case it is left out. For example, a section of a vector bundle is a section of the projection map that is part of the data of a vector bundle.

• Vector Field

  A section of the tangent bundle of a manifold.

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The main text for this course are the notes Differential Topology by Bjørn Dundas of Bergen. We shall be using the stable version from 2007. For those keen to save a little paper, I have made 2-up and 4-up versions. The magnification is .85 for the 2-up and .6 for the 4-up.

• Original
• 2-up version PDF PS.gz
• 4-up version PDF PS.gz

Other versions, including a new - but unstable - version can be found on Bjørn Dundas' webpage.

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According to the official regulations, the assessment for this course depends completely on the final exam. There will be regular problem sets and whilst these do not count towards the final grade the best preparation for the exam is to do the problem sets.

Exam date: 6th June, 9am (4hrs).

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Links to official information about this course:

• Description Norsk English
• Final exam dates (scroll down a long way)
• Previous webpages can be found from this page
• The time plan is now irrelevant.

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There will be a revision session for the course on Wednesday 3rd June from 10am until 12noon in room 734 (7th floor in the mathematics building).

I shall also hold office hours on Friday 5th June from 10am until 11am in my office (12th floor in the mathematics building).

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