# MA3402 Differential Forms on Manifolds - Autumn 2021

Schedule | Room | ||
---|---|---|---|

Lectures: | Mondays 12:15–14:00 | Berg B1* | |

Fridays 12:15–14:00 | Gamle fysikk F4* | ||

Exam: | The oral exams will most likely be held during Week 49 (beginning 6 December). See formal info here |
||

Lecturer: | Abigail Linton | ||

Office: | 1206 Sentralbygg 2 | ||

Email: | abigail [dot] linton [at] ntnu [dot] no |

*From Mon 4 October (Week 40) we will try to return to purely physical lectures, and thus the Zoom link will not generally be used.

~~*From Friday 24 September 2021 we will no longer require sign-up for lectures as the 1m distance rule is removed from teaching rooms. I encourage you to attend lectures physically, but if you cannot make it, the Zoom link in Blackboard will remain an option for the time being.~~

~~*For the moment, we are doing hybrid lectures. Before attending a lecture in person, ~~
**please sign up on the "groups" page on Blackboard**. If you don't make it to sign up in time, you can find the Zoom link in the announcements on Blackboard.

## What this course is about

Our goal is to revisit what we learned in vector calculus but now learn how to integrate on manifolds instead. In particular we will reformulate Stokes' and Green's Theorems, which have important applications in everything from topology to physics. We will also learn about de Rham cohomology and see that is surprisingly equivalent to the cohomology theories you will learn in Algebraic Topology I.

## Prerequisites

It is recommended that students have taken TMA4190 "Introduction to Topology", but this is not absolutely necessary. Students should be familiar with: open and closed subsets of n-dimensional Euclidean space, smooth maps between open subsets of Euclidean space, derivatives of smooth maps between Euclidean space, the inverse function theorem. For example you could use Chapter 2 in Tu to both study and use as a reference throughout the course. We will start off building our intuition in Euclidean space, so you will have time to learn about smooth manifolds before we use them. For an alternative reference, try Chapter 1 §§1–5 and Chapter 2 §§1–2 in Guillemin and Pollack.

## Reference Group

Our reference group is: Elias Klakken Angelsen, Markus Valås Hagen, Sigve Lysne, Emil August Hovd Olaisen.

- First meeting: Week 37 (14 September)
- Second meeting: Week 42 (20 October)
- Third meeting: 1 December

## Lecture Summary

In case you miss a lecture, or simply want to remember what we did, here is a summary of our past lectures.

Week | Topic | Reference | Comments |
---|---|---|---|

34.1 | Introduction and tangent vectors on Euclidean space | §2 of [Tu] | Monday 23 Aug in Gamle fysikk F6 |

34.2 | Multilinear functions (k-tensors) | §§3.1-3.6 of [Tu] | lecture_wk34-2.pdf |

35.1 | Wedge product & Exercises | §§3.7-3.9 of [Tu] | week35-1.pdf |

35.2 | Wedge product & Differential forms on Euclidean space | §§3.9-4.1 of [Tu] | week35-2.pdf |

36.1 | Differential forms on Euclidean space | §§4.1-4.3 of [Tu] | week36-1.pdf |

36.2 | Exterior differentiation on Euclidean space | §4.4 of [Tu] | week36-2.pdf |

37.1 | Applications to vector calculus | §§4.5-4.6 of [Tu] | week37-1.pdf |

Exercise sheet 1: | exercise1.pdf | ||

37.2 | Manifolds recap | §§5-6.3 of [Tu] | week37-2.pdf |

38.1 | Derivatives on manifolds | §6.4, §6.6 of [Tu] | week_38-1.pdf |

38.2 | Exercise class | §§3.1-4.6 of [Tu] | exercise1_solutions.pdf |

39.1 | Tangent spaces for manifolds | §6.7, §§8.1-8.2 of [Tu] | week39-1.pdf |

39.2 | Tangent spaces and differential 1-forms | §§8.3-8.6, §17.1 of [Tu] | week_39-2.pdf |

40.1 | (Co)Tangent bundles, (co)vector fields | §§12.1, 12.4, 14.1, 14.5, 17.2-17.5 of [Tu] | week 40-1 overview |

40.2 | Pullback of 1-forms & differential k-forms | §17.5, §§18.1-18.2 of [Tu] | week 40-2 overview |

41.1 | Smooth k-forms and pullback recap | §18.4, §18.6 of [Tu] | R*: §18.3 (bundles) |

Exercise sheet 2: | exercise2.pdf | Edit: Updated hint on last part of Exercise 1 | |

41.2 | Pullback of a k-form, circle example, exterior differentiation | §17.6, §18.5, §19.0 of [Tu] | R*: §18.7 (the circle's periodic 0,1-forms) |

42.1 | Exterior derivative uniqueness and under pullback | §19.1, §§19.3-19.5 of [Tu] | |

42.2 | Exercise class | exercise2solutions.pdf | |

43.1 | Orientation of vector space | §§21.1-21.3 of [Tu] | |

43.2 | Orientation of manifolds, support & partition of unity | §§21.3-21.4, §13.2 of [Tu] | |

44.1 | Orientation & forms/atlases | §§21.4-21.5 of [Tu] | |

44.2 | Manifolds with boundary & interior multiplication | §20.4, §§22.5-22.6, of [Tu] | |

45.1 | Boundary orientation, integration on Euclidean space | §22.6, §23.3 of [Tu] | |

45.2 | Integration on manifolds | §23.4 of [Tu] | |

Exercise sheet 3: | exercise3-updated.pdf | Edit: Updated sign typo in Ex2, typo in Ex3 | |

46.1 | Stokes' and Green's Theorems | §§23.5-23.6 of [Tu] | |

46.2 | De Rahm Cohomology | R*: §§24.1-24.2 of [Tu], Theorem 18.14 in [Lee] | |

47.1 | Recap & Exercise class for Sheet 3 | ||

47.2 | Exercises (Sheet 3 & more) | exercise3solutions.pdf |

*R = Other reading. These are extra sections from [Tu] that we haven't explicitly covered in lectures but that might be interesting to some of you :)

## Exams

Please sign up to your preferred slot in the schedule below! Please note that if necessary, I may contact some of you to reschedule. Contact me if you are unable to make any of these times.

**Sign up here: ** (Google sheet sign-up closed)

We will talk more about the format of the exam nearer the time, but for now, here are some notes about the oral exam.

- The exams will be held in the week 6-10 December.
- Each exam will be approximately 30 minutes, in the form of a physical oral exam in a room with a chalkboard. There will be three of us in the room: you, your lecturer (Abi) and a censor.
- The exam will focus on results/proofs and exercises from the course. Practice exercises to get used to working with the key definitions.
- Practice your oral exam skills by discussing questions/ideas with other students - it is exactly the same process!
- Come with your favourite result+proof, question+answer or exercise from the course and be prepared to talk about it for 5 minutes. The purpose of this isn't (necessarily!) to show off your creativity, but to showcase your talents! This way we won't miss out one of the topics that you are most comfortable with.
- It's ok to be nervous! Just focus on processing your thoughts aloud.

Please contact Abi if you have any specific requirements for the exam (for example if you are unable to attend the exam in person).

## References

Since there are many great books on differential forms and differential geometry, it is worth spending some time to flick through a few and find a style that works for you. Most of them will cover very similar material to what we will in lectures, though they may differ slightly in notation. As well as the campus library, you can find many books free online. We will mainly start our course following Tu.

Suggestions of books:

- [Tu] L. W. Tu,
*An Introduction to Manifolds*, second edition, Springer Verlag, 2011. - [Lee] J.M. Lee,
*Introduction to Smooth manifolds*, Springer-Verlag. - [GH] Guillemin and Haine,
*Differential Forms,*ISBN13: 9789813272774, World Scientific, 2019, Preprint here. - [GP] V. Guillemin and A. Pollack,
*Differential Topology*, Prentice Hall, 1974. - [M69] J. Milnor,
*Topology from the differentiable viewpoint*, The University Press of Virginia, 1969. - [M63] J. Milnor,
*Morse Theory*, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963. - [D] B. Dundas, Differential Topology. Version January 2013
- [S] D. Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965. - [Mu91] J.R. Munkres,
*Analysis on Manifolds*, Addison-Wesley, 1991. - [Mu75] J.R. Munkres,
*Topology: a first course*, Prentice-Hall, 1975. - [RS] Robbin and Salamon,
*Introduction to Differential Topology*, 2018. Preprint here