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: see here
Lecturer
Abigail Linton
Office: 1206 Sentralbygg 2
Email: abigail [dot] linton [at] ntnu [dot] no

*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

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: exercise1.pdf
37.2 Manifolds recap Chapter 2 of [Tu]

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
2021-09-14, Abigail Linton