Analysis on Manifolds (UU 2015, Block 1)

General Information

The main goal of this course is to develope the language of smooth manifolds and differential forms. The lecture times are:

• Tuesday, 9:00 - 10:45 (BBG 169)
• Thursday, 15:15 - 17:00 (BBG 165)

In addition, there are exercise sessions that take place right after each lecture. The precise times are:

• Tuesday, 11:00 - 12:45 (MIN 023)
• Thursday, 15:15 - 17:00 (MIN 018)

Here's a preliminary list of subjects that I plan to cover:

• Smooth manifolds and smooth maps
• Tangent vectors and differentials of maps
• Tangent bundles and vector bundles
• Submanifolds and the regular value theorem
• Vector fields and their flows
• Tensor calculus and differential forms
• The exterior derivative and the de Rham complex
• Integration on manifolds
• Manifolds with boundary and Stokes' theorem
• De Rham cohomology

In the unlikely event that time will allow it, I will try to mention these things:

• Morse functions
• The index of a vector field and the Poincare-Hopf theorem

Lecture Notes, Slides & Literature

I will not follow any particular book, but I will try to stay reasonably close to Lee's "Introduction to smooth manifolds". Most of what I say will be in this book and I will use the same notation. But I will present some things in a slightly different order and describe some subjects in a different way (e.g. tangent vectors). I have decided to type lecture notes, but I suggest that you take notes yourself nevertheless. You can download the latest version of the lecture notes and also of the slides used in the lectures here:

Download the Lecture Notes
(updated on November 4, 2015, final form for the exam)

Download the slides
(updated on October 22, 2015)

Note: Lee's book is freely available as a PDF at SpringerLink, at least from within the university network. I recommend this book as a supplement to the lecture. It is very easy to navigate and it should always be possible to find things that appeared in the lecture using the table of contents or the subject index. I will therefore not always give precise references.

If Lee's book and the lecture notes are not enough for you, here are some more books to read:

• Broecker, Jaenich - Introduction to differential topology
• Guillemin, Pollack - Differential Topology
• Bott, Tu - Differential Forms in Algebraic Topology
• Golubitsky, Guillemin - Stable Mappings and Their Singularities
• Milnor - Topology from the differentiable viewpoint

Exercise Sessions

For each exercise session there will be an exercise sheet. These exercises are not supposed to be handed and will not be graded. Nevertheless, they should be taken seriously.

• Exercise Sheet 1 (discussed on September 8, 2015)
• Exercise Sheet 2 (discussed on September 10, 2015)
• Exercise Sheet 3 (discussed on September 15, 2015)
• Exercise Sheet 4 (discussed on September 17, 2015)
• Exercise Sheet 5 (discussed on September 22, 2015)
• Exercise Sheet 6 (discussed on September 24, 2015)
• Exercise Sheet 7 (discussed on September 29, 2015)
• Exercise Sheet 8 (discussed on October 1, 2015)
• Exercise Sheet 9 (discussed on October 6, 2015)
• Exercise Sheet 10 (discussed on October 8, 2015)
• Exercise Sheet 11 (discussed on October 13, 2015)
• Exercise Sheet 12 (discussed on October 15, 2015)
• Exercise Sheet 13 (discussed on October 20, 2015)
• Exercise Sheet 14 (discussed on October 22, 2015)

Homework

In addition to the sheets there will also be homework. These have to be handed in and they will be graded. The grades for the homework will contribute 20% of the final grade. Most of the homework exercises will be taken from the sheets discussed in the sessions.

• Homework Sheet 1 (due on September 24, 2015)
• Homework Sheet 2 (updated version) (due on October 1, 2015) [old version]
• Homework Sheet 3 (due on October 8, 2015)
• Homework Sheet 4 (corrected version) (due on October 15, 2015) [old version]
• Homework Sheet 5 (due on October 22, 2015)

Final Exam

Now that it's all over and done, here are my solutions for the final exam:

Final Exam & Solutions

I'll try to get the grading done as soon as possible.