In this course we sharpen and combine our tools from linear algebra and calculus to address geometric questions such as: What is the shortest path from A to B in a curved space? What do we mean by ‘curved’ anyway? How can we understand vector fields qualitatively? And what does the graph of a holomorphic function look like in $\mathbb{R}^4$?

A natural setting for addressing such questions are manifolds: spaces that locally look like $\mathbb{R}^n$. We will clean up the the vector calculus of $\mathbb{R}^n$ and lift it to the context of manifolds using differential forms. This includes a vast generalization of the fundamental theorem of calculus: the general Stokes theorem.

In the setting of manifolds we will be able to prove the famous Gauss-Bonnet theorem: The integral of the curvature of a surface equals the total index of any vector field on the surface and this in turn equals the Euler characteristic. This illustrates the beautiful interaction of geometry, differential equations and topology that manifolds are all about.

Teacher Roland van der Veen, office 234.
Assistant: Sjabbo Schaveling, office 227.

Audience This course is aimed at a broad audience of 3rd year bachelor students, particularly all students with an interest in geometry and analysis. By working concretely in $\mathbb{R}^n$ we aim to strike a balance between learning how to prove theorems and how to do computations in curved space.

Homework and Examination Written exam plus weekly homework. Homework is to be written in latex and should be emailed to manifolds2016@gmail.com Please hand in both a .tex and a .pdf file of your work. The deadline is before 9am the friday after the homework was posted.

Final grade The final grade is computed as follows. Suppose your average homework grade is $H$ and your grade for the final exam is $E$. Define $m= \min(H,E)$ rounded down to the nearest integer. Take two identical sheets of paper, put them on top of each other and glue them together along the edges. Next take a pen and punch $m$ holes in the two layers of paper. At every hole, glue together the edges of the pieces of paper corresponding to each hole. The result is an orientable smooth $2$-manifold $M\subset \mathbb{R}^3$. Denote its Gaussian curvature by $K$. If $\int_M K>-6\pi^2$ then $m$ is your final grade. Otherwise your final grade is $\nu H+(1-\nu)E$, where $\nu$ is the volume of the unit ball bounded by $\mathbb{S}^{14}\subset\mathbb{R}^{15}$.

Prerequisites Linear Algebra 1,2, Analysis 2,3, Complex Analysis, Algebra 1. Topology is not a prerequisite.

Literature Course notes available here. Please note that the course notes will be corrected and updated during the course.
A tex file containing all the current exercises is also available
A useful reference is the book Calculus on manifolds by M. Spivak.

Lectures Thursdays 9:00-10:45, Snellius 401. Problem session Fridays 13:45-15:30, Snellius 401.

### Schedule

• Sept 8 Differentiation and manifolds
Syllabus 1.2, 2.1.
Homework: due Thursday Sept 15 before class. Exercises 1.2: 2c,3b,4c,5b.
• Sept 9
Exercises 1.2:1.
• Sept 15 Manifolds and tangent space
Syllabus 2.1, 2.2 (beginning)
Homework: due Thursday Sept 22 before class. Exercises 2.1: 4a,c. Exercises 2.2:1, 2a.
• Sept 16
Exercises 2.1:1,2.
• Sept 22 Tangent space and geodesics
Syllabus 2.2 and 2.3
Homework: due Thursday Sept 29 before class. Exercises 1.2 6b for V = W= R^2. 2.2.3abce 2.3.1
There is a mistake in 2.2.3e the point (\pi/2,0,0) is not on the manifold M, you may take the point (1,0,0) instead.
• Sept 29 Vector fields and Curvature
Syllabus 2.4 and 2.5 (beginning)
Homework: due Friday Oct 7 before the problem session 13:45. Exercises 2.4.1abc, 2.4.2, 2.5.1
• Sept 30 Exercise 2.2.4
• Oct 6 Multilinear algebra
Syllabus 2.6
Homework: due Friday Oct 14 before the problem session 13:45.: 2.6.1, 2.6.2, 2.6.3.
• Oct 13 Tensor product and Differential forms
Homework: due Friday Oct 21 before the problem session 13:45.: 2.6.6b,c,d, 2.6.7a, 2.6.8a,b,c,d.
• Oct 20 Differential forms
No Homework this week.
• Oct 27 Integration on manifolds
Syllabus 2.7 and 3.1. Homework due Monday Nov 7: 2.7.4 3.1.1 3.1.2
• Nov 3 Manifolds with boundary
• Nov 10 Stokes theorem
Syllabus 3.3. Homework due Friday Nov 18 before the problem session: 3.2.2, 3.3.1, 3.3.3, 3.3.4.
• Nov 17 Mapping Degree
Syllabus 3.4. Homework due Friday Nov 25 before the problem session: 3.3.5, 3.4.2, 3.5.1.
• Nov 24 The Gauss Bonnet theorem Homework due Dec 9 (electronically or in Roland's postvak) 2.6.9a, 3.4.3, 3.5.2, fill in the poll on the exam date below
The deadline for handing in this homework set has been extended to Dec 23. If you already handed it in you're welcome to hand in a revised version. In 3.5.2 b) S should be U and R^5 should be R^6. Part e) may be skipped.
• No class in December
• Jan 5 No class
• Jan 12 Recap
Jan 13 Problem session: Homework due Friday Jan 20, Syllabus Ch5, additional problems 1,2. Note in problem 1j the diffeomorphism may be assumed to respect the canonical projection onto M.
• Jan 19 Questions
• EXAM January 24, 10-13 Snellius room 402

An excellent summary of the main definitions and theorems by Kevin van Helden.

Sage: Using the computer to explore manifolds and other math, for free inside the browser.

Curvature and pizza.

Robot-arms, spiders and manifolds. Why every manifold is diffeomorphic to the configuration space of a robot arm in the plane (well almost)

An amazing lecture by the Fields medalist and Abel prize winner John Milnor. Notice how his approach to manifolds starts out quite similar to ours (dont miss parts 2,3).

A whimsical lecture by another Fields medalist Mikhail Gromov, discussing what a manifold is from quite exotic points of view, including that of an orangutan.