Differentiable manifolds 1.

Instructor: Roland van der Veen (snellius 234) Assistant: Kevin van Helden. Email us at manifolds17@gmail.com

Lectures: Monday 13:30-15:15, Snellius 402. Problem session: Thursday 15:30-17:15, Snellius 401.

Course description. 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? Why does a soap bubble curve like it does? How can we understand differential equations qualitatively? And what does the graph of a holomorphic function look like in R^4?
A natural setting for discussing such questions are spaces that locally look like R^n, these are called manifolds. We will clean up the the vector calculus of 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.
At the end of the course we will be able to understand 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 statement illustrates the beautiful interaction of geometry, differential equations and topology that manifolds are all about.

Lectures and examination. Weekly lectures and problem sessions. Written exam plus regular homework. Homework should be handed in (or emailed) on Monday before class. The final grade is the weighted average of the written exam and the homework grade. The weight of the homework is the area of the unit sphere in 21-dimensional space. For people that need to retake the exam, it will be an oral exam by appointment and the homework grade will be weighted by the area of the unit sphere in 22-dimensional space.

Literature: Elementary topics in differential geometry by J.A. Thorpe. Pdf version available through the library. A hard copy may be bought for 25 euro through the same site

Tentative schedule:

Additional material: