Riemannian Geometry

Riemannian Geometry is roughly speaking the study of length and angles, but generalised from the plane (and euclidean space) to curves, surfaces, and general spaces called manifolds. It is the mathematics behind Einstein's theory of general relativity, but has also found use recently in the form of statistical manifolds in machine learning.

Contrary to what is written in the module catalogue, Analysis III is not a requirement to take this course. Students should have a strong understanding of Analysis II, particularly derivatives of functions of several variables. From Dynamical Systems we need only the Picard-Lindelöff theorem. Linear algebra is only needed for basic knowledge of vectors and bases.

News

16.2. Thank you for particpating in the poll. The tutorial will be Fridays 12:00–13:45 in Seminar Room B6 C401 (same as lecture).

12.2. Here is the link to schedule the tutorial https://rallly.co/invite/MbuuXs0OFy3M . There are three options for each time slot: yes, if necessary, cannot attend. Please use accordingly.

Also, please send me an email at r.ogilvie@uni-mannheim.de so I can contact the class for announcements.

Script

Here is the script. Below is a summary of the topics we will cover in the course. For each chapter there will be a main example; I aim to keep the course as concrete and geometric as possible. I will also link to examples here.

Ch 1: Curves and Surfaces – helix and helicoid – Using the example and intutitive geometry we will motivate the questions that we want to investigate in the rest of the course.

Length of a helix

Osculating plane of a helix

Frenet frame of a helix

Curvature of Catenary

Helicoid and normal Curvature

Ch 2: Manifolds – circle – A short intrroduction to manifolds from a slightly unusual perspective. We show how to construct general spaces called manifolds by gluing together pieces of euclidean space. In particular, how can we describe directions in manifolds?

Ch 3: Metrics and Connections – 3-sphere – What extra information do we need to give a manifold in order to define length and angle? In euclidean space, we can slide vectors from one point to another without changing its direction, but in a curved space it is not clear how to do that. We need a “connection”.

Ch 4: Geodesics – hyperbolic plane – How should you define a “straight line” when space itself is curved? An example of a geodesic is the equator of a sphere.

Ch 5: Curvature – sphere – We can see that the world is sphere by looking at objects at sea or from space, but this uses height. How can we tell that the universe is curved without looking from outside the universe?

Teaching

There will be one lecture per week, Mondays 15:30–17:00 in B6 C401.

The tutorial will be on Fridays 12:00–13:30 in B6 C401. We will calculate out some of the examples, as well as discuss some exercises (which are in the script). Please attempt these yourself before the tutorial.

Exam

The assessment for this course will be an oral exam.