Introduction to Riemannian manifolds
Reading course in Riemannian geometry in fall 2024. Riemannian geometry studies curved spaces endowed with a selection of inner products which smoothly vary with respect to the base point. These objects are known as Riemannian metrics and one can for example use them to measure length and angles on curved spaces. The aim of this course is to establish the basic theory of Riemannian metrics and their associated objects on smooth (finite dimensional) manifolds. It is an advantage to be familiar with smooth manifolds and tangent spaces if you are taking the course. However, we will spend the first weeks reviewing basic material on manifolds and tangent spaces.
The course will follow chapters from various book. See below for a list of literature with links. All literature needed can be obtained as download from the university library. The reason we have so many different books is that we will mix and match chapters to avoid the more technical presentations taken by some authors in their books (which is all appropriate and good for what they want but we try a more low power approach)
We will organize weekly meetings to discuss the material and some selected exercises. In case you need study points for this course we will offer oral examinations on the material at the end of the semester.
Time plan
- We will coordinate a weekly meeting time in the week before the semester starts
Course organisers
Preliminary weekly plan
| Week | Topic | Literature |
|---|---|---|
| 1 | Definition of a (smooth, finite-dimensional) manifold using charts, examples: Sphere and product manifold; exercise: Hyperboloid model | Lee p.1-22 (without 1.16), 32-40 & XKCD #977 on Map Projections |
| 2 | The tangent space and the tangent bundle | Sec. 2.2 in Alexanders script on Lie groups |
| 3 | Vector fields and the Riemannian metric | Boothby, Chapter IV.2 until Example 2.5, and Chapter V.2 |
| 4 | Length and Riemannian distance, geodesics | |
| 5 | Connections, covariant derivatives, affine connections | |
| 6 | Geodesics of a connection and parallelism | |
| 7 | Geodesics and minimizing curves | |
| 8 | The Riemannian exponential map | |
| 9 | Complete manifolds and the Riemannian logarithm | |
| 10 | Retractions | |
| 11 | Vector transport | |
| 12 | Gradient and descent directions | |
| 13 | Riemannian Hessian and classical differential operators | |
| 14 | Exam (in case you want to get study points for the course) |
Literature
- Absil, Mahony, Sepulchre, Optimization Algorithms on Matrix Manifolds, 2008 https://press.princeton.edu/absil
- Boumal, N. An introduction to optimization on smooth manifolds, 2023, https://www.nicolasboumal.net/#book
- Boothby, W.M. An introduction to differentiable manifolds and Riemannian geometry. 2nd ed.https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/63/suppl/C.
- Lee, J.M.: Introduction to smooth manifolds, 2nd edition, Springer, https://link.springer.com/book/10.1007/978-1-4419-9982-5
- Schmeding, A.: Introduction to Lie theory Lecture notes
- Tu, L.: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer, https://link.springer.com/book/10.1007/978-3-319-55084-8