Announcements

Date
18.09. Since the water supply in all of SB-II is cut off Monday-Thursday, we will meet today in R93 (Realfagbygget)
01.10. By moving the course to 14.00-15.00 we have to move to the room 634.

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) Boumal, p. 260-262 (read geodesic = locally length minimizing curve), Lee p. 337-341
5 Connections, covariant derivatives, affine connections Tu, chapter 6
6 Christoffel Symbols, Geodesics of a connection Tu chapter 13.1 and 13.3 (Poincare example optional), 14.1-14.3 (14.4 optional), see also geodesics on the torus
7 Parallelilsm, Geodesics and minimizing curves Tu chapter 14.5-14.7, Lee (Curvature) p. 96–101
8 The Riemannian exponential map Lee (Curvature) p. 72-86 (see also Tu, 15.1-15.3)
9 Complete manifolds and the Riemannian logarithm Lee (Curvature) p.108-111, Boumal 10.2
10 Retractions Boumal, 3.6 (p. 38/39), Absil, Mahony, Sepulchre; Sec. 3.6.2 on quotient manifolds, Sec. 4.1 (maybe excluding Sec 4.1.3)
11 Vector transport Absil, Mahony, Sepulchre Sec 8.1 (choose 1-2 from the examples), see also Boumal p. 282; special case: Boumal 10.3, p.262-265: parallel transport.
12 Gradient and descent directions Boumal, 3.8 (p 42-46), if you like also 3.9
Boumal, 4.1-4.5 (p. 51-62), if you like also 4.6 – numerically interesting also the one page 4.8
13 Riemannian Hessian and classical differential operators Boumal 5.5 (p. 94-97), take 5.4 as a reminder on connections; similar to 4.8 ou can also look at 6.8
Boumal 6.2, 6.3 (p. 121-131, or continue even into 6.4 if you like
14 Exam (in case you want to get study points for the course)

Literature

2024-11-06, Ronny Bergmann